Worst-case optimal squares packing into disks

Abstract

$\newcommand{\nicefrac}{\tfrac}$We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $\delta=\nicefrac{8}{5\pi}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \nicefrac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\nicefrac{8}{5}+\varepsilon$ that cannot be packed, even if squares may be rotated.

This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square ($\nicefrac{1}{2}$), circles in a square ($\nicefrac{\pi}{(3+2\sqrt{2})}\approx 0.539)$ and circles in a circle ($\nicefrac{1}{2}$) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container;  neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.