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

Abstract

Based on numerical simulations that we have carried out, we provide evidence that for regular polygons with $σ= 6j$ sides (with $j=2,3,\dots$), $N(k)=3 k (k+1)+1$ (with $k=1,2,\dots$) 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 {\sl curved hexagonal packing} (CHP) in the circle found long time ago by Graham and Lubachevsky. 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 possible to fully characterize the general properties of the CHP configurations and to devise a {\sl deterministic} algorithm that completely ensembles a given CHP configuration once an appropriate input ("DNA") is specified. Our analysis shows that the number of CHP configurations for a given $N$ is highly degenerate in the packing fraction and it can be explicitly calculated in terms of $k$ (number of shells), of the building block of the DNA itself and of the number of vertices in the fundamental domain (because of the symmetry we work in $1/6$ of the whole domain). With the help of our deterministic algorithm we are able to build {\sl all} the CHP configurations for a polygon with $k$ shells.