From the "Brazuca" ball to Octahedral Fullerenes: Their Construction and Classification

TL;DR

This paper introduces a simple method for constructing and classifying octahedral fullerenes, using symmetry considerations and an index system to uniquely identify different structures.

Contribution

It presents a novel cut-and-patch construction method and a classification scheme for octahedral fullerenes based on symmetry rules and an index system.

Findings

A new construction method for octahedral fullerenes.

A classification scheme based on symmetry and indices.

Identification of four structural types of octahedral fullerenes.

Abstract

A simple cut-and-patch method is presented for the construction and classification for fullerenes belonging to the octahedral point groups, $O$ or $O_h$. In order to satisfy the symmetry requirement of the octahedral group, suitable numbers of four- and eight-member rings, in addition to the hexagons and pentagons, have to be introduced. An index consisting of four integers is introduced to specify an octahedral fullerenes. However, to specify an octahedral fullerene uniquely, we also found certain symmetry rules for these indices. Based on the transformation properties under the symmetry operations that an octahedral fullerene belongs to, we can identify four structural types of octahedral fullerenes.