Finite sets of operations sufficient to construct any fullerene from $C_{20}$

TL;DR

This paper demonstrates that allowing a single exceptional facet in fullerenes enables a finite set of growth operations to construct any fullerene from the dodecahedron, simplifying the combinatorial classification process.

Contribution

It introduces finite sets of growth operations sufficient for constructing all fullerenes from $C_{20}$ when an exceptional facet is permitted.

Findings

Finite sets of 7 and 11 operations are sufficient.

Operations are compositions of edge- and two edges-truncations.

Explicit pairs of operation sets and acceptable polytopes are described.

Abstract

We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene $C_{20}$). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -- eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge- and two edges-truncations.