When identical charges repel each other but are confined to the surface of a sphere, they seek the arrangement with the lowest possible energy. The idea is simple, but the problem gets harder to solve with every charge added. The so-called Thomson problem dates back to the “plum-pudding” model of the atom, which Nobel laureate Joseph J. Thomson proposed in 1904. The atomic model has long been superseded, but the mathematical problem has endured.

In 1998, mathematician Stephen Smale listed it as Problem No. 7 among his 18 major mathematical challenges for the 21st century. Exact solutions still exist only for a few small systems. The question also matters well beyond mathematics—for example, for the structure of viruses, fullerenes and colloidal shells.