Euler characteristic

In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. It is usually denoted by chi.

Related Topics:
Algebraic topology - Topological invariant - Homotopy invariant - Topological space

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The Euler characteristic of a 2-dimensional topological polyhedron can be calculated using the following formula:

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:chi=F-E+V

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where F,E and V are the numbers of faces, edges and vertices respectively.

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In particular, for any polyhedron homeomorphic to a sphere

Related Topics:
Homeomorphic - Sphere

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we have

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:chi(S^2)=F-E+V=2.

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For instance, for a cube we have 6 − 12 + 8 = 2 and for a tetrahedron we have 4 − 6 + 4 = 2.

Related Topics:
Cube - Tetrahedron

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The last formula is also called Euler's formula.

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