Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations

Abstract

The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Archimedean solids, which can be constructed from the Platonic solids by local symmetry-preserving operations (e.g. truncation) on the solid. By contrast, the results of decorations of polyhedra, e.g. by Islamic artists and by Escher, have been interpreted as decorated polyhedra-and not as new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected. In this article, we first sketch and compare the operations of Goldberg, Fuller, Caspar & Klug and Coxeter to construct polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons and all vertices have three neighbours. We point out and correct an error in Goldberg's construction. In addition, we transform the term symmetry-preserving into an exact requirement. This goal, symmetrypreserving, could also be obtained by taking global properties into account, e.g. the symmetry group itself, so we make precise the terms local and operation. As a result, we can generalize Goldberg's approach to a systematic one that uses chamber operations to encompass all local symmetry-preserving operations on polyhedra.