Points in the Plane

Abstract

Mighty is geometry; joined with art, resistless. Euripedes Geometry is the art of correct reasoning on incorrect figures. George Polya In this chapter we present some intriguing results, and their delightful proofs, about some of the simplest geometric configurations in the plane. These include figures consisting solely of points and lines, including those constructed from the lattice points in the plane. We will deal with structures such as triangles, quadrilaterals, and circles in later chapters. Pick's theorem Pick's theorem is admired for its elegance and its simplicity; it is a gem of elementary geometry. Although it was first published in 1899, it did not attract much attention until seventy years later when Hugo Steinhaus included it in the first edition of his lovely book Mathematical Snapshots [Steinhaus, 1969]. Georg Alexander Pick (1859–1942) was born in Vienna but lived much of his life in Prague. Pick wrote many mathematical papers in the areas of differential equations, complex analysis, and differential geometry. Sadly, Pick was arrested by the Nazis in 1942 and sent to the concentration camp at Theresienstadt, where he perished. A lattice point in the plane is a point with integer coordinates, and a lattice polygon is a polygon whose vertices are lattice points. A polygon is simple if it has no self-intersections. Pick's theorem gives the area A ( S ) of a simple lattice polygon S in terms of the number i of interior lattice points and the number b of lattice points on the boundary: A ( S ) = i + b /2-1.