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

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.

References in corpus (1)

• Viruses and Fullerenes - Symmetry as a Common Thread?

Cited by in corpus (5)

• Weaving patterns inspired by the pentagon snub subdivision scheme
• Generation of Local Symmetry-Preserving Operations
• On Local Operations that Preserve Symmetries and on Preserving Polyhedrality of Embeddings
• Preserving and Increasing Symmetries of Polyhedral Maps
• Generating Plane Quadrangulations and Symmetry-preserving Operations on Maps