Gülhan Sariismailoglu: Picks sats

Abstract: Pick’s Theorem is a classical result in geometry that provides a simple and elegant formula for calculating the area of polygons whose vertices lie on an integer lattice. Despite its elementary formulation, the theorem establishes a deep connection between geometry, combinatorics, and graph theory.

In this thesis, we present Pick’s Theorem and illustrate its use through a series of concrete examples involving both regular and irregular lattice polygons. Two different proofs of the theorem are given. The first is a constructive proof based on induction and the additivity of area, while the second relies on Euler’s formula for planar graphs and triangulations of lattice polygons.

Finally, we investigate a generalization of Pick’s Theorem to polygons with holes. It is shown how the original formula must be modified depending on the number of holes, and how the classical version of Pick’s Theorem appears as a special case. The results demonstrate that Pick’s Theorem is a powerful and versatile tool within lattice geometry.