In a remarkable new paper, Maryna Viazovska has put forth a proof of a most efficient way to pack unit spheres in dimension 8. The only two cases known before were dimensions 2 and 3 as in Figure 1. Dimension 8 is an especially interesting and easy case, because there is a very symmetric, very efficient way of packing the spheres together, so good that it makes it much easier to prove that you cannot do any better. Even so, the proof required new insight to identify a very symmetric "modular" function. The paper has not yet been published or refereed, but it looks right.
Figure 1. The most efficient ways to pack unit spheres in dimensions 2 and 3, proved by A. Thue over a hundred years ago and by Thomas Hales just ten years ago. Images from Morgan and Wikipedia.
The problem is closely related to finding the richest sets of 8-letter code words that are far enough from each other so as not to be confused by transmission errors, just as the centers of the spheres are required to be distance at least 2 from each other so that the spheres do not overlap.
Maryna Viazovska is at the Berlin Mathematical School and Humbolt University of Berlin.