# Packing Disks into Disks with Optimal Worst-Case Density

## TL;DR

This paper establishes that the maximum density for packing arbitrary disks into a disk is 50%, providing a precise threshold and implications for packing algorithms.

## Contribution

The authors prove the exact critical density for disk packing in a disk, combining manual and automated analysis, with applications to recursive packing algorithms.

## Key findings

- Critical density of disk packing is 0.5.
- Any set of disks with total area ≤ 0.5 can be packed.
- Sets with area > 0.5 may not be packable.

## Abstract

We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $\delta\leq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $\varepsilon>0$ there are sets of disks of area $1/2+\varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic 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.