Euler characteristic

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In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler–Poincaré 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 χ (Greek 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 connects to many other invariants.



The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula

where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic

This result is known as Euler's formula. This corresponds to the Euler characteristic of the sphere (which is 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below.

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