The face-spiral algorithm proposed by Manolopoulos et al.1
has been extensively used to construct classical fullerenes, 2
and the corresponding numbering of isomers has been recommended as a systematic nomenclature. The spiral algorithm is an important help in clarifying the structures of fullerenes, endohedral fullerenes and fullerene derivatives. Recent and accumulating evidence shows that non-classical fullerene cages with heptagon(s) or square(s) may be competitive with their classical counterparts in some situations; non-classical fullerene cages may be favored parent cages of metallofullerenes and fullerene derivatives. It is desirable to make a systematic examination of the effect of heptagon(s) and/or square(s) on the structures and properties of bare fullerenes, fullerene ions, metallofullerenes and fullerene derivatives. The original spiral algorithm does not generate non-classical fullerene isomers, but can be adapted to do so. An extended spiral algorithm can selectively generate non-classical fullerenes with one or two heptagon(s) or square(s) in the size range of interest, and has been used [3-5] to identify favored non-classical fullerenes and metallofullerenes. Energetic arguments suggest that non-classical fullerene cages may play a significant role in the formation of fullerenes and metallofullerenes.
 D. E. Manolopoulos, J.C. May, S.E. Down, Chem. Phys. Lett, 1991, 181, 105-111.
 P. W. Fowler, D. E. Manolopoulos, an Atlas of Fullerenes, Clarendon Press, 1995.
 L. H. Gan, D. Lei, P. W. Fowler, J. Comput. Chem. 2016, 37, 1907-1913
 L.-H. Gan, R. Wu, J.-L. Tian, J. Clarke, C. Gibson, P. W. Fowler, 2017, 38, 144–151
 L.-H. Gan, R. Wu, J.-L. Tian and P. W. Fowler, Phys. Chem. Chem. Phys., 2017, DOI: 10.1039/C6CP07370K.