We have studied possible ways of generating and growing the fullerenes having five-fold symmetry. Beginning with cyclopentane C₅H₁₀ and clusters C₅C₅, we obtained elementary fullerenes C₁₀ and mini-fullerenes C₂₀, and then we produced the fullerenes from C₂₀ to C₇₀, perfect (basic) and imperfect, as well as nanotubes. The basic fullerenes C₂₀, C₃₀, C₄₀, C₅₀, C₆₀, and C₇₀ have the ordinary five-fold symmetry, the intermediate ones having no such symmetry. Their imperfection is connected with extra 'interstitial' carbon dimers, the dimers playing the role of defects. One can define the imperfect fullerenes with defects as the fullerenes having topological five-fold symmetry. We have calculated the energies of both groups of fullerenes and discussed their dependence on a fullerene size and shape