More About Euler’s Characteristic

This post is going to go more in-depth about the importance of Euler’s Characteristic and how it is formed and applied today.

Let’s start by refreshing ourselves with the equation. The base equation is [Euler Characteristic]= V - E + F. Lets first dissect this equation and see where each part comes from.

Let’s first take a triangle

A vertex is defined as a point where two or more edges meet. This triangle has 3.

And the edge is defined as the joining of two vertices. This triangle has 3.

A face is defined as the flat surface of a shape. This triangle has 1.

Now let's take this information and plug it into the Euler equation.

3 - 3 + 1 = 1

So this Triangle has an Euler Characteristic of 1.

Now let’s take this same triangle and manipulate it a bit.

This shape has 8 vertices, 8 edges, and 1 face.

This gives us the equation 8 - 8 + 1 = 1

If you looked at the interior faces and edges, you get 8 vertices, 13 edges, and 6 faces.

This yields 8 - 13 + 6 = 1

In these equations, regardless of the apparent shape or how the interior was split up, the Euler Characteristic was still 1. As we have talked about in the Donut = Coffee cup post, topology is often looked at as “rubber“ geometry. This means that shapes can be manipulated and stretch, just not cut or glued. If two shapes are topological equivalent, they have the same Euler Characteristic. This is the case with our two examples. This means the triangle in the original picture, and the new shape can be manipulated into the same shape.

Note:

The generalized equation is that on any planar graph (no lines intersect), for a conected shape, then the vertices - edges + regions created (faces) = 1. (Example Below.) The only problem with this, is that in some cases, the outer region is counted as a face making the Euler Characteristic 2. This would still be the same equation and would result in the same number if you kept which regions you counted consistant.

The image on the left is an example of a planar graph with regions filled in. The image ont he right is a non conected, non-planar graph because of the intersecting lines

The Euler Characteristic is also known as a topological invariant, however, it is not perfect. This means that just because two shapes have the same number, doesn’t mean that they are equivalent. Take the example of a single line.

A single line has two vertices and one edge. This yields 2 - 1 + 0 = 1

A line is not topologically equivalent to the above shapes, however, because it can’t be manipulated into a surface.

Shapes can also have negative Euler Characteristics. Take the example of the solid double torus, or a figure eight. That has a number of -1.

So why does this Euler Characteristic matter?

Firstly, it is used to classify shapes. As stated above, topological equivalent shapes will have the same Euler Characteristics. This means that all closed 2-dimensional without a hole have a characteristic of one. Something interesting to note is a 3-d shape with a “hole“ will have an Euler’s Characteristic less than the number of holes that it has.

The Characteristic was originally used to classify polyhedra and proofs about them. The most famous one being the fact there are only 5 Platonic Solids.

For being such a simple proof, it is one more step into classifying shapes and recognizing topological equivalence between them.