Densest local packing diversity. II. Application to three dimensions

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N = 348. We analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar, as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the FCC Barlow packing. We additionally observe that the densest local packings are dominated by the spheres arranged with centers at precisely distance Rmin(N) from the fixed sphere's center.Comment: 45 pages, 18 figures, 2 table

Density of Binary Disc Packings: Playing with Stoichiometry

We consider the packings in the plane of discs of radius $1$ and $\sqrt{2}-1$ when the proportions of each type of disc are fixed. The maximal density is determined and the densest packings are described. A phase separation phenomenon appears when there is an excess of small discs.Comment: The code used in the proof is included in the arxiv (check.cpp

When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each "hole" is bounded by three pairwise tangent discs are called triangulated. Connelly conjectured that when such packings exist, one of them maximizes the proportion of the covered surface: this holds for unary and binary disc packings. For ternary packings, there are 164 pairs (r, s), 1 > r > s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to study ternary triangulated packings. We prove that the conjecture holds for 31 triplets of disc radii and disprove it for 40 other triplets. Finally, we classify the remaining cases where our methods are not applicable. Our approach is based on the ideas present in the Hales\u27 proof of the Kepler conjecture. Notably, our proof features local density redistribution based on computer search and interval arithmetic

Maximally Dense Disc Packings on the Plane

Suppose one has a collection of disks of various sizes with disjoint interiors, a packing, in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book \textit{Regular Figures} \cite{MR0165423}, L\'aszl\'o Fejes T\'oth found a series of packings that were his best guess for the maximum density for any $1> q > 0.2$. Meanwhile Gerd Blind in \cite{MR0275291,MR0377702} proved that for $1\ge q > 0.72$, the most dense packing possible is $\pi/\sqrt{12}$, which is when all the disks are the same size. In \cite{MR0165423}, the upper bound of the ratio $q$ such that the density of his packings greater than $\pi/\sqrt{12}$ that Fejes T\'oth found was $0.6457072159..$. Here we improve that upper bound to $0.6585340820..$. Both bounds were obtained by perturbing a packing that has the property that the graph of the packing is a triangulation, which L. Fejes T\'oth called a \emph{compact} packing, and we call a \emph{triangulated} packing. Previously all of L. Fejes T\'oth's packings that had a density greater than $\pi/\sqrt{12}$ and $q > 0.35$ were based on perturbations of packings with just two sizes of disks, where the graphs of the packings were triangulations. Our new packings are based on a triangulated packing that have three distinct sizes of disks, found by Fernique, Hashemi, and Sizova, \cite{1808.10677}, which is something of a surprise. We also point out how the symmetries of a triangulated doubly periodic packing can by used to create the actual packing that is guaranteed by a famous result of Thurston, Andreev, and Andreeson \cite{MR2131318}.Comment: The main graph that shows the relation to previous packings has been changed and focused on the critical portion. Also various unneeded parts have been remove

Density of triangulated ternary disc packings

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 15 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 40 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.Comment: 37 pages, SageMath code included in source (in 'code' directory