** Maximally Dense Packings **

We have recently discovered the densest known crystalline packing of congruent ellipsoids, which are periodic packings with two particles in the fundemantal cell. The resutls was published in the letter entitled *Unusually Dense Crystal Ellipsoid Packings* (* Phys. Rev. Lett.* **92 **, 255506 (2004)) by A. Donev, F. H. Stillinger, P. M. Chaikin and S. Torquato. In this Letter, we report on the densest known packings of congruent ellipsoids. The family of new packings consists of crystal arrangements of spheroids with a wide range of aspect ratios, and with density always surpassing that of the densest Bravais lattice packing, i.e., 0.7405. A remarkable maximum density of 0.7707 is achieved for maximal aspect ratios larger than \(\sqrt{3}\), when each ellipsoid has 14 touching neighbors. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

We obtained the densest known packings of superdisks and superballs, which are certain Bravais lattice packings with symmetries consistent with that of the particles. See Y. Jiao, F. H. Stillinger and S. Torquato *Optimal Packing of Superdisks and the Role of Symmetry* (* Phys. Rev. Lett.* **100 **, 245505 (2008)) for superdisk packings, where we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by \(|x_1|^{2p}+|x_2|^{2p}\le 1\) and thus contain a large family of both convex (\(p \gt 0.5\)) and concave (\(p \lt 0.5\)) particles. Our candidate maximal packing arrangements are achieved by certain families of Bravais lattice packings, and the maximal density is nonanalytic at the circular-disk point \((p=1)\) and increases dramatically as p moves away from unity. Moreover, we show that the broken rotational symmetry of superdisks influences the packing characteristics in a nontrivial way.

See Y. Jiao, F. H. Stillinger and S. Torquato *Optimal Packing of Superballs *(* Phys. Rev. E ***79 **, 041309 (2009)) for superball packings, where we provide analytical constructions for the densest known superball packings for all convex and concave cases, defined by \(|x_1|^{2p}+|x_2|^{2p}+|x_3|^{2p}\le 1\). The candidate maximally dense packings are certain families of Bravais lattice packings in which each particle has 12 contacting neighbors possessing the global symmetries that are consistent with certain symmetries of a superball. We also provide strong evidence that our packings for convex superballs (\(p\gt 0.5\)) are most likely the optimal ones. The maximal packing density as a function of p is nonanalytic at the sphere point \(p=1\) and increases dramatically as p moves away from unity. Two more nontrivial nonanalytic behaviors occur at \(p_c=1.1509\) and \(p_o =\ln{3}/\ln{4}=0.7924\) for cubic and octahedral superballs, respectively, where different Bravais lattice packings possess the same densities.

**— The featured cover story of Aug. 13, 2009 issue of Nature**

Abstract:

Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline states of matter, granular media, heterogeneous materials and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory. Previous work has focused mainly on spherical particles, very little is known about dense polyhedral packings. Torquato and Jiao formulate the generation of dense packings of polyhedra as an optimization problem, using an adaptive fundamental cell subject to periodic boundary conditions (termed as the ‘adaptive shrinking cell’ scheme). Using a variety of multi-particle initial configurations, Torquato and Jiao find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space. The densities are \(0.782…, 0.947…, 0.904…\) and \(0.836…\), respectively. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Combining the simulation results with derived rigorous upper bounds and theoretical

arguments leads to the conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analogue of Kepler’s sphere conjecture for these solids.

See S. Torquato and Y. Jiao, **Nature** **460**, 876 (2009) for details.

The packing details can be found here.

Here are some links to discussions of this work in the popular press:

– News on the American

Mathematical Society webpage

– Wolfram

Demonstration Project: Densest Tetrahedral Packing

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered “dimer” packings of tetrahedra, including the densest known packings with density \(\phi = 4000/4671 =0.856347…\). This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.

See S. Torquato and Y. Jiao, Phys. Rev. E **81**, 041310 (2010).

**— Featured on the cover of Journal of Chemical Physics**

The Platonic and Archimedean polyhedra possess beautiful symmetries and arise in many natural and synthetic structures. Dense polyhedron packings are useful models of a variety of condensed matter systems, including liquids, glasses and crystals, granular media, and heterogeneous materials. Understanding how nonspherical particles pack is a first step toward a better understanding of how biological cells pack. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory.

Recently, there has been a large effort devoted to finding dense packings of polyhedra. Although organizing principles for the types of structures associated with the densest polyhedron packings have been put forth, much remains to be done to find the maximally dense packings for specific shapes. Here, we analytically construct the densest known packing of truncated tetrahedra with packing fraction \(207/208=0.995 192 …\), which is amazingly close to unity and strongly implies the optimality of the packing. This construction is based on a generalized organizing principle for polyhedra that lack central symmetry. Moreover, we find that the holes in this putative optimal packing are small regular tetrahedra, leading to a new tiling of space by regular tetrahedra and truncated tetrahedra. We also numerically study the equilibrium melting properties of what apparently is the densest packing of truncated tetrahedra as the system undergoes decompression. Our simulations reveal two different stable crystal phases, one at high densities and the other at intermediate densities, as well as a first-order liquid-crystal phase transition.

See Y. Jiao and S. Torquato, J. Chem. Phys. **135**, 151101 (2011).

(Left) A dense packing of unlinked ring tori. (Right) A dense packing of linked ring tori.

Dense packings of nonoverlapping bodies in three-dimensional Euclidean space \(\mathbb{R}^3\) are useful models of the structure of a variety of many-particle systems that arise in the physical and biological sciences. Here we investigate the packing behavior of congruent ring tori in *R ^{3}*, which are multiply connected nonconvex bodies of genus 1, as well as horn and spindle tori. Specifically, we analytically construct a family of dense periodic packings of unlinked tori guided by the organizing principles originally devised for simply connected solid bodies [Torquato and Jiao, Phys.Rev.E 86, 011102 (2012)]. We find that the horn tori as well as certain spindle and ring tori can achieve a packing density not only higher than that of spheres (i.e., \(\pi/\sqrt{18}=0.7404…\)) but also higher than the densest known ellipsoid packings (i.e., \(0.7707…\)). In addition, we study dense packings of clusters of pair-linked ring tori (i.e., Hopf links), which can possess much higher densities than corresponding packings consisting of unlinked tori.

See R. Gabrielli, Y. Jiao, and S. Torquato, Phys. Rev. E. **89**, 022133 (2014).

Invited talk by Professor Torquato entitled “Packing Nonspherical Particles: All Shapes Are Not Created Equal” given at the March American Physical Society Meeting in Boston on February 28, 2012. See also the APS link: here.

See also: *A New Tool to Help Mathematicians Pack*

** Densest Local Packings **

**The Densest Local Packings of Identical Spheres in Three Dimensions**

We have used a novel algorithm combining nonlinear programming methods with a random search of configuration space to find the densest local packings of spheres in three-dimensional Euclidean space. Our results reveal a wealth of information about packings of spheres, including counterintuitive results concerning the physics of dilute mixtures of spherical solute particles in a solvent composed of same-size spheres and about the presence of unjammed spheres (rattlers) in the densest local structures. Read more

**The Densest Local Packings of Identical Disks in Two Dimensions**

N=15, point group D_{5h}

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. 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 a realizability condition for the pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. Read more

**The Densest Local Packings of Spheres in Any Dimension and the Golden Ratio**

The optimal spherical code problem involves the placement of the centers of N nonoverlapping spheres of unit diameter onto the surface of a sphere of radius R such that R is minimized. We prove that in any dimension, all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem. Read more