Transformation, Identification, and Inversion of Goldberg-Coxeter Fullerenes

Abstract

It is difficult to identify a G-C fullerene directly from its dimensions as its lattice is not proportional to that of its archetype in general, although they have the same three-dimensional shape. In this paper, the area scale factor of a G-C fullerene is proved to be an integer, which can be calculated from its dimensions. All the G-C transformations are k-inflations that can be easily identified and inversed, primary transformations whose area scale factors are prime numbers, or composite transformations whose area scale factors are the product of those of its sub-transformations. As the result, a method to identify any G-C fullerenes according to the area scale factor was presented.