For this fullerene any of the equivalent C5 axes (the highest order axis for this molecule), that pass through the centers of opposite faces, provides equivalent contiguous helical pathways in either direction from any atom of any pentagon (all atoms are symmetry- equivalent). Because this fullerene is achiral, the clockwise or anticlockwise direction of the helical numbering pattern is not important (caution: the direction of numbering may become important for chiral derivatives, see ref. 4). Therefore, each of the C5 axes can be the reference axis, and it is not necessary to examine the equivalent C3 or equivalent C2 axes which are of lower order. The numbering for this fullerene is given in Fig. 3.
Fig. 3. Systematic numbering of (C20-Ih)fullerene
4. International Union of Pure and Applied Chemistry, Division of Organic Chemistry, Commission on Nomenclature of Organic Chemistry. "Nomenclature for the (C60-Ih) and (C70-D5h(6))[5,6]fullerenes (IUPAC recommendations 2002)", Pure Appl. Chem. 2002, 74, 629-695.
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995