Thue's theorem

Remarks

For the hexagonal packing with radius $1$, the centres form a lattice whose fundamental parallelogram has area $2\sqrt 3$. Each cell carries one disc of area $\pi$, and this gives the density. For the upper bound, one standard proof joins the centres of a saturated packing into a Delaunay triangulation. Each triangle has sides at least $2$. One shows that the discs cover at most the fraction $\pi/\sqrt{12}$ of each triangle, with equality for the equilateral triangle of side $2$. Lagrange proved the bound for lattice packings in 1773. Axel Thue gave proofs for general packings in 1890 and 1910, and László Fejes Tóth gave the first proof regarded as complete in 1940.

Going deeper

For a student

Thue's theorem is the two-dimensional case of the sphere packing problem; its three-dimensional case is the Kepler conjecture. The local argument works in the plane because the densest local arrangement, six circles around one, fits together into a global tiling. In three dimensions this fails, which is one reason why the Kepler conjecture is so much harder.