Goldberg–Coxeter Construction

The general operation that unifies the geometry of geodesic domes, Goldberg polyhedra, and fullerene molecules.

The Goldberg–Coxeter construction (GC construction or GC operation) is a graph operation on regular polyhedral graphs of degree 3 or 4, formalized using lattices over the Gaussian and Eisenstein integers and parameterized by two integers GC(k, ℓ). Intuitively, it subdivides the faces of a polyhedron with a lattice of triangular, square, or hexagonal cells — possibly skewed — extending the ideas behind both Goldberg polyhedra and geodesic polyhedra into a single framework. In geodesic-dome terminology the subdividing unit corresponds to the "breakdown structure" or principal polyhedral triangle.

The construction knits together several strands of R. Buckminster Fuller's design-science geometry. Fuller coined the term "geodesic dome" in the 1940s (though he largely kept the underlying mathematics a trade secret), and a full geodesic dome is a geodesic polyhedron — the geometric dual of a Goldberg polyhedron. Michael Goldberg had introduced his polyhedra in 1937; Donald Caspar and Aaron Klug published the most general correct construction in 1962 while studying the geometry of viral capsids, and H.S.M. Coxeter covered similar ground in 1971. Because Caspar and Klug were first to the general result, the eponym "Goldberg–Coxeter" is an instance of Stigler's law. The 1985 discovery of buckminsterfullerene spurred renewed interest, and Michel Deza introduced the modern name and the degree-4 case around 2000.

The GC construction has become a working tool in organic chemistry (fullerenes), virology (capsid architecture), nanoparticle design, computer-aided design, and even basket weaving, demonstrating how the geometry Fuller intuited for shelter recurs across the natural and designed world.

See Also

• R. Buckminster Fuller (R. Buckminster Fuller) — coined "geodesic dome"
• Goldberg Polyhedron (Goldberg Polyhedron) — the dual family the construction generalizes
• H. S. M. Coxeter (H. S. M. Coxeter) — namesake geometer
• Geodesic Math and How to Use It (Geodesic Math and How to Use It) — Kenner's dome-math manual cited in the source

Sources

• Goldberg–Coxeter construction (Wikipedia)