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In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p in (1, 1.58]$. We conjecture that the family also exists for $p in (1.58, log_2 3 = 1.5849625ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
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Suppose one has a collection of disks of various sizes with disjoint interiors, a packing, in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book textit{Regular Figures} (MR0165423), Laszlo Fejes Toth found a series of packings that were his best guess for the maximum density for any $1> q > 0.2$. Meanwhile Gerd Blind in (MR0275291,MR0377702) proved that for $1ge q > 0.72$, the most dense packing possible is $pi/sqrt{12}$, which is when all the disks are the same size. In (MR0165423), the upper bound of the ratio $q$ such that the density of his packings greater than $pi/sqrt{12}$ that Fejes Toth found was $0.6457072159..$. Here we improve that upper bound to $0.6585340820..$. Our new packings are based on a perturbation of a triangulated packing that have three distinct sizes of disks, found by Fernique, Hashemi, and Sizova, (MR4292755), which is something of a surprise.
In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zongs recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840ldots$ to $0.3745ldots$, getting closer to the best known lower bound of $0.3673ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization.
We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {em average} radius of the balls, resulting in bounds of the familiar classical form.
Understanding granular materials aging poses a substantial challenge: Grain contacts form networks with complex topologies, and granular flow is far from equilibrium. In this letter, we experimentally measure a three-dimensional granular systems reversibility and aging under cyclic compression. We image the grains using a refractive-index-matched fluid, then analyze the images using the artificial intelligence of variational autoencoders. These techniques allow us to track all the grains translations and three-dimensional rotations with accuracy sufficient to infer contact-point sliding and rolling. Our observations reveal unique roles played by three-dimensional rotations in granular flow, aging, and energy dissipation. First, we find that granular rotations dominate the bulk dynamics, penetrating more deeply into the granular material than translations do. Second, sliding and rolling do not exhibit aging across the experiment, unlike translations. Third, aging appears not to minimize energy dissipation, according to our experimental measurements of rotations, combined with soft-sphere simulations. The experimental tools, analytical techniques, and observations that we introduce expose all the degrees of freedom of the far-from-equilibrium dynamics of granular flow.
A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $mathbb{R}^3$ is not greater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $mathbb{R}^4$ and $mathbb{R}^5$.
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