The
Euler characteristic of a
surface, sometimes called the Euler-Poincaré characteristic, is a
topological invariant which is related to the figures which may be drawn on the surface.
We illustrate by considering the simplest example: The surface of a sphere or three-dimensional polyhedron, which is topologically almost identical to an two-dimensional plane, has Euler characteristic equal to two. The characteristic may be calculated by drawing any figure on the surface in question, and calculating
C = V - E + F
where C is the Euler characteristic, V is the number of vertices in the figure, E is the number of edges and F is the number of "faces", that is, the number of regions into which the surface is divided by the figure. For example, drawing the following figure on the surface of a sphere divides the sphere into five regions (counting the area outside the square as a region):
_______
| | |
|__|__|
| | |
|__|__|
The number of vertices is 9, and there are 12 edges, showing that the Euler characteristic (9-12+5) is two as claimed.
A fairly simple process of triangulation can be used to prove that any figure drawn on this surface, no matter how complex, will give C = 2, i.e., it is an invariant of the surface. One essentially gives a construction which divides any figure into a grid of triangles, and shows that successively removing triangles leaves the Euler characteristic unchanged; this means that we need only calculate V - E + F for the figure consisting of a single triangle.
Surfaces with different topologies, such as the torus or the Klein bottle, have different Euler characteristics. (Both the surfaces mentioned have characteristic zero.) Roughly speaking, a surface with holes will have its Euler characteristic decreased by two for each hole which is present.
The fact that this characteristic is an invariant was used to prove that there are only five Platonic solids.