A Note on 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 $2\le S_n/n<3\sqrt{3}/2$, and these are the best constants; moreover, for $\Delta (n):=(3\sqrt{3}/2)n-S_n$, we have $0.727384<\mathrm{liminf}\Delta (n)/\sqrt{n}<2.121321$ and $0.727384<\mathrm{limsup}\Delta (n)/\sqrt{n}<4.165064$.

8 pages, 3 figures, some corrections made in version 2