# Euler's Formula

(There is another "Euler's Formula" about complex numbers,

## Euler's Formula

For any polyhedron that doesn't intersect itself, the

• Number of Faces
• plus the Number of Vertices (corner points)
• minus the Number of Edges

always equals 2

This is usually written:

F + V − E = 2 Try it on the cube.

A cube has 6 Faces, 8 Vertices, and 12 Edges, so:

6 + 8 − 12 = 2

## Example With Platonic Solids

Let's try with the 5 Platonic Solids:

Name   Faces Vertices Edges F+V-E
Tetrahedron 4 4 6 2
Cube 6 8 12 2
Octahedron 8 6 12 2
Dodecahedron 12 20 30 2
Icosahedron 20 12 30 2

(In fact Euler's Formula can be used to prove there are only 5 Platonic Solids)

 Why always 2? Imagine taking the cube and adding an edge (from corner to corner of one face). We get an extra edge, plus an extra face: 7 + 8 − 13 = 2 Or try to include another vertex, and we get an extra edge: 6 + 9 − 13 = 2. "No matter what we do, we always end up with 2" (But only for this type of Polyhedron ... read on!)

## Flattened Out

It may be easier to see when we "flatten out" the shapes into what is called a graph (a diagram of connected points, not the data plotting kind of graph).

A tetrahedron can be drawn like this: Note that one of the "faces" is the outside region, like this: So there are 4 regions (faces), 4 vertices and 6 lines (edges):

F + V − E   =   4 + 4 − 6   =   2

Let's try this with a cube. Here is one way to show it: There are 6 regions (counting the outside), 8 vertices and 12 edges:

F + V − E   =   6 + 8 − 12   =   2

We can discover what is going on when we build up graphs from just one vertex: With one vertex we have one region (the whole area), the one vertex and no edges: 1 + 1 − 0 = 2 Add another vertex. We still have one region, but now have two vertices and one edge: 1 + 2 − 1 = 2 Adding another vertex we can have two regions (inside and outside), three vertices and three edges: 2 + 3 − 3 = 2 Or we could have one region, three vertices and two edges (this is allowed because it is a graph, not a solid shape): 1 + 3 − 2 = 2 Adding another vertex we can have several different graphs ... ... one of those graphs is a tetrahedron: 4 + 4 − 6 = 2

There is always a balance between faces, vertices and edges.

Your turn! Turn the tetrahedron graph into a cube graph. See if you can make an octahedron graph, too.

(Isn't it amazing how we can show 3d shapes as a series of connected points!)

## The Sphere All Platonic Solids (and many other solids) are like a Sphere ... we can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit).

For this reason we know that F + V − E = 2 for a sphere

(Be careful, we cannot simply say a sphere has 1 face, and 0 vertices and edges, for F+V−E=1)

So, the result is 2 again.

## But Not Always 2 ... !

Now that you see how it works, let's discover how it doesn't work.

We join up two opposite corners of an icosahedron like this:

images/polyhedra.js?mode=icosahedron-intersected

It is still an icosahedron (but no longer convex).

In fact it looks a bit like a drum where someone has stitched the top and bottom together.

There are the same number of edges and faces ... but one less vertex!

So:

F + V − E = 1

Oh No! It doesn't always add to 2.

The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices become 1.

## Euler Characteristic

So, F+V−E can equal 2, or 1, and maybe other values, so the more general formula is:

F + V − E  =  χ

Where χ is called the "Euler Characteristic".

Here are a few examples:

Shape   χ
Sphere 2
Torus 0
Mobius Strip 0 And the Euler Characteristic can also be less than zero.

This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:

F + V − E = −2

In fact the Euler Characteristic is a basic idea in Topology (the study of the Nature of Space).

## Donut and Coffee Cup (Animation courtesy
Wikipedia User:Kieff)

Lastly, this discussion would be incomplete without showing that a Donut and a Coffee Cup are really the same!

Well, they can be deformed into one another.

We say the two objects are "homeomorphic" (from Greek homoios = identical and morphe = shape)

Just like the platonic solids are homeomorphic to the sphere.

And your body is homeomorphic to a torus if you pinch your nose closed.

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