Dense Ellipsoid Packings

We Ref. [1], we report our discovery of the densest known crystalline packing of congruent ellipsoids (see Figure 1), which are periodic packings with two particles in the fundamental cell. This family of new packings consists of crystalline arrangements of ellipsoids with a wide range of aspect ratios whose densities are all above that of the densest Bravais lattice packing of monodisperse spheres, i.e., \( \phi_2 = 0.7405 \) for the face-centered-cubic lattice (see Figure 2). Notably, we observe a remarkably high maximal density of \( \phi_2 = 0.7707 \) for ellipsoid aspect ratios \( \alpha \gt \sqrt{3} \) and when each ellipsoid is contact with 14 neighboring particles. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

Figure 1: Schematic of an ellipsoid with semiaxes \(a,b\) and \(c\) (into page) with aspect ratios \( \alpha=b/a \) and \( \beta=c/a=1\) (ellipsoid of revolution). We refer to the ratio of the largest to smallest semiaxes as the “maximal aspect ratio” and denote it with \(\delta\).

Figure 2: The density of the laminate crystal packing of ellipsoids as a function of the aspect ratio \(\alpha=1\) (\beta=1). The point \( \alpha=1 \) corresponding to the FCC lattice sphere packing is shown, along with the two sharp maxima in the density for prolate ellipsoids with \(\alpha=\sqrt{3}\) and oblate ellipsoids with \(\alpha=1/\sqrt{3}\), as illustrated in the insets. The presently maximal achievable density is highlighted with a thicker line, and is constant for \(\delta\gt\sqrt{3}\). Figure and caption from Ref. [1].

1. A. Donev, F. H. Stillinger, P. M. Chaikin, and S. Torquato, Unusually Dense Crystal Packings of Ellipsoid, Physical Review Letters, 92, 255506 (2004).

Maximally Dense Superdisk Packings

Almost all studies of the densest particle packings consider convex particles. In Ref. [1], we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by \(|x_1|^{2p} + |x_2|^{2p} \lt 1 \) (see Figure 1) and thus contain a large family of both convex (\( p \gt 0.5 \)) and concave ( \(0\lt p\lt 0.5\) ) particles. Our candidate maximal packing arrangements are achieved by certain families of Bravais lattice packings (see Figure 2), 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.

Figure 1: Superdisks with different deformation parameters \(p\). Figure and caption taken from Ref. [1].

Figure 2: Candidate optimal packings of superdisks with \(p=1.8\) (left), and \(p=2.0\) (right). Figure taken from Ref. [1].

Maximally Dense Superball Packings

In Ref. [2], 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.

Figure 3: Superballs with different deformation parameters \(p\). (a) \(p=0.5\), (b) \(p=0.75\), (c) \(p=1..0\), and (d) \(p=2.0\). Figure taken from Ref. [2].

Figure 4: Candidate optimal packings of superballs with \(p=1.8\) (top left), \(p=2.0\) (top right), \(p=0.8\) (bottom left), and \(p=0.55\) (bottom right). Figure taken from Ref. [2].

1. Y. Jiao, F. H. Stillinger and S. Torquato, Optimal Packings of Superdisks and the Role of Symmetry, Physical Review Letters, 100, 245504 (2008).
2. Y. Jiao, F. H. Stillinger and S. Torquato, Optimal Packings of Superballs, Physical Review E, 79, 041309 (2009). Please see Erratum.

Dense Packings of Tetrahedra, Icosahedra, Dodecahedra, and Octahedra

Figure 1: The five Platonic solids are the tetrahedron (P1), icosahedron (P2), dodecahedron (P3), octahedron (P4) and cube (P5). The 13 Archimedean solids are the truncated tetrahedron (A1), truncated icosahedron (A2), snub cube (A3), snub dodecahedron (A4), rhombicosidodecahedron (A5), truncated icosidodecahedron (A6), truncated cuboctahedron (A7), icosidodecahedron (A8), rhombicuboctahedron (A9), truncated dodecahedron (A10), cuboctahedron (A11), truncated cube (A12), and truncated octahedron (A13). The cube (P5) and truncated octahedron (A13) are the only Platonic and Archimedean solids, respectively, that tile space. Figure and caption taken from Ref. [1].

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. Prior work in this field has focused mainly on spherical particles, but very little is known about dense polyhedral packings. In Refs. [1] and [2], we formulate the generation of dense packings of polyhedra as an optimization problem by using an adaptive fundamental cell subject to periodic boundary conditions; a method termed the “adaptive shrinking cell” scheme (see Ref. [2] for details).

Figure 2: Portions of the densest packing of (a) tetrahedra obtained from our simulations, and the optimal lattice packings of the (b) icosahedra, (c) dodecahedra, and (d) octahedra to which our simulations converge. Figure and caption taken from Ref. [1].

Using a variety of multi-particle initial configurations, we find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space (see Figures 1 and 2). The densities of these packings 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 which is the analogue of Kepler’s sphere conjecture for these solids.

Dense Periodic Packings of Regular Tetrahedra

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. In Ref. [3], 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 \(\phi=4000/4671=0.856347…\) (see Figure 3). 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.

