Knowra Euler characteristic Euler characteristic The Euler characteristic is a topological invariant that, for a convex polyhedron, equals the number of vertices minus edges plus faces. It extends to broader spaces through homology and cell decompositions.
Euler's polyhedron formula : For every convex polyhedron, the number of vertices minus edges plus faces equals two. This is the characteristic’s classical form for convex polyhedra.
Vertex : A point where edges meet in a graph or the corner of a polygon or polyhedron. Vertices provide the zero-dimensional terms in the classical count.
Sphere : A surface homeomorphic to the boundary of a three-dimensional ball. Its Euler characteristic is two, matching the boundary of any convex polyhedron.
Genus (mathematics) : A topological measure of the number of handles on a connected, closed, orientable surface. For such surfaces, genus determines the Euler characteristic through a simple formula.
Cell complex : A space built by attaching cells of different dimensions according to specified boundary maps. Counting cells with alternating signs defines the characteristic beyond polyhedra.
Edge : A one-dimensional line segment or connection between vertices in a geometric or combinatorial structure. Edges are subtracted in the polyhedral formula and counted as one-dimensional cells.
Torus : A surface shaped like the boundary of a solid doughnut, with one hole. Its Euler characteristic is zero, contrasting with the sphere’s value.
Gauss–Bonnet theorem : A theorem equating the integral of Gaussian curvature on a compact surface with a multiple of its Euler characteristic. It connects this discrete invariant to geometry and curvature.
Simplicial complex : A collection of vertices, edges, triangles, and higher-dimensional simplices joined along shared faces. Its simplex counts give a direct way to calculate the invariant.
Face : A flat polygonal region on the boundary of a polyhedron. Faces supply the positive two-dimensional terms in the classical count.
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