# Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations

## TL;DR

This paper reviews and unifies various methods for constructing symmetric polyhedra, extending Goldberg's approach to include all local symmetry-preserving operations, with implications for chemistry, biology, and mathematics.

## Contribution

It generalizes Goldberg's method to a systematic framework that encompasses all local symmetry-preserving operations on polyhedra.

## Key findings

- Unified approach to symmetry-preserving operations.
- Clarified differences between Goldberg, Caspar-Klug, and Coxeter methods.
- Extended Goldberg's construction to a comprehensive systematic method.

## Abstract

Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug -- not knowing about Goldberg's work -- in 1962 in a biological context. In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. In 1971 Coxeter published a survey article that refers to these constructions. Subsequently, the literature often refers to the Goldberg-Coxeter construction. This construction is actually that of Caspar and Klug. Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra.