For additional proofs, see Twenty Proofs of Euler's Formula by David Eppstein.
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It is common to construct soccer balls by stitching together pentagonal and hexagonal pieces, with three pieces meeting at each vertex (see for example the Adidas Telstar).
According to Euler's theorem these 12 pentagons are required for closure of the carbon network consisting of n hexagons and C60 is the first stable fullerene because it is the smallest possible to obey this rule.
The coefficient before is in fact the Euler characteristic of the sphere (for the torus it vanishes).
The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron.
Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.
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Chromatic number of a graph, sometimes (Chi) is used, which is also used for the Euler characteristic
The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).
In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.