Mathematicians like to generalize concepts into higher dimensions. Sometimes this is easy.

If you want to efficiently pack squares in two dimensions, you arrange them like a checkerboard. To squeeze together three-dimensional cubes, you stack them like moving boxes. Mathematicians can easily extend these arrangements, packing cubes in higher-dimensional space to perfectly fill it.

Packing spheres is much harder. Mathematicians know how to pack circles or soccer balls together in a way that minimizes the empty space between them. But in four or more dimensions, the most efficient packing scheme is a complete mystery. (With the exception of dimensions 8 and 24, which were solved in 2016.)

“It sounds so simple,” said Julian Sahasrabudhe, a mathematician at the University of Cambridge. “There could be 20 different ways of approaching it. And that seems to be what’s happened — there’s lots of different ideas.”

The known optimal sphere packings in 2, 3, 8 and 24 dimensions look like lattices, full of patterns and symmetry. But in every other dimension, the best packings might be totally chaotic.

“That, I think, is a very tantalizing aspect of it. It’s really very open,” said Akshay Venkatesh, a mathematician at the Institute for Advanced Study. “We just do not know.”

Last December, Sahasrabudhe, together with his Cambridge colleague Marcelo Campos, Matthew Jenssen of King’s College London and Marcus Michelen of the University of Illinois, Chicago, provided a new recipe for how to densely pack spheres in all arbitrarily high dimensions. It’s the first significant advance on the general sphere-packing problem in 75 years.

“It’s a beautiful piece of mathematics,” said Yufei Zhao, a mathematician at the Massachusetts Institute of Technology. “There are new, groundbreaking ideas.”

Improving the Baseline

The tightest way to arrange circles on a two-dimensional plane is in a hexagonal pattern, with circles placed at the corners and center of each hexagon. Such a grid fills a bit over 90% of the plane.

In 1611, the physicist Johannes Kepler thought about the best way to pack three-dimensional spheres.  For the base layer, he packed the spheres in a hexagonal arrangement, like the circles.

He then placed a second layer of spheres over the first, filling the gaps. But then there’s a choice to make. The third layer can go directly above the first layer:

Or it can be offset:

In both cases, the pattern then repeats. And in both cases, the spheres fill the exact same amount of space: about 74%.

In 1831, Carl Friedrich Gauss, one of the most prominent mathematicians of the 19th century, showed that Kepler’s configurations are the best possible lattices — repeating gridlike configurations — but he wasn’t able to rule out the possibility that some irregular arrangement could do better. (It was finally ruled out around the turn of the millenium.)

In higher dimensions, mathematicians were at a loss. Then, in 2016, Maryna Viazovska used the existence of symmetries particular to eight-dimensional space to prove that particular lattices are optimal. She also worked with collaborators to extend the proof to 24 dimensions. She earned the 2022 Fields Medal, the highest prize in mathematics, for this work.

“What I love about [sphere packing] is the way it’s a thread connecting lots of different areas in mathematics, in computer science and in physics,” said Henry Cohn, a mathematician at Microsoft Research who worked with Viazovska on the 24-dimensional proof.

These known optimal packings — in one, two, three, eight and 24 dimensions — don’t seem to generalize to higher dimensions. In higher dimensions, mathematicians don’t know what percentage of space an optimal arrangement would fill. Instead, they try to approximate it.

In any dimension, if you start with a very large box and successively fill it with balls — sticking one anywhere you find a large enough opening — then the spheres will occupy at least $latex frac{1}{2^d}$ of the box, where d is the dimension of the space. So in two dimensions, they will fill at least 1/4 of the space, and in three dimensions, they will fill at least 1/8 of the space, and so forth. In relatively small dimensions, mathematicians often know of specific packings that do much better than this general bound. (For example, Kepler’s three-dimensional packing fills 74% of the space — significantly more than the minimal 12.5%.) But the $latex frac{1}{2^d}$ baseline is useful because it applies to all dimensions.