putting the spheres back into tetrahedra

This post has its roots in a discussion during the coffee-time, inspired by an almost empty jar of condensed milk. It seemed obvious that spherical jars do not pack as good as cylindrical jars. But how good do they pack? Some scribbling ensued.

Realising that solving the problem from the first principles might take much more than quarter an hour, I looked up the problem of close-packing of spheres, seemingly one of easier packing problems, in Wikipedia. I thought it was solved ages ago. I was wrong (or only partially right, if we are optimists.)

Sir Walter Raleigh posed so-called Cannonball Problem around 1587 to the English polymath Thomas Harriot, his scientific adviser on the expedition to the New World. Later, the correspondence of Harriot with Johannes Kepler influenced the Kepler conjecture (1611), which says that no arrangement of equally sized spheres filling space has a greater average density than that of the face-centered cubic (fcc) packing and hexagonal close packing (hcp) arrangements. Gauss proved in 1831 that the average density of close-packed spheres is π/√18 ≃ 74%. (Compare that with circle packing — or cylinder packing — density of π/√12 ≃ 90.7%.)

In 1998 Thomas Hales announced that the proof by exhaustion of Kepler conjecture was complete. It took four years for a twelve-strong panel of referees to agree with 99% certainty that Hales’s proof is correct. (How did they do that?) However a complete formal proof is still to be produced — Hales’s Flyspeck Project is said to last at least twenty more years.

What is the best container for close-packed spheres? At first, I naïvely thought that, if one can stack spheres as tetrahedral pyramids, maybe tetrahedral boxes? Provided, of course, that one knows what to do with tetrahedral boxes. Another discovery awaited me. Aristotle wrote in 350 BC in his treatise On the Heavens:

It is agreed that there are only three plane figures which can fill a space, the triangle, the square, and the hexagon, and only two solids, the pyramid and the cube.

It is somewhat reassuring that until recent I shared with great Aristotle the belief that one can fill space with regular tetrahedra. On the other hand, it is not. It was known at least since fifteenth century that these polyhedra do not tile space. Interestingly, the optimal packing for regular tetrahedra remains to be found. The best known packing of 85.63% was achieved as recently as 2010. Of course it is much better than 74% density of spheres, but maybe there is still room for improvement.

Even so, as this diagram shows, one can quite nicely stack the balls in more convenient cubic boxes, which do tile space for sure.

What, if any, is the moral of the story? Not every statement starting with “It is agreed that...” is true. The old problem does not mean it is a solved problem. You might as well have a go at it. Or, at least, read in Wikipedia about it.

Update (25.05.2023): The Flyspeck Project was completed in 2014 and a formal proof of the Kepler conjecture was published in 2015.