Fullerenes and Related Objects

Face-regular polyhedra, considered in the paper by Brinkmann and Deza, are defined as polyhedra with similar surroimdings of equally sized rings. The similar 

Related ieosahedral-symmetry fuUeroids are studied by Friedrichs and Deza. Fulleroids have all vertices of degree 3, but sizes of faces other than 5 (and 6) are allowed. Those smaller polyhedral fulleroids with ieosahedral symmetry and no more than one size faee other than 6 are comprehensively identified. The combinatorial structural characterization of ieosahedral fullerenes is addressed by Quinn et and shown to be of use in constructions for large such cages. 

These techniques presumably extend to the ieosahedral symmetry ftilleroids. 

The study of higher genus fiilleroids, particularly with translational symmetry, has been a topic of some interest over the last decade or so. Terrenes and Terrones identify some partieularly favorable finite structures of genus up to a dozen or so. King also presents some combinatorial group-theoretic aspects of the extended translationally symmetric high-genus such structures. 

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