Circle packing in regular polygons

Abstract

We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon and we have carried out intensive numerical experiments spanning several polygons (the largest number of sides considered here being $16$) and up to $200$ circles ($400$ circles in the special cases of the equilateral triangle and the regular hexagon) . Some of the configurations that we have found possibly are not global maxima of the packing fraction, particularly for $N\gg 1$, due to the great computational complexity of the problem, but nonetheless they should provide good lower bounds for the packing fraction at a given $N$. This is the first systematic numerical study of packing in regular polygons, which previously had only been carried out for the equilateral triangle, the square and the circle.

arXiv comment38 pages, 20 figures

References in corpus (16)

• Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms
• Grain Boundary Scars and Spherical Crystallography
• Two-Dimensional Matter: Order, Curvature and Defects
• Interacting Topological Defects on Frozen Topographies
• Crystalline Order on a Sphere and the Generalized Thomson Problem
• Basic Understanding of Condensed Phases of Matter via Packing Models
• Maximum and Minimum Stable Random Packings of Platonic Solids
• A comparative study of crumpling and folding of thin sheets
• Defect Motifs for Constant Mean Curvature Surfaces
• Morphological Phases of Crumpled Wire
• Crystalline Order On Riemannian Manifolds With Variable Gaussian Curvature And Boundary
• Confinement of semiflexible polymers
• Measuring order in the isotropic packing of elastic rods
• Plastic Deformation of 2D Crumpled Wires
• Unpacking of a crumpled wire from two-dimensional cavities
• On the Packing of Stiff Rods on Ellipsoids Part I -- Geometry

Cited by in corpus (2)

• Hyperuniformity scaling of maximally random jammed packings of two-dimensional binary disks
• Echoes of the hexagon: remnants of hexagonal packing inside regular polygons