Euler Characteristic

23.07.10

The Euler characteristic is a number which describes a topological space's shape or structure no matter how it is bent. For the plane (and sphere), the Euler characteristic is 2 (well known for those of my classmates who took the course TMA4165 - dynamical systems) and I will now show you how you can use this number for a more… practical situation (rather than calculating the index at the point infinity).



So the scenario is this.

Say that you want to build a system of fences. You know how many pieces of fence you have available (edges, essentially) and you know how many areas you want to enclose. How many fence posts do you then need, and is this number dependent of how you arrange your fence?

We will now use the "definition" of the Euler characteristic which is
$$\chi = V - E + F$$
where V is the number of vertices, E is the number of edges and F is the number of areas you are enclosing. Note that the outside is to be included in this number.

So say now that you have 12 pieces of fence and want to enclose 5 regions. You will then need
$$2 + 12 - 5 = 9$$
posts to do so.

Try it yourself. Create a fence system and verify that the Euler characteristic is indeed two for any combination you might come up with.