Circle packing in arbitrary domains

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 some 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 Nurmela and Östergärd [Discrete Comput. Geom. 18(1), 111–120 (1997)], Amore and Morales [Discrete Comput. Geom. 70, 249–267 (2022)], and Amore [Phys. Fluids 35(2), 027130 (2023)]. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.

Packing Spheres into a Minimum-Height Parabolic Container

“…However, as was highlighted in review papers [12][13][14][15], challenging packing problems in irregular containers are much less investigated. Several publications consider ellipsoidal containers [16,17], while there are only a few works focusing on packing for multiply connected domains and cardioids [18] and paraboloids [19]. Sphere packing into irregular containers arises in, e.g., material science [20,21] and nanotechnology [22].…”

“…This approach was applied to a problem of packing circles into an ellipse and minimizing the ellipse size. Packing algorithms applied to different-shaped twodimensional domains are studied in [18], including rectangles, ellipses, crosses, multiply connected domains and even cardioid shapes. The authors introduce a novel approach centered around the concept of "image" disks, enabling the study of packing within fixed containers.…”