How Many Fullerenes Are There

Within the simple definition of the previous section there are in general many isomeric cages for a given number of carbon atoms. It has been argued that the number of fullerene isomers of C should increase asymptotically as explicit 

Several methods of enumeration have been developed for the fullerene problem, and there is now general agreement on the isomer counts for n 150, which is more than adequate for present chemical purposes. Two classes of method can be envisaged one that takes a vertical slice of fullerenes of some particular type or symmetry and one that takes a horizontal slice of all fullerenes at a given Both have advantages, and one method of each type will be summarized here. 

A method for counting the fullerenes within each symmetry can be based on a 60-year-old mathematical construction that has proved useful for classifying polyhe-dra, viruses, and geodesic domes. The original work by Goldberg, as rediscovered by Caspar and Klug in the context of viruses, implicit in the geodesic dome constructions of Buckminster Fuller, and reviewed by Coxeter, applies to structures of icosahedral (A or I) symmetry. 

For icosahedral symmetry, the whole net is completely specified by one lattice vector, i.e, by an ordered pair of integers a, ft). The atom count of the fullerene is proportional to the area of the planar net, and so it is easily shown that at least one icosahedral fullerene C exists for 

The code (a, b) gives information about the overall symmetry fullerene cages with = fl or = 0 have the untwisted 4 symmetry, other combinations a b 0 have the chiral 7 symmetry and exist as enantiomeric pairs. 

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