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.
