packing disks tightly in the plane Given a discrete point set $S$ in ${\bf R}^2$ with a specified base-point $p_0 \in S$, label the remaining points as $p_1, p_2, \dots$ in order of increasing distance from $p_0$ (with ties resolved indifferently), and let $d_n(S,p_0)$ be the distance between $p_0$ and $p_n$.
What is known about the infimum of $d_n(S,p_0)$ as $S$ and $p_0$ vary, if $S$ is required to have the property that no two of its points are less than 1 apart?  An equivalent statement of the problem replaces each point-set with a packing of the plane by disjoint disks of radius 1/2.  It would be very nice if the infimum of $d_n$ was achieved by the hexagonal packing of the plane, but my intuition says that for some $n$ this is not the case; I wonder if this is known.
 A: There's an intuitive way to see that $d_7$ is not achieved by the hexagonal packing: put five disks (evenly spaced), rather than six, around disk 0.  Now add five disks into the gaps between the first five disks.  It is easy to see that disks 6 through 10 in this packing are closer to the center than disk 7 in the hexagonal packing.
A: Hopkins, Stillinger and Torquato give putative minima for $d_n$ for $n\le348$. In many cases these are improvements over a triangular lattice packing.
From the abstract:

The densest local packings of $N$ identical nonoverlapping spheres within a radius $r_{\min}(N)$ of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space. The knowledge of $r_{\min}(N)$ in $d$-dimensional Euclidean space $R^d$ allows for the construction both of a realizability condition for pair-correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings in $R^d$. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding $r_{\min}(N)$ for selected values of $N$ up to $N=348$ and use this knowledge to construct such a realizability condition and an upper bound. We additionally analyze the properties and characteristics of the maximally dense packings, finding significant variability in their symmetries and contact networks, and that the vast majority differ substantially from the triangular lattice even for large N. Our work has implications for packaging problems, nucleation theory, and surface physics.

