How can identical particles be crammed together as densely as possible? A combination of theory and computer simulations shows how the answer to this intricate problem depends on the shape of the particles.
We all know from experience with luggage just how difficult it is to pack objects efficiently into a limited space. These difficulties are even greater when huge numbers of objects, such as grains of wheat, are involved. From Luke the evangelist's reference to “a good measure, pressed down, shaken together and running over” all the way to modern disclaimers that contents may have settled during shipping, nobody has been able to analyse how to achieve the densest possible packings. On page 876 of this issue, Torquato and Jiao describe1 remarkable computer-simulation results that show how subtle this problem can be, while offering new hope for understanding important cases.
Identical, perfect spheres are among the simplest objects to pack. It is not difficult to guess how — just look at the way cannonballs are stacked at war memorials. But theoretical analysis of the problem presents profound difficulties that were overcome only recently by Thomas Hales2, nearly four centuries after the answer was conjectured by Johannes Kepler3. Of course, most granular materials do not consist of spherical grains, and this complicates matters tremendously. For most grain shapes we cannot guess or even closely approximate the answer, let alone prove it, and it is difficult to develop even a qualitative understanding of the effects of grain shape on packing density.
Instead of perfect spheres, Torquato and Jiao study1 packings of the five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron (see Fig. 1 on page 877). These are the simplest and most symmetrical polyhedra. Needless to say, nobody expects the grains in physical materials to have these precise shapes, but they are a beautiful test case for understanding the effects of corners and edges and the role of symmetry.
The cube-packing problem is easy — cubes can fill space completely. But the densest packings of the other Platonic solids are much less obvious. They do not tile space — fill space with no gaps or overlaps — despite Aristotle's incorrect assertion4 that tetrahedra do. In their simulations, Torquato and Jiao find a striking difference between two cases: the octahedron, dodecahedron and icosahedron have central symmetry (that is, each remains unchanged by a 180° rotation about a point at its centre). But the tetrahedron does not have central symmetry, and this turns out to be the crucial distinction.
In the centrally symmetric cases, Torquato and Jiao show that the highest-density packings belong to the simplest kind, called Bravais lattice packings (Fig. 1a), although this constraint is never directly imposed on their simulations. In such arrangements, all the particles are perfectly aligned with each other, and the packing is made up of lattice cells that each contain only one particle. The densest Bravais lattice packings had been determined previously5,6, but it had seemed implausible that they were truly the densest packings, as Torquato and Jiao's simulations and theoretical analysis now suggest. By contrast, for the tetrahedron it has long been known that Bravais lattice packings are far from optimal, and in this case the authors achieve a record density: they find a non-Bravais lattice configuration (Fig. 1b) that fills up 78.20% of the space available (an improvement over the previous record7 of 77.86%, or 36.73% for Bravais lattices8).
To find their packings, Torquato and Jiao use a powerful simulation technique. Starting with an initial guess at a dense packing, they gradually modify it in an attempt to increase its density. In addition to trying to rotate or move individual particles, they also perform random collective particle motions by means of deformation and compression or expansion of the lattice's fundamental cell. With time, the simulation becomes increasingly biased towards compression rather than expansion. Allowing the possibility of expansion means that the particles are initially given considerable freedom to explore different possible arrangements, but are eventually squeezed together into a dense packing.
The new tetrahedron packing is a variant of an ingenious construction found by Chen last year7. Using physical models, she observed that tetrahedra pack well when arranged in a form Torquato and Jiao call 'wagon wheels': wheels of five tetrahedra sharing an edge, with the wheels joined in pairs at right angles (see Fig. 3a on page 878). How best to arrange these pairs of wagon wheels is not clear, but Chen used a computer algebra system (a software program that manipulates mathematical formulae) to optimize their placement and achieved a density of 77.86%, which is a vast improvement over the previous record9 of 71.75%. The authors' simulations1 suggest that Chen's solution was nearly, but not quite, optimal.
Although Torquato and Jiao's improvement on Chen's packing is noteworthy, perhaps the most intriguing implication of their work is the apparent optimality of Bravais lattice packings for the centrally symmetric Platonic solids (as well as generalizations such as Archimedean polyhedra). This part of the work may seem less exciting, because the densest packings turned out to be known already. However, in a field with few clear organizing principles, this latest insight into the part played by symmetry might take on the role of a twenty-first-century Kepler conjecture. It will surely inspire many future research papers, and with any luck we won't have to wait 400 years for a full understanding of it.
Torquato, S. & Jiao, Y. Nature 460, 876–879 (2009).
Hales, T. C. Ann. Math. 162, 1065–1185 (2005).
Kepler, J. Strena Seu de Nive Sexangula [A New Year's Gift of Hexagonal Snow] (Godfrey Tampach, 1611).
Aristotle On the Heavens Book III, Pt 8 (transl. Guthrie, W. K. C.) (Harvard Univ. Press, 1939).
Minkowski, H. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II, 311–355 (1904).
Betke, U. & Henk, M. Comput. Geom. 16, 157–186 (2000).
Chen, E. R. Discrete Comput. Geom. 40, 214–240 (2008).
Hoylman, D. J. Bull. Am. Math. Soc. 76, 135–137 (1970).
Conway, J. H. & Torquato, S. Proc. Natl Acad. Sci. USA 103, 10612–10617 (2006).
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Physical Review E (2010)