Though the dimensions and formation mechanisms of geodesic domes, the protein capsids of virus，and the C60 molecular cage vary significantly, they all belong to the same category of the icosahedral shells and consequently follow the same geometrical rules. Geodesic domes and virus capsids consist of triangular faces of almost the same size (quasi-equivalence)forming 12 penta-connected 35 and 10（T-1)hexa-connected 36 ver- tices, where T=h2+hk十k2is the triangular number or the number of small triangles in an icosahedral face, and h, k are the hexagonal coordinates between two 35 vertices. On the other hand，the fullerene Cn，including the archetype C60，consists of 12 pentagonal faces and 10(T-1)hexagonal faces forming only tri-connected vertices，now h，k become thehexagonal coordinates between the centers of two pentagonal faces. Obviously，these two kinds of icosahedral shells are duals with each other，or，in other words，their faces and vertices are interchanged. The geometrical characteristics of these icosahedral shells are discussed and the great variety of fullerenes is emphasized.
