ABSTRACT
Abstract .......................................................................................................146 4.1 Introduction .......................................................................................146 4.2 Structures Including the Fullerene C20 ..............................................149 4.3 Structures Including the Fullerene C24 ..............................................154 4.4 Structures Including the Tetrahedron ................................................155 4.5 Omega Polynomial in 3-Periodic Networks ......................................160 4.6 Conclusion .........................................................................................167 Acknowledgments .......................................................................................168 Keywords ....................................................................................................168 References ...................................................................................................168
Multi-shell nanostructures populate the Nanoworld in a wide variety of structures, spongy or filled ones. As constructive units, small cages can be used to design complex structures, of rotational or translational symmetry. In this chapter, the attention was focused on the design of multishell cages based on the Platonic solids and their transforms, obtained by using simple map operations. It is shown that the primary hyperstructures can self-arrange in even more complex arrays, expanded linearly (with 1-periodicity) or spherically, the majority of which belonging to quasicrystals. To prove the consistency of such molecular constructions, the calculation of genus (for spongy structures) and figure sum (for filled ones) was performed. Topological characterization of some 3-periodic networks is given in terms of Omega polynomial.
