## The Euler Characteristic

The Euler characteristic for a surface is given by F-E+V, or the number of faces - the number of edges+the number of vertices. The Euler characteristic is a topological invariant for a surface, so that surfaces with different Euler characteristics cannot be homeomorphic.

Any closed surface with no holes has Euler characteristic 2:

The five Platonic solids are shown above. Each has Euler characteristic 2.

Shape | Number of Vertices, V | Number of Edges, E | Number of Faces, F | Euler Characteristic F-E+V |

Tetrahedron | 4 | 6 | 4 | 2 |

Cube | 8 | 12 | 6 | 2 |

Octahedron | 6 | 12 | 8 | 2 |

Dodecahedron | 10 | 30 | 12 | 2 |

Icosahedron | 12 | 30 | 20 | 2 |

Finding the Euler Characteristic of more complicated shapes is well, more complicated. In general a shape must be cut up and the number of vertices, edges and faces found. It can be a complicated process.