Sphere Packing Problem: From Kepler’s Conjecture to Hales’ Proof

How a Wasp’s Nest Reveals the Mathematics of Sphere Packing

Right above my head, wasps were building this home out of neat hexagons.

If you place circles at the vertices of those hexagons and add one more circle in the center, you get the densest possible packing of circles in a plane. Lagrange proved this back in 1773 — more precisely, he proved it was the densest among lattice packings; the proof for all packings was given by László Fejes Tóth only in 1940. The circles cover nearly 90% of the plane.

Kepler’s Conjecture and the 3D Sphere Packing Problem

If instead you place three-dimensional spheres at the vertices of those hexagons, you get the densest possible packing in three-dimensional space. This conjecture was formulated by Kepler as far back as 1611. The packing occupies 74% of space — the centers of spheres in each new layer sit in the gaps between the spheres of the layer below.

Proving it took nearly four hundred years. Only in 2017, after two decades of refinement by Thomas Hales, was a complete proof officially accepted by the mathematical community.

Solved Dimensions and Open Questions in Sphere Packing

To date, mathematicians have proven the optimal sphere packing density in one, two, three, eight, and twenty-four dimensions. For all other dimensions, the question remains open.

But wasps and bees have known something about this for a very long time. The only question is — who do you ask about the five-dimensional case?