Why do C60 molecules have 12 pentagonal rings?

The $C_{60}$ molecule, known as buckminsterfullerene, consists of 60 carbon atoms arranged in a closed, spheroidal cage. Its structure is mathematically defined as a truncated icosahedron, which requires exactly 12 pentagonal rings to form a closed-shell network.12

The geometry of the molecule is derived from an icosahedron, which has 20 triangular faces. By cutting off the 12 vertices of the icosahedron, the triangular faces are transformed into 20 hexagons, while the 12 removed vertices become the 12 pentagonal faces. These 12 pentagons are essential for the closure of the carbon network; without them, the structure would not form a complete, stable cage.2

While the number of hexagonal rings can vary in larger fullerenes, a closed-shell carbon network must contain exactly 12 pentagons. In $C_{60}$, these pentagons are positioned such that no two share an edge, which contributes to the molecule's exceptional stability. Smaller cages are less stable because their constituent pentagons are forced to share edges.21