Editing Gauss–Bonnet theorem (section)

==Combinatorial analog==
There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following one. Let {{mvar|M}} be a finite 2-dimensional [[pseudo-manifold]]. Let {{math|''χ''(''v'')}} denote the number of triangles containing the vertex {{mvar|v}}. Then

:<math> \sum_{v\,\in\,\operatorname{int}M}\bigl(6 - \chi(v)\bigr) + \sum_{v\,\in\,\partial M}\bigl(3 - \chi(v)\bigr) = 6\chi(M),\ </math>

where the first sum ranges over the vertices in the interior of {{mvar|M}}, the second sum is over the boundary vertices, and {{math|''χ''(''M'')}} is the Euler characteristic of {{mvar|M}}.

Similar formulas can be obtained for 2-dimensional pseudo-manifold when we replace triangles with higher polygons. For polygons of {{mvar|n}} vertices, we must replace 3 and 6 in the formula above with {{math|{{sfrac|''n''|''n'' − 2}}}} and {{math|{{sfrac|2''n''|''n'' − 2}}}}, respectively.
For example, for [[quadrilateral]]s we must replace 3 and 6 in the formula above with 2 and 4, respectively. More specifically, if {{mvar|M}} is a closed 2-dimensional [[digital manifold]], the genus turns out <ref>{{Cite journal |last1=Chen |first1=Li |last2=Rong |first2=Yongwu |date=August 2010 |title=Digital topological method for computing genus and the Betti numbers |journal=Topology and Its Applications |language=en |volume=157 |issue=12 |pages=1931–1936 |doi=10.1016/j.topol.2010.04.006 |doi-access=free }}</ref>

:<math> g = 1 + \frac{M_5 + 2 M_6 - M_3}{8}, </math>

where {{math|''M''<sub>''i''</sub>}} indicates the number of surface-points each of which has {{mvar|i}} adjacent points on the surface. This is the simplest formula of Gauss–Bonnet theorem in three-dimensional digital space.