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

TL;DR

This paper introduces a novel tessellation based on Voronoi cells called L-triangles, providing a new, elementary proof of Thue's theorem on the densest circle packing in the plane, connecting local and global packing properties.

Contribution

It presents a new tessellation method using Voronoi cells and L-triangles to simplify the proof of Thue's theorem, linking local configurations to global packing density.

Findings

New tessellation approach for circle packings

Reduction of global packing problem to local configurations

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.