Mathematicians have long known that it’s possible to pack at most 12 identical spheres around and touching a 13th.

In 1943, Hungarian mathematician L. Fejes Tóth conjectured that the optimal arrangement of the surrounding spheres is a highly symmetric pattern based on the 12-faced geometric shape known as a dodecahedron. Sean T. McLaughlin of the University of Michigan in Ann Arbor recently proved this conjecture and described his approach at last month’s Joint Mathematics Meetings in Washington, D.C.

Like an earlier proof of Kepler’s sphere-packing conjecture by Michigan mathematician Thomas C. Hales (SN: 8/15/98, p. 103: https://www.sciencenews.org/sn_arc98/8_15_98/fob7.htm), McLaughlin’s effort required extensive computer calculations involving more than 2,000 possible types of arrangements.