# Euler's Formula

*(There is another "Euler's Formula" about complex numbers, this page is about the one used in Geometry and Graphs)*

## 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 can be written: **F + V − E = 2**

Try it on the cube: A cube has 6 Faces, 8 Vertices, and 12 Edges, so: 6 + 8 − 12 = |

## 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)*

| |

Or try to include another vertex, 6 + | |

"No matter what we do, we always end up with 2" (But only for this type of Polyhedron ... read on!) |

## 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 can **not** 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 this works, let's discover how it **doesn't** work.

What if we joined up two opposite corners of an icosahedron?

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.

Now, 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 get reduced to 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.