20/07/2022

Viana discusses sphere packaging in a column in Folha.

Foto: Pexels

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Take a large number of identical coins and lay them flat on a table. How can you arrange them so that as many as possible fit in the same space? By testing, it's easy to see that the best arrangement is hexagonal, where each coin touches 6 neighbors. Bees discovered this millions of years ago, and they use it to build honeycombs that can hold as much honey as possible in the hive.

The hexagonal arrangement occupies 90% of the table's area. But proving that it's impossible to achieve more isn't easy. Lagrange proved in 1773 that the hexagonal configuration is the best among all regular arrangements. But it wasn't until 1942 that the Hungarian László Tóth managed to extend the proof to any ("messy") arrangements.

The question of packing spheres is similar: how to store identical balls in a container in such a way that the largest possible number fits? In 1611, the astronomer Johann Kepler pointed out that the hexagonal arrangement in layers, like how market vendors 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 extending it to any arrangement 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 verify. The controversy was only resolved in 2017, when Hales wrote and ran an algorithm to automatically verify the proof by computer.

Beyond dimensions 2 (coins) and 3 (balls), mathematicians also study sphere packing in higher dimensions. And it's not just out of curiosity; there are also practical applications. For example, in information theory, the study of error-correcting codes—which allow for more robust communication—leads to sphere packing problems in very high dimensions.

To read the full text, visit the newspaper's website.