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Pick's theorem
First published in 1899, 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 polygons – polygons 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
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.
Related categories
• GEOMETRY
• POLYGONS
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