Revisiting the hexagonal lattice: on optimal lattice circle packing

TL;DR

This paper provides a simplified proof that the hexagonal lattice achieves the highest circle packing density among all lattices in two dimensions, highlighting the importance of well-rounded lattices in such optimization problems.

Contribution

It offers a straightforward proof of the optimality of the hexagonal lattice and emphasizes the significance of well-rounded lattices in lattice packing problems.

Findings

Hexagonal lattice has the highest density among 2D lattices.

The problem can be restricted to well-rounded lattices for simplicity.

The density function simplifies on well-rounded lattices.

Abstract

In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.