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Concept
Euler characteristic
Summary
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, 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 (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. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra.
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 equation, stated by Euler in 1758,
is known as Euler's polyhedron formula. It corresponds to the Euler characteristic of the sphere (i.e. ), and applies identically to spherical polyhedra. An illustration of the formula on all Platonic polyhedra is given below.
The surfaces of nonconvex polyhedra can have various Euler characteristics:
For regular polyhedra, Arthur Cayley derived a modified form of Euler's formula using the density D, vertex figure density and face density
This version holds both for convex polyhedra (where the densities are all 1) and the non-convex Kepler–Poinsot polyhedra.
Projective polyhedra all have Euler characteristic 1, like the real projective plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0, like the torus.
Planar graph#Euler's formula
The Euler characteristic can be defined for connected plane graphs by the same formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face.
Official source
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