Icosahedral symmetry breaking: C\(_{60}\) to C\(_{84}\), C\(_{108}\) and to related nanotubes (Q2810252)

Fullerenes and related nanotubes are often used as carriers for other molecules in their interior. Symmetry alone admits several possibilities of defining special positions within fullerenes. Therefore the systematic description of such cases would be of sufficient interest. The article is the third one in series of articles by the authors in which structures of fullerenes, its geometrical and topological properties are investigated by analytical methods. Here, it is stated that breaking of the icosahedral symmetry of C\(_{60}\) to the subgroup \(A_1\times A_1\) is the most complicated of the three possible methods of constructing nanotubes through the repeated application of the symmetry-breaking mechanism. In previous papers by the authors, an appropriate number of rings of hexagons and pentagons was inserted between the upper and lower halves of the C\(_{60}\) shell. In the present case, a number of complete loop of a spiral of hexagons needs to be inserted between the upper and lower parts of C\(_{60}\).