Gruber P. Convex and Discrete Geometry

260 Convex Polytopes
Euler’s Polytope Formula for d = 3
Theorem 15.1. Let P be a proper convex polytope in E 3 with v vertices, e edges and
f facets, i.e. f (P) = (v, e, f ). Then
v − e + f = 2, v≤ 2 f − 4, f ≤ 2v − 4.
Euler’s original proof lacked an argument of the type of Jordan’s curve theorem
for polygons which then was not available. Since the curve theorem for polygons is
easy to show (see, e.g. Benson [96]), this criticism of Euler’s proof should not be
taken too seriously. The first rigorous proof seems to have been that of Legendre
[639]. Before presenting it, we show the following lemma, actually a special case
of the Gauss–Bonnet theorem. By a convex spherical polygon on the 2-dimensional
unit sphere S 2 , we mean the intersection of S 2 with a closed convex polyhedral cone
with apex o.
Lemma 15.1. The area of a convex spherical polygon on S 2 , the boundary of which
consists of n great circular arcs and with interior angles α1,...,αn at the vertices,
is α1 +···+αn − (n − 2)π.
Proof. Since the polygon may be dissected into convex spherical triangles, it is sufficient
to prove the lemma for convex spherical triangles. Let T be a spherical triangle
on S 2 with interior angles α, β, γ . Consider the three great circles containing the
edges of T . Each pair of these great circles determines two spherical 2-gons, one
containing T , the other one containing −T . The interior angles of these six 2-gons
are α, α, β, β, γ, γ . Hence these 2-gons have areas (Fig. 15.1)
α
4π = 2α, 2α, 2β,2β,2γ,2γ.
2π
The 2-gons cover each of T and −T three times and the remaining parts of S 2 once.
Thus we have for the area A(T ) of T ,
α
β
Fig. 15.1. Area of a spherical triangle
γ