Euler and the Kids

euler-o-ballBeing on parental leave, somewhat naturally, gave me more time to spend in the kids’ room. After a chance encounter with Mr Pythagoras there some time ago, it was now time  for a scheduled meeting with Mr Euler. The configuration in the image to this post is known to young parents as an Oball. To mathematicians, it is known as the truncated icosahedron. An interesting fact is that because it is constructed entirely from pentagons and hexagons, it has to have exactly 12 pentagons. Let’s see how Mr Euler can convince us of this necessity.

We start out with Euler’s well-known polyhedron formula: If V, E, and F are the numbers of vertices, edges, and faces of a polyhedron then V-E+F always equals two. The sum of angles between edges meeting at a given vertex cannot exceed 2\pi and, therefore, the number of edges meeting at a vertex is always equal to three. This gives us the following relationship between the number of vertices and the number of angles:

2E = 3V.

Now, let’s look for a relationship between the number of faces and the number of edges. We’ll call the number of pentagons F_5 and the number of hexagons F_6. This gives us F=F_5+F_6 for the total number of faces. Each pentagon is bounded by five edges and each hexagon is bounded by six edges. If we add these numbers for all faces, we count every edge twice (exactly two faces meet in an edge). Hence, we get the following relationship between the number of faces and the number of edges:

2E = 5F_5 + 6F_6.

This is all the information we need. We can put it together as follows:

6V-6E+6F=12, (six times Euler’s polyhedron formula)

\Rightarrow 4E-6E+6F_5+6F_6=12, (using that 3V=2E and F=F_5+F_6)

\Rightarrow 6F_5+6F_6-2E=12,

\Rightarrow 6F_5+6F_6-5F_5-6F_6=12, (using that 2E = 5F_5 + 6F_6)

\Rightarrow F_5=12.

Now, whenever you see a polyhedron made from pentagons and hexagons alone, you can say immediately that it has 12 pentagons. Why? Because it is built into the fabric of reality and mathematics allows us a glimpse at it.

You can find slightly different presentations of this in other places on the web (like here and here) but also in wonderful books like Shaping Space (I’ve advertised this before).


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