# Euler Characteristic Exploration

**Objective:**
Compute the Euler characteristic for some polyhedra and some other surfaces.

In 1750, the Swiss mathematician Leonhard Euler noticed a remarkable formula involving the number of faces F, edges E, and vertices V of a polyhedron. It is now called the Euler characteristic, and is written with the Greek letter <math>\chi</math>:

The Euler characteristic is <math>\chi</math> = V - E + F

For the Platonic solids, the Euler characteristic is always 2.

## Polyhedra

The Euler characteristic of a shape is the value of V - E + F and is usually written as <math>\chi = V - E + F</math>.

- Escher's Ivory Ball Study shows a Rhombic dodecahedron, with twelve rhombic faces. Compute V, E, F and <math>\chi</math> for this polyhedron.
- The dual of the Rhombic dodecahedron is the Cuboctahedron, which is a cube with its corners cut off, or equivalently, an octahedron with its corners cut off.

Compute V, E, F, and <math>\chi</math> for the Cuboctahedron. - Compute V, E, F and <math>\chi</math> for the tessellation by 45°-60°-90° triangles in Concentric Rinds.
- Compute V, E, F and <math>\chi</math> for the Deltoidal Icositetrahedron.

The Euler characteristic of a Polyhedron is always 2.

## Surfaces

Here, you will compute the Euler characteristic of tessellations that are not on the sphere:

- Compute V, E, F and <math>\chi</math> for this square lattice:
- Compute V, E, F and <math>\chi</math> for this graph:
- Compute V, E, F and <math>\chi</math> for this picture of a torus:

**Handin:**
A sheet with answers to all questions.