The polyhedron shape is the "truncated icosahedron."
With the hexagons and pentagons divided it becomes equal to the 3 frequency icosa (Garden Dome 3 type).
Divided once again is similar to the Garden Dome 3-X1 type which provides complete great circles (hemisphere lines) and a 1/2 sphere dome

See through view of the wireframe model,
looking straight down at a pentagon:

Compare with the top view of the Garden Dome 3:

The Buckyball divided twice and orientation of a half sphere. Does not illustrate the potential of more "roundedness" that is possible with this dome (each triangle on a seperate face plane).
This is the Garden Dome 3-X1 design, but turned for orientation of half-sphere. Check Out -
The Geometry Junkyard