If you want to efficiently pack squares in two dimensions, you arrange them like a checkerboard. To squeeze together three-dimensional cubes, you stac

To Pack Spheres Tightly, Mathematicians Throw Them at Random

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2024-04-30 16:00:07

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.