The Voronoi Cell in a saturated Circle Packing and an elementary proof of Thue´s theorem

Abstract

The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying Jensen's inequality on convex functions to receive a local estimate which is globally valid. Based on the concept of Voronoi cells, we will introduce a new tessellation into so-called L-triangles which can be related to fundamental parallelograms of lattice circle packings. Therefore a globally disordered circle packing can be reduced to locally ordered configurations: We will show how the theorem of Lagrange on lattice circle packings can be applied to non-lattice circle packings. Thus we receive a new proof of Thue's theorem.