Periodic Planar Disk Packings

Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any strictly jammed packings, whose graph does not consist of all triangles and the torus lattice is the standard triangular lattice, is at most $\frac{n}{n+1}\frac{\pi}{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.Comment: 26 pages, 13 figure

On Thickness and Packing Density for Knots and Links

We describe some problems, observations, and conjectures concerning thickness and packing density of knots and links in \sp^3 and $\R^3$. We prove the thickness of a nontrivial knot or link in \sp^3 is no more than $\frac{\pi}{4}$, the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in $\R^3$ or \sp^3 is generally less than $\frac{\pi}{\sqrt{12}}$, the density of the hexagonal packing of unit disks in $\R^2$.Comment: 6 pages; to appear in Contemporary Mathematics volume edited by Calvo, Millett & Rawdo

A Solidification Phenomenon in Random Packings

We prove that uniformly random packings of copies of a certain simply-connected figure in the plane exhibit global connectedness at all sufficiently high densities, but not at low densities