A convex polyhedron of twelve pentagons and any number of hexagons with icosahedral symmetry — the geometric dual of the geodesic sphere and the shape underlying buckminsterfullerene.

Goldberg Polyhedron

The pentagon-and-hexagon "buckyball" polyhedron that is the mathematical dual of Fuller's geodesic sphere.

A Goldberg polyhedron is a convex polyhedron whose every face is a pentagon or a hexagon, with exactly three faces meeting at each vertex and overall rotational icosahedral symmetry. First described by the mathematician Michael Goldberg in 1937, these solids always contain exactly twelve pentagons (a consequence of Euler's polyhedron formula), with the hexagon count varying. Individual forms are denoted GP(m, n) by a "knight's move" construction across the surface: the dodecahedron is GP(1,0) and the truncated icosahedron — the classic soccer-ball shape — is GP(1,1).

The concept is directly tied to R. Buckminster Fuller's work: a Goldberg polyhedron is the dual polyhedron of a geodesic sphere. Where Fuller's geodesic domes triangulate a sphere into a mesh of triangles meeting at vertices, the dual operation replaces those vertices with faces, yielding the twelve-pentagon, many-hexagon pattern. This is the same geometry that describes the carbon molecule buckminsterfullerene (C60, the truncated icosahedron) and, more broadly, the "buckyballs" and viral capsid shells studied in chemistry and virology.

Goldberg polyhedra can be generated systematically through Conway polyhedron operators such as the chamfer and whirl, and they extend beyond icosahedral symmetry to octahedral and tetrahedral variants built from squares or triangles instead of pentagons. The geometer George Hart and others have popularized and extended the family, which remains a bridge between Fuller's design-science geometry, pure polyhedral combinatorics, and molecular structure.

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Goldberg polyhedron (Wikipedia)

Goldberg Polyhedron

Goldberg Polyhedron

See Also

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