Take a large number of identical coins and place them on a table. How to organize them in such a way that they fit as many as possible? Testing, it is easy to convince yourself that the best is the hexagonal arrangement, in which each coin touches 6 neighbors. Bees discovered this millions of years ago, and they use it to build combs with as much honey as they can store in the hive.
The hexagonal arrangement occupies 90% of the table area. But proving you can’t get more isn’t easy. Lagrange proved in 1773 that the hexagonal configuration is the best of all regular arrangements. But it was not until 1942 that the Hungarian László Tóth managed to extend the test to any (“messy”) arrangements.
The question of packing balls is similar: how to store identical balls in a container in such a way that they fit as many as possible? In 1611, the astronomer Johann Kepler pointed out that the hexagonal layered arrangement, as marketers display fruit in their stalls, occupies 74% of the volume, and conjectured that this would be the maximum possible. In 1831, Gauss proved Kepler’s conjecture for regular arrangements, but the extension to any arrangements took almost 400 years.
The first proof was given by the American Thomas Hales in 1998, but the work was very long (250 pages!) and contained a huge amount of calculations that no one could check. The controversy was not resolved until 2017, when Hales wrote and ran an algorithm to automatically verify the evidence by computer.
In addition to dimensions 2 (coins) and 3 (balls), mathematicians also study the packing of spheres in higher dimensions. And it’s not just out of curiosity, there are practical applications as well. For example, in information theory, the study of error-correcting codes —which allow more robust communication— leads to problems of packing spheres in very high dimensions.
The problem is that very little was known about dimensions greater than 3. At least until 2016, when young Ukrainian mathematician Maryna Viasovska found the most effective arrangement for spheres in dimension 8 (25% occupancy of the hypervolume). Soon afterwards, with collaborators, Viasovska extended his method to dimension 24! For these works, she has just become the second woman in history to win the Fields Medal, the most prized award in mathematics.
For the other dimensions, the problem remains unresolved. For a while..
