A large class of geometric objects can be built up from a finite number of triangles, so that any pair which touch, do so along a single edge, or at a single vertex, the triangle need not be flat nor the edges should be straight. These objects are called triangulable. A surface is a topological space, which is triangulable, connected (is all in one piece), and without any edge. For example, 2-spheres, torus, a sphere with n handles attached to it, Klein bottle and projective plane are surfaces. A Mobius strip is not a surface in this sense, as it has an edge. For any surface S, it is possible to have more than one decomposition (called triangulation or cell division), as we have noted for a 2-sphere. A cell division is called regular if every face has the same number of edges and every vertex lies on the same number of edges. For any surface S and any triangulation of S the number V-E+F can be shown to be independent of the triangulations of S. it is known as the Euler characteristic of the surface and is denoted by X(S). The Euler characteristic is a topological invariant of the surfaces. It can be shown that the Euler characteristic is 2, 0, 2-2n, 1, 0 according as S is a sphere, a torus, a sphere with n handles, a projective plane or a Klein bottle. Euler’s formula was known to Descartes and even to the Platonists, since they knew of the following corollary to the Euler’s formula:

Theorem

The 2-sphere has precisely 5 regular cell divisions.

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