Euler's Formula

Euler's Formula is a relationship between the numbers of faces, edges and vertices (corners) of a convex polyhedron (a 3-D shape with flat faces and straight edges that doesn't have any dents in it).

It states that the number of vertices plus the number of faces minus the number of edges always equals \(2\).

We write Euler's Formula as

where \(V\) is the number of vertices, \(F\) is the number of faces and \(E\) is the number of edges of the polyhedron.

The Platonic Solids

Now, let's try Euler's formula on the rest of the Platonic solids.

The Cube (or Hexahedron)

The Octahedron

The Dodecahedron

The Icosahedron

Are There Solids that Euler's Formula Doesn't Work For?

Yes, Euler's formula is only guaranteed to work for convex polyhedrons. Here's a polyhedron that isn't convex. It's called a small stellated dodecahedron:

The small stellated dodecahedron (try saying that 10 times quickly) has 12 vertices, 12 faces and 30 edges. So,

which is nothing like \(2\). So, Euler's theorem doesn't work for this solid. This is because it isn't convex: it has lots of dents.

Euler Characteristic

Mathematicians love to generalise things. One generalisation of Euler's formula is the Euler Characteristic. The Euler characteristic can take on all sorts of different values (not just 2) and we can find the Euler characteristic of non-convex polyhedra and other solids that are not polyhedra. The Euler characteristic is defined to be

Mathematicians are interested in the Euler characteristic as it doesn't change if you deform a solid: twist or bend it, squeeze it or shrink it. It is what we call a topological invariant. Let's have a look at a few examples of Euler characteristics

Note: we know that the Euler characteristic of a sphere is \(2\) because we can pump up any of the Platonic solids to form a sphere, and their Euler characteristics are all \(2\). We say that the Platonic solids are homeomorphic to a sphere because we can deform them (by pumping them up) to form a sphere.