Sunday, August 23, 2009

The Torquato-Jiao conjecture

Torquato-Jiao coverA recent issue of Nature (Aug. 13) includes a conjecture of fundamental importance in geometry. Torquato and Jiao conjecture that "the densest packing of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings." In plain language, this means that for most of these solids (those that remain unchanged when rotated about their central point, which is all but the tetrahedron) the most compact arrangement of in space is a regular repeat, where each particle (solid) is arranged in the same orientation. This is analogous to the situation with spheres, in which case Johannes Kepler proposed in 1611 that one could not do better than to pack them "in the same way that grocers stack oranges or melons." This conjecture took almost 400 years to be proved, and that story is the subject of a popular book, "Kepler's Conjecture," by George G. Szpiro.

What Torquato and Jiao did was to seek the densest packing for each of the solids by computational modeling of their positions in space. You can read the paper for details, but I think of it as virtual shaking of a virtual box to maximize "settling of contents." Their trick was to allow deformation of the fundamental cell shape. What they observed was that for all of those solids with central symmetry, the densest packing was a lattice packing. Their conjecture is that this is a general truth, one that can be proved, in the mathematical sense. I have no doubt about the power of their method. They achieved higher densities than were previously known for many of these solids, and it seems very unlikely that something that is generally true from their results and so intrinsically appealing could be wrong, but mathematical proof is something else entirely. Let me quote Szpiro on the difference between conjecture and proof (pg. 7):

"This is the best way to stack melons. In other words, it is the densest packing. Market vendors know it, you and I know it, and Harriot and Kepler knew it, but mathematicians refused to believe it. And it took 387 years to convince them of this fact." Torquato and Jiao have now proposed something quite elegant that nobody knew before. It is tempting to think that this conjecture is somehow connected to a deeper truth and will one day not only be proved, but will play a central role in a deeper theory of symmetry.
Five Platonic and 13 Archimedean solidsThis figure (which is a corrected version of Fig. 1 from the Torquato and Jiao paper) shows the five Platonic and 13 Archimedean solids.