Icosahedral symmetry breaking: C₆₀to C₈₄, C₁₀₈and to related nanotubes

Abstract

This paper completes the series of three independent articles [Bodner et 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 C60 can 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 by A1 × A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A1 × A1. Thirteen of the A1 × A1 orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60 passing 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.; Zhang et al. (1993). J. Chem. Phys. 98, 3095-3102], there are only two that can be identified with breaking of the H3 symmetry to A1 × A1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.

On symmetry breaking of dual polyhedra of non-crystallographic group H 3

“…As was shown by Bodner et al (2013Bodner et al ( , 2014Bodner et al ( , 2015, the decomposition of vertices of DðÞ polytopes into orbits of the two-dimensional subgroups of H 3 can be presented geometrically.…”

“…Extending the structures of such nets, rolling them onto a cylinder and seamlessly closing the ends with the corresponding caps result in the structures of closed tubes. Here the difference from the approach of Bodner et al (2013Bodner et al ( , 2014Bodner et al ( , 2015 lies in the additional rotation of 'pancakes' (rings of polygons) determined in (ii) to satisfy the two-dimensional tilings shown in Figs. 16(b),20(b).…”

“…Recognizing the importance of the H 3 group in the assembly of icosahedral molecules, such as viruses and fullerenes, in this paper, we examine in detail the dual polyhedra of non-crystallographic type V H 3 ðÞ. The main goal is to extend the structures of V H 3 ðÞ to tube-like structures, such as carbon nanotubes, considering a well known mechanism described by Bodner et al (2013Bodner et al ( , 2014Bodner et al ( , 2015. A connection between the family of V H 3 ðÞ polyhedra and the structures of spherical viruses, biomolecules and fullerenes warrants the study of chosen polytopes in mathematical generality.…”

“…During the past decades, there were several attempts to generate a complete list of fullerene isomers: the spiral algorithm (Manolopoulos et al, 1991), the algorithm using folding nets (Yoshida & Osawa, 1995) and the algorithm based on the stitching of patches bounded by a zigzag (Brinkmann & Dress, 1997;Brinkmann et al, 2012). However, as here only the polytopes with H 3 symmetry are considered, a remarkable approach to a structural extension of the fullerene C 60 (the truncated icosahedron) explicitly described by Bodner et al (2013Bodner et al ( , 2014Bodner et al ( , 2015 serves as a powerful tool for our research. In these papers, the vertices of C 60 are split between the parallel circular/polygonal orbits depending on the symmetry of a two-dimensional subgroup G 0 & H 3 .…”

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

“…Such a construction leads to fullerenes, for example C 70 , C 78 , C 84 , as well as closedend carbon nanotubes, once a large enough number of orbits is inserted (Bodner et al, 2013(Bodner et al, , 2014(Bodner et al, , 2015. Indeed, it can be used to close an open-end nanotube of particular diameter, or to construct a closed-end nanotube by breaking the symmetry of the considered polytope.…”