Kepler’s Year

As some readers may already know, 2009 is the International Year of Astronomy, commemorating 400 years since Galileo raised his telescope to the stars and Kepler put forward his laws of planetary motion in his treatise Astronomia Nova. This put the Copernican idea of heliocentrism on a more rigorous scientific footing, and would eventually pave the way for Newton to formulate his theory of gravity.

Even aside from these achievements, Kepler is particularly fascinating character in the history of science. His work always seemed to be strongly guided by a Platonic view of a perfectly structured, geometrically harmonic universe. For example, in his Mysterium Cosmographicum – a lesser-known precursor to Astronomia Nova – Kepler devised an elegant, if incorrect, model of the six known planets as following orbits along spheres inscribed within and circumscribed around five platonic solids. Another example is his extensive exploration of tesselations and sphere-packings, which he often used to explain the structure of various materials, such as snowflakes.

A well-known result of this is Kepler’s conjecture on sphere packings – namely, that the maximum possible volume fraction of a container that can be filled by equally-sized spheres is that of face-centered cubic packing, slightly more than 74%. This was only proved by Thomas Hales recently, using a good deal of computational machinery.

A related problem – part of Hilbert’s list of some of the most important problems in mathematics – involves finding the densest packings of regular polyhedra. While cubes and truncated octahedra can tile space (so their densest packing has a volume fraction of 100%), it turns out that none of the other five Platonic solids or thirteen Archimedean solids can. A good deal of recent work has focused on trying to find the densest possible packing of regular tetrahedra, mainly by construction. In two very nice recent papers, however, Sal Torquato and his student Yang Jiao have come up with a computational scheme for generating very, very dense packings of polyhedra, calculating the densest known packings of the Platonic and Archimedean solids. I won’t go into the details of the algorithm – the follow-up paper does a very nice job of explaining it. The basic idea is to start with a randomized ‘dilute’ configuration of the polyhedra in a box of some shape and either randomly move a randomly chosen polyhedron by a small amount, or deform the box by some amount, only allowing changes that increase the volume fraction while still preventing polyhedra from overlapping. One can imagine that iterating this many times would result in very, very dense packings — the hope, of course, is that these are the densest possible polyhedral packings. This remains to be proven.

The cool thing that Torquato and Jiao found is that for all the Platonic and Archimedean solids possessing central symmetry, the densest possible packings they found had volume fractions equal to the volume fractions of Bravais lattice packings of the same solids, to within a few hundredths of a percent. A nice Kepler-style argument using inscribed spheres gives upper bounds on the volume fractions of these densest possible packings – these are larger by only ~3-13%. Taken together, these results hint at a possible “Kepler Conjecture” for polyhedral packings: namely, that the densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings. Very cool — this suggests that quite complicated polyhedral packings might be able to be understood using some very simple rules.

On the other hand, Torquato and Jiao found that for the two polyhedra not possessing central symmetry — the tetrahedron and the truncated tetrahedron — the volume fractions of the densest lattice packings grossly underestimated the volume fractions of the densest packings they found using simulations. This leads to a converse conjecture: in particular, that the densest possible packing of any convex, congruent polyhedron without central symmetry is not a Bravais lattice packing — rather, it is significantly more complicated. (A nice side result is that the densest packing they found possesses no long-range order — more on this later.) It is not clear at this point what rules, if any, would dictate what the densest possible packing of these polyhedra look like.

One word of caution: as I mentioned before, because this approach is computational, none of these densest-known packings has been proved to be the densest-possible. Because the algorithm requires some choice of the starting ‘dilute’ configuration, it is possible that this choice will influence the final structure the algorithm settles at. In fact, Torquato and Jiao already found this to be the case in their search for the densest possible tetrahedral packing, as they note in their second paper — using a different initial condition, they found a densest packing ~4% more dense than the one they initially reported in the first paper.

While they don’t mention it, I think these results are particularly interesting in the context of amorphous systems, such as glasses. For example,why can a simple liquid metal be supercooled below its freezing point without crystallizing, potentially forming a glass? F. C. Frank put forward a very nice explanation for this. Considering the atoms of the liquid metal as spheres interacting via a non-directional Lennard-Jones potential, it turns out that the local energy density can be minimized by forming “locally-preferred” tetrahedral clusters. These then come together to form polytetrahedral Frank-Kasper phases, because forming these clusters requires less energy than forming crystalline clusters. The only problem with these phases is that they cannot tile space and are geometrically ‘frustrated‘ — the system is not in the crystalline state of lowest possible free energy, but is rather trapped a local free energy minimum in phase space. A significant amount of work has focused on trying to understand these kinds of structures, and connecting the geometric frustration inherent in these phases to the physical properties of supercooled liquids and glasses. It is not too surprising, then, that definitively finding the densest possible packing of tetrahedra (and analyzing the physical properties of such a structure) could help flesh out these connections — and Torquato and Jiao’s work seems to point the way.