Periodic Planar Disk Packings

TL;DR

This paper investigates conditions for locally maximally dense packings of equal disks in a torus, proposing conjectures on density limits for jammed packings and presenting classes where these hold.

Contribution

It introduces conjectures on the maximum density of jammed disk packings in a torus and provides classes of packings where these conjectures are validated.

Findings

Conjecture on density upper bound for jammed packings.

Identification of classes of packings satisfying the conjecture.

Conditions for local maximal density in torus packings.

Abstract

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.