A Simple Proof of Thue's Theorem on Circle Packing (1009.4322v1)

Abstract: A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration.

A Simple Proof of Thue's Theorem on Circle Packing

The paper presents a streamlined proof of Thue's theorem, which asserts that the regular hexagonal packing is the densest arrangement of circles in the plane. Specifically, the authors provide a concise argument using concepts and techniques from computational geometry and combinatorial optimization to establish that the hexagonal lattice achieves a density of $\pi/\sqrt{12} \approx 0.90690$.

In the field of geometry, circle packing is analogous to the three-dimensional sphere packing problem initially proposed by Kepler in 1611. Here, the focus is on arranging non-overlapping unit circles in the plane, maximizing the density—defined as the proportion of area covered by the circles within a container. Historical contributions by Lagrange, Gauss, and Thue have laid the groundwork for understanding these configurations, with the hexagonal lattice recognized as optimal, albeit Thue's original proof was thought to lack completeness.

The authors leverage the concept of saturation in circle configurations, highlighting its significance for density evaluation. A saturated configuration is one that cannot be extended without violating non-overlap constraints. Consequently, examining these saturated configurations suffices for density optimization of the circle packings.

The paper utilizes Delaunay triangulation to explore saturated circle configurations, emphasizing the triangular nature of these configurations in computational geometry. Delaunay triangulation involves subdividing a set of points into triangles, where no point lies within the circumcircle of any triangle. Importantly, for saturated configurations, a Delaunay triangulation exists, facilitating the analysis of circle packing densities.

A critical result detailed in the paper stipulates bounds on the angles within these Delaunay triangles; specifically, the largest internal angle $\theta$ satisfies $\frac{\pi}{3} \leq \theta < \frac{2\pi}{3}$. Leveraging this geometric insight, the authors establish through the sine law and the circumradius concept that the density of any triangle configuration adheres to the bounding value $\pi/\sqrt{12}$, with equality uniquely achieved by regular triangles with side length of 2.

The methodology culminates in demonstrating that the density of any finite union of Delaunay triangles respects this maximum, thereby substantiating Thue's theorem via a straightforward proof constructed on geometric fundamentals and density calculations.

The implications of this research are twofold. Practically, it provides a robust tool for evaluating and designing circle configurations in computational tasks requiring dense packing, such as material sciences and telecommunications. Theoretically, it refines the understanding of geometric packing problems, reinforcing the utility of computational geometry in addressing classical optimization puzzles. Future research may explore extensions within higher dimensions or alternative geometric constructs, potentially invigorating new insights into lattice theory and packing problems.