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| Pick's theorem | |
|---|---|
| Name | Pick's theorem |
| Caption | Lattice polygon with interior and boundary points |
| Field | Geometry |
| Introduced | 1899 |
| Named after | Georg Alexander Pick |
Pick's theorem Pick's theorem gives an exact formula for the area of a simple lattice polygon in terms of the number of lattice points on its boundary and in its interior. The theorem connects combinatorial counts of integer-coordinate points with Euclidean area and appears in contexts ranging from Georg Alexander Pick's original work to modern studies in Ehrhart theory, Diophantine approximation, and Minkowski-style geometry. It is elementary to state yet links to diverse topics such as Euler characteristic, Blichfeldt's theorem, and Gauss's lattice point investigations.
Let P be a simple polygon whose vertices lie on the integer lattice Z^2 (a lattice polygon). Denote by I the number of integer lattice points strictly in the interior of P and by B the number of integer lattice points on the boundary of P. Pick's theorem states that the area A(P) equals A(P) = I + B/2 − 1. This identity ties the continuous invariant area to discrete quantities I and B and applies to polygons that are homeomorphic to a disk, exclude self-intersections, and have vertices at points of Z^2.
Several proofs exploit triangulation, induction, and linearity under set union. A standard proof triangulates P into lattice triangles whose vertices are lattice points; using unimodular transformations from SL(2,Z) reduces to proving the formula for a unit triangle and showing additivity across triangle unions. An alternative proof uses Euler's formula for planar graphs by embedding the lattice polygon into a planar graph whose faces correspond to unit cells of the integer grid; this approach invokes ideas related to Euler characteristic and planar graph combinatorics. A third method derives Pick's theorem from results in the geometry of numbers, invoking Minkowski's convex body theory and Blichfeldt's theorem to count lattice points in convex sets. Connections with Ehrhart polynomials give an algebraic-combinatorial viewpoint: one interprets A(P) as the leading coefficient of the Ehrhart polynomial for lattice dilates of P, recovering the linear term and constant via lattice-point enumerations reminiscent of Ehrhart's theorem.
Pick's theorem generalizes in several directions. In higher dimensions, no direct analogue with linear boundary correction exists; instead one obtains results in Ehrhart theory expressing the volume of a lattice polytope via coefficients of its Ehrhart polynomial, linking to Stanley's work on lattice polytopes. For polygons on other lattices or in affine transforms, the formula holds after applying appropriate determinant factors from SL(2,R)-transformations; unimodular maps preserve the lattice structure and thus the relation. Extensions include versions for polygons on triangulated surfaces associated with Farey sequence combinatorics and adaptations for polygons with holes, where the formula acquires additional additive terms equal to the number of holes. Further development relates Pick-type counts to the Pick–Minkowski framework and to computational algebraic geometry approaches used in toric varieties and Newton polygons.
Pick's theorem is used in computational geometry for exact area computation of lattice polygons with integer coordinates, facilitating robust algorithms in computer graphics and computational number theory. In Diophantine geometry, it provides bounds on solutions to linear inequalities with integer coefficients by translating area constraints into lattice point counts, relating to Gauss’s circle problem heuristics. In algebraic geometry, the combinatorics of lattice polygons guided by Pick's relation underpin properties of plane curve Newton polygons in Bernstein–Kushnirenko theorem contexts. Education and recreational mathematics exploit the theorem for problem solving in mathematical olympiads and for instructive connections between geometry and number theory found in texts influenced by Hilbert's classical problems.
For a unit square with vertices at (0,0), (1,0), (1,1), (0,1), B = 4 and I = 0, so A = 0 + 4/2 − 1 = 1, matching the area. For a right triangle with vertices (0,0), (2,0), (0,1), lattice counts give B = 4 (three vertices and one lattice point on the hypotenuse), I = 0, yielding A = 0 + 4/2 − 1 = 1, and direct geometric area is (2·1)/2 = 1. More elaborate examples include convex lattice polygons like the standard hexagon formed by lattice vectors; computing B and I by direct enumeration yields area via Pick's formula without recourse to coordinate geometry. In practice, one computes B by counting lattice points along each boundary segment using gcd differences of coordinates (a consequence of Bézout-type lattice step counts), and I by lattice enumeration or via rearrangement: I = A − B/2 + 1.
Pick's theorem was introduced by Georg Alexander Pick in 1899 in the context of planar lattice investigations. The result quickly attracted interest and was incorporated into studies of lattice point geometry influenced by earlier work of Carl Friedrich Gauss on lattice point problems and by developments in the geometry of numbers from Hermann Minkowski. Subsequent expositions and proofs appeared in classical texts and in the evolving literature on Ehrhart theory and convex geometry, linking Pick's compact relation between area and lattice counts to broader themes explored by figures such as Eugène Ehrhart, Richard Stanley, and Hermann Weyl.