Euler Characteristic Exploration

From EscherMath

In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron:

He found that the Euler characteristic of the surface V - E + F depends on the surface

Let's check this formula on some of the shapes below.

Tetrahedron

A tetrahedon is a simple shape that is made up of 4 triangles. Below you see a picture with labels on the vertices (V) and edges (E).

Number of vertices = V = _______

Number of edges E = ______

Number of faces F = ______

Now find V - E + F =

Octahedron

A tetrahedon is a simple shape that is made up of 8 triangles. Below you see two pictures, the one on the left is given with labels on the vertices (V) and edges (E).

Number of vertices = V = _______

Number of edges E = ______

Number of faces F = ______

Now find V - E + F =

Cube

A cube is a simple shape that is made up of 6 squares. Below you see a picture with labels on the vertices (V) and edges (E).

Number of vertices = V = _______

Number of edges E = ______

Number of faces F = ______

Now find V - E + F =

Dodecahedron

A dodecahedron is made up of pentagons (5-gons). There are 12 pentagons in one dodecahedron.

Above you see a drawing of a dodecahedron and a tombstone in the form of a dodecahedron.

Number of vertices = V = _______

Number of edges E = ______

Number of faces F = ______

Now find V - E + F =

Icosahedron

An icosahedron is made up of triangles. There are 20 triangles in one icosahedron.

Above you see a drawing of a icosahedron and a game piece (like dice) in the form of a icosahedron.

Number of vertices = V = _______

Number of edges E = ______

Number of faces F = ______

Now find V - E + F =

Pattern?

What kind of pattern do you notice?

Going Further

The Euler characteristic of a shape is the value of V - E + F and is usually written as χ = V − E + F.

1. Compute V, E, F and χ for the tessellation by 45°-60°-90° triangles in Concentric Rinds.
2. Compute V, E, F and χ for the Deltoidal Icositetrahedron.
3. Compute V, E, F and χ for this picture of the 26-fullerene molecule:
4. Compute V, E, F and χ for this picture of a torus: