After just a few months of work, a complete newcomer to the world of sphere packing has solved one of its biggest open problems.

In math, the search for optimal patterns never ends. The sphere-packing problem — which asks how to cram balls into a (high-dimensional) box as efficiently as possible — is no exception. It has enticed mathematicians for centuries and has important applications in cryptography, long-distance communication and more.

It’s deceptively difficult. In the early 17th century, the physicist Johannes Kepler showed that by stacking three-dimensional spheres the way you would oranges in a grocery store, you can fill about 74% of space. He conjectured that this was the best possible arrangement. But it would take mathematicians nearly 400 years to prove it.

In higher dimensions, mathematicians still don’t know the answer. (With the strange exceptions of dimensions 8 and 24.) Over the years, they’ve come up with better packings. But these improvements have been small and relatively rare.

Now, in a short manuscript posted online in April, the mathematician Boaz Klartag has bested these previous records by a significant margin. Some researchers even believe his result might be close to optimal.

A newcomer to this area of study, Klartag achieved his packing method — which works in all arbitrarily high dimensions — by resuscitating an old technique that experts had abandoned decades earlier. The work taps into several long-running debates about the nature of optimal packings in high dimensions. Should they be ordered or disordered? And how snug can they possibly get?

“This is really an amazing breakthrough,” said Gil Kalai, a mathematician at the Hebrew University of Jerusalem. “It’s something that’s excited mathematicians for nearly 100 years.”

...read more at quantamagazine.org