**
Toward Maximally Dense Ellipsoid Packings
**

**
Maximally Dense Superdisk Packings
**

**Maximally
Dense Superball Packings**

**Dense
Packings of Platonic and Archimedean Solids**

**Dense
Packings of Regular Tetrahedra**

**A Dense
Packing of Truncated Tetrahedra**

**Dense Periodic Packings of Tori**

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

See Y. Jiao, F. H. Stillinger and S. Torquato

**--- 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
Sal Torquato and
Yang 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

See
R. Gabbrielli, 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: http://absuploads.aps.org/presentation.cfm?pid=10202.

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

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