  A fullerene graph can be embedded in a piecewise linear 2-manifold with each
non-hexagonal carbon ring corresponding to a cone vertex. Adjacent two or three
such vertices can be combined as a cluster cut out from a parent cone round a
parent vertex. The locations for the combined vertices are distributed on their
parent cone with certain symmetries. With proper combinations, a fullerene can
be simplified to a parent structure easily to be parameterized. And then, with
additional parameters for the combined vertices in each cluster, the fullerene
can also be parameterized. Such parameters may be different for fullerenes from
type to type, but they are able to characterize the specific shape for each
type.
