[Paper Review] Maximally Dense Disc Packings on the Plane

Robert Connelly, Maurice Pierre|arXiv (Cornell University)|Jul 1, 2019

Point processes and geometric inequalities4 citations

TL;DR

This paper improves the upper bound for the radius ratio $ q $ in maximally dense disk packings on the plane from Fejes Tóth's 0.6457 to 0.6585, using a novel three-disk-size triangulated packing discovered by Fernique, Hashemi, and Sizova. The method involves perturbing a symmetric, triangulated packing with three distinct disk radii to achieve higher density than previous two-size packings, approaching Florian's theoretical upper bound for density.

ABSTRACT

Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures (MR0165423), László Fejes Tóth found a series of packings that were his best guess for the maximum density for any $1 > q > 0.2$. Meanwhile Gerd Blind in (MR0275291, MR0377702) proved that for $1 \ge q > 0.72$, the most dense packing possible is $π/\sqrt{12}$, which is when all the disks are the same size. In Regular Figures, the upper bound of the ratio $q$ such that the density of his packings is greater than $π/\sqrt{12}$ that Fejes Tóth found was $0.6457072159...$. Here we improve that upper bound to $0.6585340820...$. Our new packings are based on a perturbation of a triangulated packing that has three distinct sizes of disks, found by Fernique, Hashemi, and Sizova (MR4292755), which is something of a surprise.

Motivation & Objective

• To improve the upper bound on the radius ratio $ q $ for which non-uniform disk packings exceed the hexagonal lattice density $ \pi/\sqrt{12} $.
• To identify and utilize a new class of triangulated disk packings with three distinct disk sizes to achieve higher density than prior two-size packings.
• To systematically explore perturbations of symmetric, triangulated packings to approach Florian's theoretical upper bound on packing density.
• To classify and analyze the symmetry groups of newly discovered packings for potential use in constructing further dense periodic arrangements.

Proposed method

• Perturb a known three-disk-size triangulated packing (from Fernique, Hashemi, and Sizova) to maximize density while preserving contact graph triangulation.
• Use geometric constraints derived from triangle and rhombus area formulas involving disk radii $ r_1=1 $, $ r_2=p $, $ r_3=q $ to enforce non-overlapping and contact conditions.
• Apply the area constraint equation $ x(1,q,p)y(1,q,p) = p\sqrt{2p+1} $ to ensure blue disks remain in contact during perturbation.
• Derive a polynomial equation in $ p $ and $ q $ by squaring and simplifying the area equality, then isolate the relevant factor to determine valid radius ratios.
• Calculate the overall packing density $ \Delta(p,q) = A(p,q)/A_T(p,q) $, where $ A(p,q) $ is total disk area and $ A_T(p,q) $ is total torus area, under the derived constraint.
• Use symbolic computation (via Wolfram Alpha) to expand and factor the resulting high-degree polynomial to identify the valid solution curve for $ q $.

Experimental results

Research questions

• RQ1 What is the highest possible radius ratio $ q $ for which a non-uniform disk packing can exceed the hexagonal lattice density $ \pi/\sqrt{12} $?
• RQ2 Can a triangulated packing with three distinct disk sizes yield higher density than known two-size packings?
• RQ3 How close can perturbed triangulated packings come to Florian's theoretical upper bound $ s(q) $ for packing density?
• RQ4 What symmetry groups are realized by three-disk-size triangulated packings, and can they be used to generate new dense arrangements?

Key findings

• The upper bound for the radius ratio $ q $ in packings with density greater than $ \pi/\sqrt{12} $ is improved from 0.6457072159 to 0.6585340820.
• The new upper bound is achieved using a perturbation of a three-disk-size triangulated packing, which is a significant departure from Fejes Tóth’s prior two-size-based constructions.
• The density of the new packings approaches Florian’s theoretical upper bound $ s(q) $ more closely than any previous construction for $ q > 0.65 $.
• The constraint equation derived from area equality ensures that blue disks remain in contact throughout the perturbation, preserving the packing’s validity.
• The polynomial factorization reveals a valid solution curve defined by $ 2p^4 + (4q+3)p^3 + (2q^2 - 2q + 1)p^2 - (5q^2 + 6q)p + q^2 = 0 $, which defines the permissible radius ratios.
• The overall packing density $ \Delta(p,q) $ is computed as a ratio of total disk area to total torus area, with the constraint ensuring non-overlapping and contact conditions.