Sphere packing problem solved in 8 and 24 dimensions

Optimal stacking of oranges

In the 17th century, Johannes Kepler conjectured that the most space-efficient way to pack spheres is to arrange them in the usual way that we see oranges stacked in the grocery store. However, this conjecture stubbornly resisted proof until 1998, when University of Pittsburgh mathematician Thomas Hales, assisted by Samuel Ferguson (son of mathematician-sculptor Helaman Ferguson), completed a 250-page proof, supplemented by 3 Gbyte of computer output.

However, some mathematicians were not satisfied with Hales’ proof, as it relied so heavily on computation. So Hales embarked on project Flyspeck, which was to construct a completely detailed, machine-checkable formal proof of the conjecture. Finally, on 10 August 2014, Hales announced that the project was complete.

Hales’ proof of the Kepler conjecture was limited to three dimensions. Even prior to Hales’ original proof, mathematicians had explored generalizations of Kepler’s conjecture in higher dimensions.

So the mathematical community reacted with considerable interest when, in March 2016, Ukrainian mathematician Maryna Viazovska posted a solution, with proof, of the sphere packing problem in dimension 8, and, just a week later, also in dimension 24.

Based on computational analysis, mathematicians for some time had suspected that the optimal packing in dimension 8 was the E8 lattice, one of the fundamental objects in Lie algebras and groups, and that the optimal packing in dimension 24 was the Leech lattice. But until Viazovska’s papers, there was no proof.

The E8 lattice has been explored by researchers for many reasons. It also has connections to string theory in physics. The Leech lattice has been employed in coding theory, because it is capable of detecting up to four errors in a 24-bit word and correcting up to three errors.

Viazovska’s proof for dimension 8 employed the theory of modular forms, which are complex analytic functions satisfying a certain type of functional equation with respect to the actions of the modular group. Her proof was praised by Princeton mathematician Peter Sarnak, who said, “It’s stunningly simple, as all great things are. … You just start reading the paper and you know this is correct.”

Just one week after her original paper was posted on arxiv.org, she posted a second paper, co-authored with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko, that employed a similar approach to solve the dimension 24 case.

Additional background and details on Viazovska’s work can be read in a nice article by Erica Klarreich in Quanta Magazine, on which part of the above post was based. Viazovska’s technical paper for dimension 8 is available here, and her paper for dimension 24, co-authored with the four other mathematicians, is available here.