A Note on Rectangle Covering with Congruent Disks

TL;DR

This paper establishes bounds on the maximum rectangle area covered by n unit disks, providing precise asymptotic behavior of the difference between the ideal and actual covered area.

Contribution

It proves tight bounds for the maximum rectangle area covered by n congruent disks and analyzes the asymptotic behavior of the area difference.

Findings

Bounds for S_n/n are between 2 and 3√3/2.

Asymptotic limits of Δ(n)/√n are approximately between 0.727 and 4.165.

Provides best possible constants for rectangle covering with congruent disks.

Abstract

In this note we prove that, if $S_n$ is the greatest area of a rectangle which can be covered with $n$ unit disks, then $2\leq S_n/n<3 \sqrt{3}/2$, and these are the best constants; moreover, for $\Delta(n):=(3\sqrt{3}/2)n-S_n$, we have $0.727384<\liminf\Delta(n)/\sqrt{n}<2.121321$ and $0.727384<\limsup\Delta(n)/\sqrt{n}<4.165064$.