Editing Pick's theorem (section)

==Generalizations==
[[File:Pick_theorem_hole.svg|thumb|upright=0.6|{{color|red|{{math|''i'' {{=}} 2}}}}, {{color|green|{{math|''b'' {{=}} 12}}}}, {{color|grey|{{math|''h'' {{=}} 1}}}}, {{math|''A'' {{=}} {{color|red|''i''}} + {{sfrac|{{color|green|''b''}}|2}} + {{color|grey|''h''}} − 1 {{=}} 8}}]]
Generalizations to Pick's theorem to non-simple polygons are more complicated and require more information than just the number of interior and boundary vertices.{{r|gs|rosenholtz}} For instance, a [[Polygon with holes|polygon with {{mvar|h}} holes]] bounded by simple integer polygons, disjoint from each other and from the boundary, has area{{r|sankri}}
<math display=block>A = i + \frac{b}{2} + h - 1.</math>
It is also possible to generalize Pick's theorem to regions bounded by more complex [[planar straight-line graph]]s with integer vertex coordinates, using additional terms defined using the [[Euler characteristic]] of the region and its boundary,{{r|rosenholtz}} or to polygons with a single boundary polygon that can cross itself, using a formula involving the [[winding number]] of the polygon around each integer point as well as its total winding number.{{r|gs}}

[[File:reeve_tetrahedrons.svg|thumb|upright=1.25|Reeve tetrahedra showing that Pick's theorem does not apply in higher dimensions]]
The [[Reeve tetrahedra]] in three dimensions have four integer points as vertices and contain no other integer points, but do not all have the same volume. Therefore, there does not exist an analogue of Pick's theorem in three dimensions that expresses the volume of a polyhedron as a function only of its numbers of interior and boundary points.{{r|reeve}} However, these volumes can instead be expressed using [[Ehrhart polynomial]]s.{{r|br2|ehrhart}}