How is C60 related to the geometry of a truncated icosahedron?

The $C_{60}$ molecule, also known as buckminsterfullerene, is characterized by a highly symmetrical, closed-cage structure that corresponds to the geometry of a truncated icosahedron.12

This geometric configuration is defined by the following features:
* Vertices and Faces: The structure consists of 60 vertices and 32 faces, comprising 12 pentagons and 20 hexagons.12
* Construction: The shape is mathematically derived by truncating (slicing off) the 12 vertices of a regular icosahedron. This process transforms the icosahedron’s 20 triangular faces into hexagons, while the 12 removed vertices become the 12 pentagonal faces.3
* Symmetry: In this arrangement, each of the 12 pentagons shares its edges exclusively with the hexagonal faces, meaning no two pentagons share a common edge.3

This unique, ball-like geometry—often compared to a soccer ball—is responsible for the exceptional stability of the $C_{60}$ cluster. The name "buckminsterfullerene" was chosen to honor R. Buckminster Fuller, whose geodesic dome designs utilized similar geometric principles.123