Pick's theorem

Pick's theorem, first published in 1899, is a theorem that was brought to broad attention as recently as 1969 through Hugo Steinhaus's popular book Mathematical Snapshots. Pick's theorem gives an elegant formula for the area of lattice polygonspolygons that have vertices located at the integral nodes of a square grid or lattice that are spaced a unit distance from their immediate neighbors. Pick's theorem says that the area of such a polygon can be found simply by counting the lattice points on the interior and boundary of the polygon. The area is given by


   i + (b/2) – 1


where i is the number of interior lattice points and b is the number of boundary lattice points.


The Austrian mathematician Georg Pick (1859–1942) after whom the result is named, was born in 1859 in Vienna and perished during World War II in the Theresienstadt concentration camp. Over the past few decades, beginning with a paper by J. E. Reeve in 1957, various generalizations of Pick's theorem have been made to more general polygons, to higher-dimensional polyhedra, and to lattices other than square lattices. Most recently, mathematicians have become interested in the theorem because it provides a link between traditional Euclidean geometry and the modern subject of digital (discrete) geometry.