# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.05837