Editing Angular defect (section)

==Descartes's theorem==
Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is [[homeomorphism|homeomorphic]] to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4{{pi}} radians).  The polyhedron need not be convex.<ref>[[René Descartes|Descartes, René]], ''Progymnasmata de solidorum elementis'', in ''Oeuvres de Descartes'', vol. X, pp. 265–276</ref>

A generalization says the number of circles in the total defect equals the [[Euler characteristic]] of the polyhedron. This is a special case of the [[Gauss–Bonnet theorem]] which relates the integral of the [[Gaussian curvature]] to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the curvature is zero (the surface is locally [[Isometry|isometric]] to a Euclidean plane) and the integral of curvature at a vertex is equal to the defect there (by definition).

This can be used to calculate the number ''V'' of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect (which is <math>2\pi</math> times the Euler characteristic). This total will have one complete circle for every vertex in the polyhedron. 

A converse to Descartes' theorem is given by [[Alexandrov's uniqueness theorem]], according to which a metric space that is locally Euclidean (hence zero curvature) except for a finite number of points of positive angular defect, adding to <math>4\pi</math>, can be realized in a unique way as the surface of a convex polyhedron.