We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction 100/117 ≈ 0.8547, that are simple in the sense that they are transitive and their repeating units involve only four tetrahedra. The construction of the packings was inspired from results of a numerical search that yielded a similar packing. We present an analytic construction of the packings and a description of their properties. We also present a transitive packing with a repeating unit of two tetrahedra and a packing fraction

We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well-studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of asphericity, providing an atlas of crystal structures that might be observed in systems of nanoparticles with tetrahedral symmetry.

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

Maximally random jammed (MRJ) sphere packing is a prototypical example of a system naturally poised at the margin between underconstraint and overconstraint. This marginal stability has traditionally been understood in terms of isostaticity, the equality of the number of mechanical contacts and the number of degrees of freedom. Quasicontacts, pairs of spheres on the verge of coming in contact, are irrelevant for static stability, but they come into play when considering dynamic stability, as does the distribution of contact forces. We show that the effects of marginal dynamic stability, as manifested in the distributions of quasicontacts and weak contacts, are consequential and nontrivial. We study these ideas first in the context of MRJ packing of -dimensional spheres, where we show that the abundance of quasicontacts grows at a faster rate than that of contacts. We reexamine a calculation of Jin et al. (Phys. Rev. E 82, 051126, 2010), where quasicontacts were originally neglected, and we explore the effect of their inclusion in the calculation. This analysis yields an estimate of the asymptotic behavior of the packing density in high dimensions. We argue that this estimate should be reinterpreted as a lower bound. The latter part of the paper is devoted to Bravais lattice packings that possess the minimum number of contacts to maintain mechanical stability. We show that quasicontacts play an even more important role in these packings. We also show that jammed lattices are a useful setting for studying the Edwards ensemble, which weights each mechanically stable configuration equally and does not account for dynamics. This ansatz fails to predict the power-law distribution of near-zero contact forces, ( ) ∼ .

It was conjectured by Ulam that the ball has the lowest optimal packing fraction out of all convex, three-dimensional solids. Here we prove that any origin-symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls. We also show that in dimensions 4, 5, 6, 7, 8, and 24 there are origin-symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be.Comment: Final submitted manuscrip