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.

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