Packings of equal disks in a square torus

TL;DR

This paper investigates the densest packings of equal disks in a square torus, proposing conjectures, defining efficiency constants, and analyzing bounds, revealing differences from planar packings and introducing new mathematical tools.

Contribution

The paper introduces conjectured optimal packings in a square torus, defines a new efficiency constant based on continued fractions, and compares bounds for packing density errors with planar cases.

Findings

Conjectured densest packings for certain disk counts in a square torus.

A new efficiency constant for packings inspired by Markov's constant.

Error bounds for density approximation are tighter in the torus case, on the order of 1/N.

Abstract

Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a square torus that we conjecture to be the most dense for certain numbers of packing disks, using continued fractions to approximate $1/\sqrt{3}$ and $2-\sqrt{3}$. We also define a constant to measure the efficiency of a packing motived by a related constant due to Markov for continued fractions. One idea is to use the unique factorization property of Gaussian integers to prove that there is an upper bound for the Markov constant for grid-like packings. By way of contrast, we show that an upper bound by Peter Gruber for the error for the limiting density of a packing of equal disks in a planar square, which is on the order of $1/\sqrt{N}$, is the best possible, whereas for our examples for the square torus, the error for the limiting density is on the order of $1/N$, where $N$ is the number of packing disks.