Pick's theorem: Difference between revisions

Revision as of 20:55, 12 December 2004

Revision as of 19:02, 6 January 2005

Line 12:

Note that the theorem as stated above is only valid for ''simple'' polygons, i.e. ones that consist of a single piece and do not contain "holes". For more general polygons, the "− 1" of the formula has to be replaced with "− χ(''P'')", where χ(''P'') is the [[Euler characteristic]] of ''P''.

The result was first described by [[Georg Alexander Pick]] in [[1899]]. It can be generalized to three dimensions and higher by [[Ehrhart polynomial]]s. The formula also generalizes to surfaces of [[polyhedron|polyhedra]].

===Proof===