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Dense packing of particles has provided important models to study the structure of matter in various systems such as liquid, glassy and crystalline phase, etc. The simplest sphere packing models are able to represent and capture salient…

The exploration of the densest sphere packings is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings…

Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low temperature phases of matter. In celebrated work,…

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure…

Spontaneous self-assembly of hard convex polyhedra are known to form orientationally disordered crystalline phases, where particle orientations do not follow the same pattern as the positional arrangement of the crystal. A distinct type of…

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided…

Spherical particles confined to a sphere surface cannot pack densely into a hexagonal lattice without defects. In this study, we use hard particle Monte Carlo simulations to determine the effects of continuously deformable shape anisotropy…

A new tetrahedral structure model was developed and the geometrical structure of jammed disordered packings of monodisperse spheres with different friction coefficients was systematically characterized. An intrinsic structure feature is…

The structure of the densest crystal packings is determined for a variety of concave shapes in 2D constructed by the overlap of two or three disks. The maximum contact number per particle pair is defined and proposed as a useful means of…

Dense packings composed of irregular polyhedral particles are investigated by numerical simulations under quasistatic triaxial compression. The Contact Dynamics method is used for this investigation with 40 000 particles. The effect of…

Dense, disordered packings of particles are useful models of low-temperature amorphous phases of matter, biological systems, granular media, and colloidal systems. The study of dense packings of nonspherical particles enables one to…

Structural organization and correlations are studied in very large packings of equally sized acrylic spheres, reconstructed in three-dimensions by means of X-ray computed tomography. A novel technique, devised to analyze correlations among…

Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly…

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…

Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the…

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…

Motivated by the relation between particle shape and packing, we measure the volume fraction $\phi$ occupied by the Platonic solids which are a class of polyhedron with congruent sides, vertices and dihedral angles. Tetrahedron, cube,…

We study the optimal packing of hard spheres in an infinitely long cylinder, using simulated annealing, and compare our results with the analogous problem of packing disks on the unrolled surface of a cylinder. The densest structures are…

During the last few years several new results on packing problems were obtained using a blend of tools from semidefinite optimization, polynomial optimization, and harmonic analysis. We survey some of these results and the techniques…

In 1900, as a part of his 18th problem, Hilbert proposed the question to determine the densest congruent (or translative) packings of a given solid, such as the unit ball or the regular tetrahedron of unit edges. Up to now, our knowledge…