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Let P be a convex polyhedron in 3-space with F faces, E edges, and V vertices.

Let P denote its boundary (the union of the faces). Denote the faces by f₁, ..., f_F and suppose f_i has n_i edges.

a) By counting in two ways the number of incident pairs (f, e) with e an edge of f, show that n₁ + n₂ + ... + n_F = 2E. Suppose the origin is an interior point of P. Let S² be the unit sphere and define : P S² by (x) = x/|x|, so is radial projection.

b) Show that is a homeomorphism (a continuous bijection with continuous inverse). Show (f_i) is a geodesic polygon in S² with n_i edges.

Note: Be sure you explain why each edge e projects to a geodesic in S² . Where do you use that P is convex and that the origin is an interior point?

c) Express the area Ai of (f_i) using the Gauss-Bonnet theorem. Sum over i = 1, ..., F and simplify, to get a relation between F , V , and n₁ + .... + n_F . Finally use a) to prove

Eulers formula: V E + F = 2.