A3 lattice

The A₃ Dynkin diagram:

gives the densest way to pack spheres in 3 dimensions, with each touching 12 others:

Conjectured by Kepler in 1611, this was proved in 1998 by Thomas Hales.

For more on Hales' proof, which required massive amounts of computation and was later made completely formal using computerized proof assistant, see the Wikipedia article Kepler conjecture.

I created the Dynkin diagram here, and I hereby release it into the public domain. Greg A L drew the picture of stacked spheres and put it on Wikicommons under a Creative Commons Attribution-Share Alike 3.0 Unported license.