Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio

We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have the usual regular square or hexagonal pattern. However, for 1495 values of n in the tested range n =< 5000, specifically, for n = 49, 61, 79, 97, 107,... 4999, we prove that the optimum cannot possibly be achieved by such regular arrangements. The evidence suggests that the limiting height-to-width ratio of rectangles containing an optimal hexagonal packing of circles tends to 2-sqrt(3) as n tends to infinity, if the limit exists.Comment: 21 pages, 13 figure

On Thickness and Packing Density for Knots and Links

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in \sp^3 and $\R^3$. We prove the thickness of a nontrivial knot or link in \sp^3 is no more than $\frac{\pi}{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in $\R^3$ or \sp^3 is generally less than $\frac{\pi}{\sqrt{12}}$, the density of the hexagonal packing of unit disks in $\R^2$.Comment: 6 pages; to appear in Contemporary Mathematics volume edited by Calvo, Millett & Rawdo

On packing spheres into containers (about Kepler's finite sphere packing problem)

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.Comment: 13 pages, 2 figures; v2: major revision, extended result, simplified and clarified proo

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

Based on numerical simulations that we have carried out, we provide evidence that for regular polygons with $\sigma= 6j$ 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 $\pi/3$ 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.Comment: 25 pages, 12 figures, 4 table

Circle packing in arbitrary domains

We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles, ellipses, crosses, multiply connected domains and even to the cardioid. For many of the cases that we have studied no previous result was available. The fundamental idea in our approach is the introduction of "image" disks, which allows one to work with a fixed container, thus lifting the limitations of the packing algorithms of \cite{Nurmela97,Amore21,Amore23}. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.Comment: 26 pages, 17 figure

Electrical networks and Stephenson's conjecture

In this paper, we consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We then construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. As an application, we will affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome

Phase behavior of hard circular arcs

By using Monte Carlo numerical simulation, this work investigates the phase behavior of systems of hard infinitesimally thin circular arcs, from an aperture angle θ → 0 to an aperture angle θ → 2π, in the twodimensional Euclidean space. Except in the isotropic phase at lower density and in the (quasi)nematic phase, in the other phases that form, including the isotropic phase at higher density, hard infinitesimally thin circular arcs autoassemble to form clusters. These clusters are either filamentous, for smaller values of θ, or roundish, for larger values of θ. Provided the density is sufficiently high, the filaments lengthen, merge, and straighten to finally produce a filamentary phase while the roundels compact and dispose themselves with their centers of mass at the sites of a triangular lattice to finally produce a cluster hexagonal phas