Well... there a little more complicated than that, the mollecular formula of a Buckminterfullerene is C60 so there are sixty carbon atoms in each sphere each is joined to three other carbon atoms with one double bond per each carbon atom.
This is a picture of the skeletal formula of a Buckminsterfullerene which is difficult to depict as its three dimensional.
If you were to count all the shapes you would find 20 hexagons and 12 pentagons.
Each carbon atom has a free electron so it is free to move and carry and conduct electricity.
Each molecule has a diameter of 0.71 nano meters, but despite it's large size it has wave-particle duality (has both wave and particle like qualities)
Although Buckminsterfullerenes sound like something that can only be found in labs; C60 molecules can be found in small quantities in soot.