Goldberg polyhedra are convex polyhedra first described by Goldberg (1937) and classified in more detail by Hart (2013) for which each face is a regular pentagon or regular hexagon, exactly three faces meet at each vertex, and the rotational symmetry is that of a regular icosahedron.
Goldberg polyhedra can be constructed with planar equilateral (but not in general equiangular) faces, though in general the corresponding vertices do not lie on a sphere (i.e., the solid has no circumsphere).
In the notation of Hart (2013, p. 126), the positive integers and indicate pentagon-to-pentagon "60-degree knights moves," meaning first take steps in one direction, then turn to the left and takie additional steps. Some special cases are summarized in the following table.
notation | polyhedron |
regular dodecahedron | |
chamfered dodecahedron (truncated rhombic triacontahedron) | |
truncated icosahedron |
Fullerenes of type I (isomorphic to the skeletons of -Goldberg polyhedra) and type II (isomorphic to the skeletons of -Goldberg polyhedra) are implemented in the Wolfram Language as BuckyballGraph[n, "I"] and BuckyballGraph[n, "II"], respectively.