Surfaces, part 4

The Euler Characteristic

The Euler characteristic of a surface is the unique number V + R - A where V = the number of vertices, R = the number of regions, and A = the number of arcs for any tiling of the surface. It is usually denoted by the symbol c, hence c = V + R - A. A tiling of a surface is accomplished by dividing the surface into a series of connected regions as shown in the diagram which follows. Each of the individual edges of the regions is an arc. For the particular surface shown V = 10, A = 15, R = 6. Thus c = 10 + 6 - 15 = 1 for this surface.

Deforming a surface without tearing it does not change its Euler characteristic. The Euler characteristic of a sphere is the same as that of a cube. Similarly, the donut and coffee cup share the same Euler characteristic. Surfaces which cannot be deformed from one into the other have different Euler characteristics. A sphere for instance cannot be deformed into a donut. As a result their Euler characteristics are different. Cutting holes in a surface or gluing distinct boundaries together (provided they are not closed loops) also changes the Euler characteristic of a surface. On the contrary, cutting a surface to form two boundaries and regluing them with a half twist does not change the Euler characteristic. Consequently a tube and a möbius band have the same characteristic even though they have different orientability. Knotting a surface does not change the Euler characteristic either. As a result, a donut and a knotted donut also have the same characteristic.