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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. 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. == Polyhedra == The Euler characteristic 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 is known as Euler's polyhedron formula.〔Richeson 2008〕 It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on some 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 ''d''''v'', 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euler characteristic」の詳細全文を読む スポンサード リンク
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