Circle packing in arbitrary domains

TL;DR

This paper presents a versatile algorithm for dense packing of congruent disks in arbitrary 2D domains, including complex shapes, by introducing 'image' disks to handle fixed containers, with potential extension to higher dimensions.

Contribution

The novel algorithm enables dense packings in diverse and complex domains using 'image' disks, overcoming previous limitations and applicable to multiple dimensions.

Findings

Successfully applied to various 2D domains like rectangles, ellipses, and cardioids.

No prior results available for many studied cases.

Algorithm's extension to higher dimensions is straightforward.

Abstract

We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles, ellipses, crosses, multiply connected domains and even to the cardioid. For many of the cases that we have studied no previous result was available. The fundamental idea in our approach is the introduction of "image" disks, which allows one to work with a fixed container, thus lifting the limitations of the packing algorithms of \cite{Nurmela97,Amore21,Amore23}. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.