Revisiting the hexagonal lattice: on optimal lattice circle packing

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 benet 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. The classical circle packing problem asks for an arrangement of nonoverlapping circles in R 2 so that the largest possible proportion of the space is covered by them. This problem has a long and fascinating history with its origins in the works of Albrecht Durer and Johannes Kepler. The answer to this is now known: the largest proportion of the real plane, about 90.7%, is covered by the arrangement of circles with centers at the points of the hexagonal lattice. The