Circle packing on spherical caps

TL;DR

This study explores the densest arrangements of congruent disks on spherical caps of various sizes, revealing how curvature influences packing density, defect structures, and the evolution of configurations from flat to spherical geometries.

Contribution

It provides the first extensive numerical analysis of disk packings on spherical caps, bridging the gap between planar and spherical packing problems with new insights into defect behavior and configuration evolution.

Findings

Packing fraction varies with number of disks and cap size.

Topological defects become more negative with increased curvature.

Hexagonal packing configurations evolve and sometimes disappear as curvature increases.

Abstract

We have studied the packing of congruent disks on a spherical cap, for caps of different size and number of disks, $N$. This problem has been considered before only in the limit cases of circle packing inside a circle and on a sphere (Tammes problem), whereas all intermediate cases are unexplored. Finding the preferred packing configurations for a domain with both curvature and border could be useful in the description of physical and biological systems (for example, colloidal suspensions or the compound eye of an insect), with potential applications in engineering and architecture (e.g. geodesic domes). We have carried out an extensive search for the densest packing configurations of congruent disks on spherical caps of selected angular widths ($\theta_{\rm max} = \pi/6$, $\pi/4$, $\pi/2$, $3\pi/4$, $5 \pi/6$) and for several values of $N$. The numerical results obtained in the present work have been used to establish (at least qualitatively) some general features for these configurations, in particular the behavior of the packing fraction as function of the number of disks and of the angular width of the cap, or the nature of the topological defects in these configurations (it found that as the curvature increases the overall topological on the border tends to become more negative). Finally we have studied the packing configurations for $N = 19$, $37$, $61$, $91$ (hexagonal numbers) for caps ranging from the flat disk to the whole sphere, to observe the evolution (and eventual disappearance) of the curved hexagonal packing configurations while increasing the curvature.