[Paper Review] A Densest ternary circle packing in the plane

Thomas Fernique|arXiv (Cornell University)|Dec 4, 2019

Quasicrystal Structures and Properties4 references4 citations

TL;DR

This paper proves that a specific ternary circle packing with radii $1$, $r \simeq 0.834$, and $s \simeq 0.651$—roots of specific algebraic polynomials—is the densest possible circle packing in the plane. Using a novel potential-based method with interval arithmetic and triangulation, it establishes that the target packing's density $\delta \simeq 0.9093$ is optimal, confirming that compact packings maximize density when allowed by algebraic constraints.

ABSTRACT

We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually tangent circles. Compact packings are believed to maximize the density when there are possible. We prove that it is indeed the case for these sizes. The proof should be generalizable to other sizes which allow compact packings and is a first step towards a general result.

Motivation & Objective

• To prove that a specific ternary circle packing with algebraic radii is the densest possible in the plane.
• To extend the known principle that compact packings maximize density to ternary circle systems.
• To develop a generalizable method for verifying density maximality in circle packings with multiple sizes.
• To provide a rigorous computational proof using interval arithmetic and triangulation-based potential functions.

Proposed method

• Decompose the plane into a triangulation $\mathcal{T}$ of circle centers in the packing.
• Define excess $E(T) = \delta \cdot \text{area}(T) - \text{cov}(T)$ for each triangle $T$ to measure local density deviation.
• Introduce a potential function $U(T)$ decomposed into vertex and edge potentials $U_v(T)$, $U_e(T)$ to bound $E(T)$ from below.
• Prove the global inequality $\sum_{T \in \mathcal{T}} U(T) \geq 0$ via finite configurations using vertex- and edge-based sums.
• Establish the local inequality $E(T) \geq U(T)$ via differential geometry for $\varepsilon$-tight triangles and computer-assisted dichotomy for non-tight triangles.
• Use interval arithmetic throughout to ensure numerical rigor and avoid floating-point errors.

Experimental results

Research questions

• RQ1 Is the compact ternary circle packing with radii $1$, $r \simeq 0.834$, and $s \simeq 0.651$ the densest possible circle packing in the plane?
• RQ2 Can the principle that compact packings maximize density be extended to ternary circle systems?
• RQ3 Does the proposed potential-based method with interval arithmetic reliably verify density maximality in complex circle packings?
• RQ4 How do algebraic constraints on circle radii influence the existence and optimality of compact packings?
• RQ5 Can the method be generalized to other compact packings with multiple circle sizes?

Key findings

• The target packing achieves a density of $\delta \simeq 0.9093$, exceeding the hexagonal packing density of $\pi/\sqrt{12} \simeq 0.9067$.
• The proof confirms that this compact packing is globally optimal, meaning no denser arrangement exists for these radii.
• The method successfully verifies the global inequality $\sum U(T) \geq 0$ using only finitely many configurations, ensuring computational feasibility.
• The use of $\varepsilon$-tight triangles prevents infinite recursion in the computer-assisted proof, enabling termination in under 2 minutes.
• The computed lower bound on the largest $q < 1$ for which a packing with circles in $[q,1]$ exceeds the hexagonal density is improved to $q \geq 0.6510501858$.
• The method is generalizable: it can be adapted to other compact packings by adjusting vertex potentials and constraints, suggesting a path toward a general theorem.