1.1 Euler's theorem We begin by pro

1.1 Euler's theorem We begin by proving a beautiful theorem ofEuler concerning polyhedra. As we shall see, the statement and proof of the theorem motivate many of the ideas of topology. Figure 1.1 shows four polyhedra. They look very different from one another, Figure 1.1 yet if for each one we take the number of vertices (v), subtract from this the number of edges (e), then add on the number of faces (f), this simple calcula-tion always gives 2. Could the formula v -e + I = 2 be valid for all polyhedra? The ans wer is no, but the result is true for a large and interesting class. We may be tempted at first to work only with regular, or maybe convex, polyhedra, and v - e + I is indeed equal to 2 for these. However, one of the examples in our illustration is not convex, yet it satisfies our formula and we would be unhappy to have to ignore it. In order to find a counterexample we need to be a little more ingenious. If we do our calculation for the polyhedra shown in Figs 1.2 and 1.3 we o