Adventures in Topology: Euler Characteristics and Platonic Solids

Topology is a branch of Mathematics that can be thought of as the simplest form of geometry. It doesn’t concern itself with the specific shape of an object, but the ways in which the different pieces of the object connect to one another. To a topologist, if two shapes can be stretched, squeezed, and smoothly morphed to look like one another, then they are the same shape. The only caveat is that no cutting or tearing can occur during this process. The old saying goes that a topologist can’t tell the difference between a donut and a teacup:

Similarly a cube is the same as a sphere which is the same as a tetrahedron. Aside from visually morphing shapes, topologists want to find more quantitative measures of ‘sameness.’ This is provided to them by a number known as the Euler Characteristic. Before we get into that, however, we will need a few definitions. When we draw a triangle, we will call the inside of it a face, the three lines on the outside edges, and the points where the edges meet vertices. Next, we will call the process of covering a shape in interlocking triangles a triangulation. With these definitions, let’s examine the triangulation of shapes a topologist would consider ‘the same’:

Now these are all supposed to be the same shape, right? Yet the number of vertices, edges and faces is different in each triangulation. This shouldn’t discourage us, however, we’re only one more definition away from finding our first topological invariant, the Euler Characteristic. Its definition is:

You’ll notice that the Euler characteristic for each of these shapes is 2. It will always be 2 for any shape that is topologically equivalent to a sphere! In fact, one doesn’t even have to use triangles; any polygons will do!

Ok, but what about other shapes, like donuts and teacups? It can be shown that the Euler Characteristic is intimately related to the genus of the shape in question. The genus of a shape is just a fancy mathematician’s way of saying the number of holes in the object. For instance, the genus of a sphere is 0, whereas for a donut it is 1. We won’t get into things like orientability and convexity, but for typical shapes topology tells us that:

This is a remarkable result. It tells us that if we take any (almost) shape in three-dimensional space, cover it with little interlocking polygons, count the number of faces, edges and vertices, and combine them to form the Euler Characteristic, and do some arithmetic then we will have discerned the number of holes the shape has!

Now a fantastic application of the Euler Characteristic is that we can use it to find out how many unique, regular solids exist. In this case, regular means that the polygons that form the faces all have equal sides and angles. Consider one such figure, and let’s say that each face has n edges. At each vertex k faces meet one another. Furthermore, every edge is shared by 2 faces. Lastly we note that the genus of a regular solid must be zero, and hence it’s Euler Characteristic is 2. These relationships can be written as:

,          ,          

This is a simple system of linear equations in three variables, and can be rearranged into a relationship for F in terms of n and k:

The denominator is crucial here, since the number of faces must be positive. Since both n an k have to be greater than two, we can make a list of all their possible values based on this equation:

If we denote each possibility as {n,k}, then we have the following shapes:

1. {3,3}: tetrahedron

2. {3,4}: octahedron

3. {3,5}: icosahedron

4. {4,3}: cube

5. {5,3}: dodecahedron

These five are the only regular solids that can exist in three-dimensional space. In fact, they were known to the ancient Greeks, who refered to them as the Platonic solids.

I hope I’ve been able to convey how fascinating topological ideas are, and what a fascinating branch of mathematics Topology is. For those of you who are a bit more mathematically inclined, I would recommend the text book by Munkres as an excellent introduction to the subject.

Till our next mathematical adventure!


 

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