Edited by Patrick Fowler, The University of Sheffield, Sheffield, United Kingdom, and accepted by the Editorial Board January 7, 2014 (received for review June 10, 2013)
The Greeks described two classes of convex equilateral polyhedron with polyhedral symmetry, the Platonic (including the tetrahedron, octahedron, and icosahedron) and the Archimedean (including the truncated icosahedron with its soccer-ball shape). Johannes Kepler discovered a third class, the rhombic polyhedra. Some carbon fullerenes, inorganic cages, icosahedral viruses, protein complexes, and geodesic structures resemble these polyhedra. Here we add a fourth class, “Goldberg polyhedra.” Their small (corner) faces are regular 3gons, 4gons, or 5gons, whereas their planar 6gonal faces are equilateral but not equiangular. Unlike faceted viruses and related carbon fullerenes, the icosahedral Goldberg polyhedra are nearly spherical. The reasoning and techniques presented here will enable discovery of still more classes of convex equilateral polyhedron with polyhedral symmetry.
The three known classes of convex polyhedron with equal edge lengths and polyhedral symmetry––tetrahedral, octahedral, and icosahedral––are the 5 Platonic polyhedra, the 13 Archimedean polyhedra––including the truncated icosahedron or soccer ball––and the 2 rhombic polyhedra reported by Johannes Kepler in 1611. (Some carbon fullerenes, inorganic cages, icosahedral viruses, geodesic structures, and protein complexes resemble these fundamental shapes.) Here we add a fourth class, “Goldberg polyhedra,” which are also convex and equilateral. We begin by decorating each of the triangular facets of a tetrahedron, an octahedron, or an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” We obtain the unique set of internal angles in each planar face of each polyhedron by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, and the variables are a subset of the internal angles in 6gons. Like the faces in Kepler’s rhombic polyhedra, the 6gon faces in Goldberg polyhedra are equilateral and planar but not equiangular. We show that there is just a single tetrahedral Goldberg polyhedron, a single octahedral one, and a systematic, countable infinity of icosahedral ones, one for each Goldberg triangle. Unlike carbon fullerenes and faceted viruses, the icosahedral Goldberg polyhedra are nearly spherical. The reasoning and techniques presented here will enable discovery of still more classes of convex equilateral polyhedra with polyhedral symmetry.
Author contributions: S.S. and J.M.G. designed research, performed research, contributed new analytic tools, analyzed data, and wrote the paper.
Conflict of interest statement: The University of California, Los Angeles may file a patent application for this work.
This article is a PNAS Direct Submission. P.F. is a guest editor invited by the Editorial Board.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1310939111/-/DCSupplemental.
