Viewpoint The tetrahedral dice are cast . . . and pack densely

Tetrahedra are special among the platonic solids. They are the simplest polyhedra and the ones most unlike spheres. Surprisingly, much of our knowledge about the packing properties of tetrahedra is very recent: the past year has witnessed a sudden proliferation of novel, and often surprising, findings. Using Monte Carlo simulations, Haji-Akbari et al.[1] found that, upon compression, systems of hard tetrahedra spontaneously form a very dense quasicrystalline structure. Now, in a paper in Physical Review Letters, Alexander Jaoshvili, Massimo Porrati, and Paul Chaikin of New York University and Andria Esakia at Virginia Polytechnic Institute, both in the US, report their experiments on (almost) tetrahedral dice, which shed new light on the disordered structures that result when tetrahedra particles are poured into a (large) container [2]. Before discussing tetrahedral packing, it is useful to consider first the venerable (yet still not fully solved) problem of sphere packing. In 1611 Kepler proposed that the densest packing of spheres could be achieved by stacking close-packed planes of spheres. In such a packing, the spheres occupy π/ √ 18 ≈ 74.05% of space. The Kepler conjecture was (almost certainly) proven in 1998 by Thomas Hales. However, that does not mean that we know all there is to know about sphere packings: in addition to regular packing, spheres (and, in fact, most hard particles) also exhibit a much less understood packing, namely, random close packing (RCP). The quantitative study of random close packing started with J. D. Bernal’s experiments on the packing of ball bearings [3]. His experiments (and those of many others) suggested that it is impossible to compress disordered sphere packings beyond a volume fraction of approximately 64%. However, this observation does not necessarily imply that there exists a well-defined density of random close packing. It could just as well be that the rate at which the disordered hard-sphere packings can be compacted becomes very small around a volume fraction 64%—small, but not zero. If that were the case, RCP would not have a clear mechanical definition (that is, pouring and shaking may not lead to a well-defined RCP state). Indeed, in 2000, Torquato, Truskett, and Debenedetti [4] argued on the basis of computer simulations that states with a density above 64% can always be obtained by increasing the local order in a “random” sphere packing. This observation implies that the “mechanical” route to random close packing may be ill defined.