Beyond graphite fullerenes and schwarzites

If five-rings are possible, why not seven-rings The presence of seven-rings in the (three-connected) network must lead to hyperbolic frameworks. 

Once the average ring-six exceeds six, hyperbolic carbon structures must result, which lie on periodic h)q)erbolic surfaces, and form three-dimensional extended frameworte. A number of theoretical studies have indicated that these hyperbolic structures, called schwarzites in honour of the mathematician Schwarz, should be more stable than the fullerenes [28]. In 

So far, theoretically predicted schwarzites do not display this universality, although a number of predictions give an area per C atom close to that found in graphite and C60 (Fig. 2.22). These theoretical frameworks are the result of complex numerical quantum mechanical calculations. The apparent conservation of surface density, irrespective of the curvatures of the surface, is clearly not a direct consequence of standard physics. It will be very interesting to compare the surface densities of actual schwarzites (although they have yet to be prepared in the laboratory) with those of fullerenes and graphite. Given the usefulness of this principle in the study of tetrahedral frameworks, our bet is that they too will lie on the dotted line in Fig. 2.22. 

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