[Paper Review] Relaxed Disk Packing

Herbert Edelsbrunner, Mabel Iglesias-Ham+1 more|arXiv (Cornell University)|May 13, 2015

Genomics and Chromatin Dynamics7 references3 citations

TL;DR

This paper investigates relaxed disk packings—configurations of equal-sized disks in the plane that are neither fully packed nor fully covering—by optimizing the probability that a random point lies in exactly one disk. Using lattice-based configurations and Voronoi/Delaunay geometry, it proves that the regular hexagonal grid (12-hour clock configuration), where each disk overlaps six neighbors in 30° arcs, maximizes this probability at 0.928..., outperforming other lattice arrangements.

ABSTRACT

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.

Motivation & Objective

• To identify optimal configurations of equal-sized disks in the plane that lie between packing and covering, avoiding rigid constraints on overlap.
• To define and optimize a new quality measure: the probability that a random point lies in exactly one disk.
• To prove that among all lattice configurations in R², the regular hexagonal grid maximizes this probability.
• To establish the 12-hour clock configuration as the unique optimal lattice arrangement under this criterion.

Proposed method

• Model disk configurations using lattices generated by two linearly independent vectors in R², with centers forming a periodic point set.
• Use Voronoi domains and Delaunay triangulations to analyze local geometry and proximity relations in the lattice.
• Define the probability that a random point lies in exactly one disk as a function of the disk radius and lattice parameters.
• Classify configurations into three cases based on how many neighboring disks intersect the boundary of a given disk’s disk.
• Apply geometric and trigonometric analysis to compute the probability function in each case, using angles and distances between lattice generators.
• Optimize the probability function across all lattice types by solving for critical points under the non-obtuse generators condition.

Experimental results

Research questions

• RQ1 What lattice configuration of equal-sized disks in R² maximizes the probability that a random point lies in exactly one disk?
• RQ2 Does the regular hexagonal grid (12-hour clock configuration) achieve this maximum among all lattice arrangements?
• RQ3 How does the probability vary across different lattice types and disk radii, particularly in intermediate regimes between packing and covering?
• RQ4 Can the optimal configuration be characterized geometrically by symmetry and overlap angles?
• RQ5 Is the 12-hour clock configuration optimal among all possible configurations of congruent disks, not just lattices?

Key findings

• The regular hexagonal grid configuration, where each disk overlaps six neighbors in 30° arcs, achieves the highest probability of 0.928... for a random point lying in exactly one disk.
• Among lattice configurations, the 12-hour clock configuration is proven optimal, with the probability decreasing for all other lattices.
• In Case 1 (two overlapping arcs), the maximum probability is 0.755..., achieved when ||b|| = √2||a|| and γ = arccos(1/(2√2)).
• In Case 2 (four overlapping arcs), the maximum probability is 0.910..., achieved when ||a|| = ||b|| and γ = arccos(√2 - 1).
• In Case 3 (six overlapping arcs), the maximum probability is 0.928..., achieved only in the regular hexagonal lattice with γ = π/3 and ϱ_L = ||a||/(2 cos(π/12)).
• The optimal configuration corresponds to the 12-hour clock arrangement, where each disk overlaps its six neighbors symmetrically with 30° arc overlaps.