in mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the euler characteristic (or euler&ndash;poincarÃ© 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 \chi (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 \chi was classically defined for the surfaces of polyhedra, according to the formula :\chi=v-e+f \,\! 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 :v - e + f = 2. \,\! this equation is known as euler's polyhedron formula.richeson 2008 it corresponds to the euler characteristic of the sphere (i.e. Ï = 2), and applies identically to spherical polyhedra. an illustration of the formula on some polyhedra is given below. {| class="wikitable" |- !name !image !verticesv !edgese !facesf !euler characteristic:v &minus; e + f |- align=center |tetrahedron |50px |4 |6 |4 |2 |- align=center |hexahedron or cube |50px |8 |12 |6 |2 |- align=center |octahedron |50px |6 |12 |8 |2 |- align=center |dodecahedron |50px |20 |30 |12 |2 |- align=center |icosahedron |50px |12 |30 |20 |2 |} the surfaces of nonconvex polyhedra can have various euler characteristics; {| class="wikitable" |- !name !image !verticesv !edgese !facesf !euler characteristic:v &minus; e + f |- align=center |tetrahemihexahedron |100px |6 |12 |7 |1 |- align=center |octahemioctahedron |100px |12 |24 |12 |0 |- align=center |cubohemioctahedron |100px |12 |24 |10 |&minus;2 |- align=center |great icosahedron |100px |12 |30 |20 |2 |} 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 d_f: :d_v v - e + d_f f = 2 d. this version holds both for convex pol
Algebraic_topology Polyhedral_combinatorics Topological_graph_theory