New sphere-packing record stems from an unexpected source

Summary

The advantage of Rogers’ method was that you didn’t have to start with a particularly efficient lattice to get an efficient sphere packing. You just had to choose the right ellipsoid. But this introduced a new complication. Unlike a sphere, which is completely defined by a single number — its radius — an ellipsoid is defined by several axes of different lengths. The higher the dimension, the greater the number of directions you can stretch your ellipsoid in, and the more options you have for what your starting ellipsoid will look like. “In higher dimensions, you have no idea how to grow it. You have too much freedom,” Klartag said. Mathematicians ultimately returned to Minkowski’s approach, choosing to focus on finding the right lattices. They became more specialized in lattice theory and moved away from Rogers’ focus on geometry. This strategy led to improvements in high-dimensional sphere packing. But for the most part, they only improved on Rogers’ packing by a relatively small margin. Mathematicians still hoped for a bigger leap. For decades, they didn’t get it. It would take an outsider to end the stagnation. An Outside Perspective Klartag, a mathematician at the Weizmann Institute of Science, was always intrigued by lattices and sphere packing. He just never had the time to learn much about them. He works in geometry, not lattice theory, and he usually studies convex shapes — shapes that don’t jut inward. These shapes involve all sorts of symmetries, particularly in high dimensions. Klartag is convinced that this makes them extremely powerful: Convex shapes, he argues, are underappreciated mathematical tools. Boaz Klartag long suspected that methods from the field of convex geometry could be useful for sphere-packing problems. He just never had the time to test out his hunch — until now. Then last November, after completing a major project in his usual area of study, he noticed his calendar was uncharacteristically clear. “I thought, I’m 47 years old, all my l...