Euler characteristic

In mathematics, the Euler characteristic of a shape is a number that describes a topological space, so that anything in the space will have the same number. It is calculated by taking the number of points in the shape, the number of lines in the shape, and the number of faces of the shape. One can find the Euler characteristic with this formula:

[math]\displaystyle{ \chi=V-E+F \,\! }[/math]

where V is the point count, E the line count, and F the face count. For most common shapes (convex polyhedron), the Euler characteristic is 2.

Name	Image	Vertices (Points) V	Edges (Lines) E	Faces F	Euler characteristic: V − E + F
Tetrahedron		4	6	4	2
Hexahedron or cube		8	12	6	2
Octahedron		6	12	8	2
Dodecahedron		20	30	12	2
Icosahedron		12	30	20	2

For spheres, the Euler characteristic is also 2.

Euler Characteristic Media

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  Vertex, edge and face of a cube

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  Tetrahemihexahedron

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  Octahemioctahedron

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  Cubohemioctahedron

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  Small stellated dodecahedron

Related pages

• Platonic solid