1) Buckministerfullerine (a buckyball). The shape is known a truncated icosahedron, which is a type of Archimedean solid. This is the structure of carbon-60, which is your basic fullerene.
2) An eicosahedron nested inside of a dodecahedron. These two Platonic solids are duals. The image is from the banner of the web page for the Dept. of Chemistry and Biochemistry at the University of Maryland.
3) A soccer ball.
Buckminister Fuller at Black Mountain College. Buckminister Fuller was at Black Mountain College in the summer of 1948, when this picture was taken, and 1949, when he was director.