Echoes of the hexagon: Remnants of hexagonal packing inside regular polygons

Abstract

We provide evidence that for regular polygons with σ = 6 j sides (with j = 2 , 3 , …), N ( k ) = 3 k ( k + 1 ) + 1 (with k = 1 , 2 , …) congruent disks of appropriate size can be nicely packed inside these polygons in highly symmetrical configurations, which apparently have maximal density for N sufficiently small. These configurations are invariant under rotations of π / 3 and are closely related to the configurations with perfect hexagonal packing in the regular hexagon and to the configurations with curved hexagonal packing (CHP) in the circle found a long time ago by Graham and Lubachevsky [“Curved hexagonal packings of equal disks in a circle,” Discrete Comput. Geometry 18(2), 179–194 (1997)]. The packing fraction, i.e., the portion of accessible volume (area) occupied by multiple solid objects, has a role in determining the properties of granular materials and fluids. At the basis of our explorations are the algorithms that we have devised, which are very efficient in producing the CHP and more general configurations inside regular polygons. We have used these algorithms to generate a large number of CHP configurations for different regular polygons and numbers of disks; a careful study of these results has made it possible to fully characterize the general properties of the CHP configurations and to devise a deterministic algorithm that completely ensembles a given CHP configuration once an appropriate input is specified.