Packing Squares into a Disk with Optimal Worst-Case Density

TL;DR

This paper determines the exact maximum density for packing squares into a disk, proving that any set of squares with total area up to approximately 0.509 can be packed, resolving a longstanding problem in geometric packing.

Contribution

The paper establishes the precise critical density for packing squares into a disk, using a novel combination of manual analysis and computer-assisted interval arithmetic, which was previously unresolved.

Findings

Critical density for packing squares into a disk is approximately 0.509.

Sets of squares with total area up to 0.509 can always be packed into a unit disk.

The proof combines manual analysis with computer-assisted interval arithmetic.

Abstract

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 $\delta=\frac{8}{5\pi}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\frac{8}{5}+\varepsilon$ 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 $\left(\frac{1}{2}\right)$, circles in a square $\left(\frac{\pi}{(3+2\sqrt{2})}\approx 0.539\right)$ and circles in a circle $\left(\frac{1}{2}\right)$ 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.