Icosahedral symmetry breaking: C60to C84, C108and to related nanotubes

Abstract

This paper completes the series of three independent articles [Bodneret al.(2013).Acta Cryst. A69, 583–591, (2014),PLOS ONE, 10.1371/journal.pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space {\bb R}^3 as a mechanism of generating higher fullerenes from C60. The icosahedral symmetry of C60can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted byA1×A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebraA1×A1. Thirteen of theA1×A1orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60passing through the centers of two opposite edges between two hexagons on the surface of C60. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992).Nature (London),355, 428–430; Fowler & Manolopoulos (2007).An Atlas of Fullerenes. Dover Publications Inc.; Zhanget al. (1993).J. Chem. Phys.98, 3095–3102], there are only two that can be identified with breaking of theH3symmetry toA1×A1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.