Circle packing on spherical caps
Paolo Amore

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, . 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 (, , , , ) and for several values of . The numerical results obtained in the…
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Taxonomy
TopicsMathematics and Applications
