Returning to the proof of Euler's theorem, thickening the trees T and r gave a decomposition of P into two discs with a common boundary and there-fore, by sending the points of one disc into the northern hemisphere and sending the points of the other south, a way of defining a homeomorphism from the poly-hedron P to the sphere.1t is possible to produce an argument in the opposite direc-tion (we shall do so in Chapter 7) and show that if Pis topologically equivalent to the sphere then P satisfies hypotheses (a) and (b) oftheorem (l.1)t, and therefore Euler's theorem holds for P. So if P and Q are polyhedra which are both homeo-morphic to the sphere, and if we call v - e + f the Euler number of a poly-hedron, then we know from the above discussion that P and Q have the same Euler number, namely 2.