::Euler characteristic


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In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–PoincarĂ© characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by <math> \chi </math> (Greek lower-case letter chi).

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra.

Euler characteristic sections
Intro   Polyhedra   Topological definition  Properties  Examples   Relations to other invariants   Generalizations   See also   References   Further reading    External links   

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