We have studied possible ways of generating and growing the fullerenes having four-fold symmetry. Beginning with cyclobutane C 4 H 8 and clusters C 8 , we obtained elementary fullerenes C 8 and mini-fullerenes C 16 , which produce the following fullerenes from C 24 to C 64 , perfect (basic) and imperfect, as well as nanotubes. The imperfection is connected either with extra 'interstitial' or 'vacancy' carbon dimers, both types of dimers playing the role of defects. Only the basic fullerenes C 24 , C 32 , C 40 , C 48 , C 56, and C 64 have the ordinary four-fold symmetry in the corresponding column of the periodic system of fullerenes, the intermediate fullerenes having no such symmetry. Considering the latter as imperfect due to defects, one can define them as the fullerenes conserving topological four-fold symmetry. We have calculated their energies and discussed possible reasons for their dependence on a fullerene size and shape.