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
