Truncated icosahedral geometry

At present the simplest carbon cluster to prepare is the buckminsterfullerene C60, the geometry of which is classified as truncated icosahedral , which belongs to the icosahedral symmetry group, Ih. As a consequence, the frontier orbitals of C60 consist of a full pentadegene-rate HOMO (hu) and two triply degenerate LUMOs (tiu and tlg), Figure 13. 

The other important symmetry-related discovery was the quasicrystals. Both the truncated icosahedral structure of buckminsterfullerene and the regular but nonperiodic network of the quasicrystals are related to fivefold symmetry. In spite of this intimate connection between them at an intellectual level, their stories did not cross. The conceptual linkage between them is provided by Fuller s physical geometry and this is also what relates them to the icosahedral structure of viruses (see, Section 9.5.2 on Icosahedral Packing). 

The properties of hypothetical closed carbon cages have been the snbject of discussion for many years, and the highly symmetric truncated icosahedral (soccer ball) geometry of the archetypal species buckminsterfullerene (Ceo) had particular appeal. It was suggested that Ceo would be stable and possibly superaromaticbut the deliberate synthesis of such a complex molecule remained too daunting a challenge. ... 

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