As an example of the utility of plane diagrams, we use them to represent the slicing of a surface along a curve on the surface. Then repeat the experiment, this time cutting the surface starting one-third of the way across. Try to find a pattern, and find a plane diagram for the general n -holed torus. In fact, the last problem is just a consequence of the classification theorem , which states that any surface is either a sphere, or the connected sum of tori, or the connected sum of projective planes. On any surface, we can draw graphs, or networks.
Edges meet at vertices, and divide the surface up into faces.
For a graph on a surface, let V be the number of vertices, E be the number of edges, and F be the number of faces. This quantity is what is called an invariant , and has been dubbed the Euler characteristic , . We can calculate the Euler characteristic of a surface by looking at its plane diagram. For example, the characteristic of a sphere is 2, while that of a torus is 0.