Ideal polyhedral models for single-walled nanotubes

In this paper, we survey existing geometric structures which have been proposed by the authors as possible models for nanotubes. Atoms assemble into molecules following the laws of quantum mechanics, and in general computational approaches to predicting the molecular structure can be arduous and involve considerable computing time. Fortunately, nature favours minimum energy structures which tend to be either very symmetric or very unsymmetric, and which therefore can be analyzed from a geometrical perspective. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature and the present authors have proposed a number of exact geometric models. Here we review a number of these recent developments relating to the geometry of nanotubes, with exact polyhedral constructions, and in particular for the three materials, carbon, boron and silicon. We summarise the main formulae and give illustrative numerical results for these three nanotubes.