Spherical networks

One can construct networks of constant positive curvature by approximating the surface of a sphere—starting with a dodecahedron and adding hexagons. (Euler's theorem implies that at any stage there must always be exactly 12 pentagonal faces.) The following are examples with 20, 60, 80, 180 and 320 nodes:

The object with 60 nodes is a truncated icosahedron—the shape of a standard soccer ball, as well the shape of the fullerene molecule C_{60}. (Note that in C_{60} one of the connections at each node is always a double chemical bond, since carbon has valence 4.) Geodesic domes are typically duals of such networks—with three edges on each face.