Fullerenes 1

Euler Theorem on Polyhedra 1
v – e + f = χχ = 2(1 – g) (3)
χχ = Euler’s Euler s characteristic
v = number of vertices,
e = number of edges,
f = number of faces,
g = genus ; (g = 0 for a sphere; 1 for a torus). torus
A consequence:
consequence
A sphere can not be tessellated only by hexagons.
Fullerenes need 12 pentagons (for closing the cage) and (N/2-10) (N/2 10) hexagons. hexagons
In the opposite, a tube and a torus allow pure hexagonal nets.
1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. Petropolitanae
Comment. Acad. Sci. I. Petropolitanae
1758, 4, 109-140.
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