Figure 3: In the top panel [(a) and (b)], we show the two different configurations of the densest packings of four tetrahedra (two dimers) with \( \phi=12250/14319=0.855506… \) within their corresponding rhombohedral fundamental cells of the two-parameter family. In the bottom panel [(c) and (d)], we show the two different configurations of the densest packings of four tetrahedra (two dimers) with \(\phi=4000/4671=0.856347…\) within their corresponding rhombohedral fundamental cells of the three-parameter family.

Dense Periodic Packings of Truncated Tetrahedra

The Platonic and Archimedean polyhedra possess beautiful symmetries and arise in many natural and synthetic structures. 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. In Ref. [4], 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 (see Figure 4). 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.

Figure 4: Structures along the continuous deformation path that bring the Conway-Torquato packing of truncated tetrahedra with \(\phi=23/24\) to the putative optimal packing with \(\phi=207/208\). The associated deformation parameter \(\gamma\) for each packing is (a) \(\gamma=0\) (the Conway-Torquato packing), (b) \(\gamma=2/27\), (c) \(\gamma=4/27\), and (d) \(\gamma=2/9\) (our current putative optimal packing). Figure and caption taken from Ref. [4].

This research has garnered significant attention: our work in Ref. [1] was the featured cover story of the August 13th, 2009 issue of Nature and was the subject of multiple discussions in the popular press; including Nature News and Views, Nature Podcasts, Nature: Making the Paper, Princeton News, American Mathematical Society, and a Wolfram Demonstration Project. Additionally, our work in Ref. [4] was featured on the cover of the October 21st, 2011 issue of The Journal of Chemical Physics.

1. S. Torquato and Y. Jiao. Dense Packings of the Platonic and Archimedean Solids, Nature, 460, 876 (2009). Please see Erratum and Corrigendum.
2. S. Torquato and Y. Jiao. Dense Packings of Polyhedra: Platonic and Archimedean Solids, Physical Review E, 80, 041104 (2009). Please see Erratum.
3. S. Torquato and Y. Jiao, Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra, Physical Review E, 81, 041310 (2010).
4. Y. Jiao and S. Torquato, Communication: A Packing of Truncated Tetrahedra That Nearly Fills All of Space and its Melting Properties, Journal of Chemical Physics, 135, 151101 (2011).

Dense Periodic Packings of 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. In Ref. [1], we investigate the packing behavior of congruent ring tori in \(\mathbb{R}^3\), 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 detailed in Ref. [2]. 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 Figure 1). For more information on the aforementioned organizing principles, please see slides from the invited talk “Packing Nonspherical Particles: All Shapes Are Not Created Equal” given by Professor Torquato at the American Physical Society March Meeting in Boston on February 28, 2012. The slides are also available on the APS website. For more information,

Figure 1: (left) A dense packing of unlinked ring tori. (right) A dense packing of linked ring tori.

1. R. Gabbrielli, Y. Jiao, and S. Torquato, Dense Periodic Packings of Tori, Physical Review E, 89, 022133 (2014).
2. S. Torquato and Y. Jiao, Organizing Principles for Dense Packings of Nonspherical Hard Particles: Not All Shapes are Created Equal, Physical Review E, 86, 011102 (2012).

Densest Local Packings of \(d\)-dimensional Spheres

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 in Ref. [1]. The knowledge of \(R_{min}(N)\) \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) 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 Rd. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding \(R_{min
}(N)\) selected values of \(N\) up to \(N=348\) (see Figure 1) 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.

Figure 1: A conjectured optimal packing (point group \(D_{5h}\)) for \(N=15\) and \(R_{min}(15)=1.873123…\) with encompassing sphere of radius \(R_{min}(15)+0.5=2.373123…\). Figure and caption taken from Ref. [1].

In Ref. [2], we extend our analysis to three-dimensional spheres and obtain the densest local packings of \(N\) identical nonoverlapping spheres within a radius \(R_{min}(N)\) of a fixed central sphere of the same size for selected values of \(N\) up to \(N=1054\). We analyze the properties and characteristics of the densest local packings in \(\mathbb{R}^d\) and employ knowledge of the \(R_{min}(N)\), using methods applicable in any dimension \(d\) 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. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance \(R_{min}(N)\). In particular, we find two “maracas” packings at \(N = 77\) and \(N = 93\), each consisting of a few unjammed spheres free to rattle within a “husk” composed of the maximal number of spheres that can be packed with centers at respective \(R_{min}(N)\). 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.

Lastly, in Ref. [3], we consider the connection of our densest local packing problem to the spherical code problem which 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 specifically 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.

1. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Densest Local Sphere-Packing Diversity: General concepts and application to two dimensions, Physical Review E, 81, 041305 (2010).
2. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Densest Local Packing Diversity. II. Application to Three Dimensions, Physical Review E, 83, 011304 (2011).
3. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Spherical Codes, Maximal Local Packing Density, and the Golden Ratio, Journal of Mathematical Physics, 51, 043302 (2010).