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


hexahedron

Try it on the cube:

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

so:

6 + 8 − 12 = 2


To see why this works, imagine taking the cube and adding an edge
(say from corner to corner of one face).

We get an extra edge, plus an extra face:

7 + 8 − 13 = 2

hexahedron extra face
   

Likewise when we include another vertex
(say halfway along a line) we get an extra edge, too.

6 + 913 = 2.

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

hexahedron extra vertex

 

Example With Platonic Solids

Let's try with the 5 Platonic Solids (Note: Euler's Formula can be used to prove that there are only 5 Platonic Solids):

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

The Sphere

sphere like icosahedron

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!

Now that you see how this works, I am going to show you 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 sphere 2
Torus torus 0
Mobius Strip mobius strip 0


cubohemioctahedron

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

torus becomes coffee cup

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.

(Animation courtesy of Wikipedia User:Kieff)