The problem of packing solid objects as closely together as possible has long fascinated mathematicians. In 1611, German mathematician and astronomer Johannes Kepler proposed that the densest packing for spheres was in a lattice — as a grocer stacks a pyramid of oranges. Salvatore Torquato, a theoretical physicist at Princeton University in New Jersey, and his graduate student Yang Jiao now make an analogous conjecture for the packing of all but two of the classic Platonic and Archimedean solids as compactly as possible.

The five Platonic solids have identical faces of regular polygons: an example is the tetrahedron, with its four triangular faces and four rather sharp vertices. The thirteen Archimedean solids are polyhedra with two or more types of polygon as faces: an example is the cuboctahedron, with eight triangular faces and six square ones.

Torquato was initially interested in the properties of randomly packed spheres — such as grains of sand on a beach. But he soon moved on to tetrahedra. In 2006, he and Princeton mathematician John Conway constructed a packing, which was anything but orderly, of congruent tetrahedra in three-dimensional space with a density (the fraction of the total volume taken up by the solids) of 72% (J. H. Conway and S. Torquato Proc. Natl Acad. Sci. USA 103, 10612–10617; 2006). “At the time, it was the world record for tetrahedra,” Torquato says.

But the record was short-lived. Two years later, Elizabeth Chen at the University of Michigan, Ann Arbor, showed that it was possible to pack tetrahedra at 77.8% density. “We concocted these things in our minds using mathematical analysis. But Chen actually used physical models of tetrahedra, which is extremely useful because these are complicated objects,” Torquato says.

So he decided to use computer simulation to test many more ways of packing tetrahedra. The result was a new record: 78.2%.

Once he had the computer program working, Torquato tested it on other polyhedra. “What we found was really surprising,” he says. “We found that all of the other Platonics, which have central symmetry, like to pack in an ordered lattice arrangement as do spheres.”

The theory held true for most Archimedean solids as well: those with central symmetry could be most densely packed in a lattice pattern. But the ones without central symmetry, such as the truncated tetrahedron — with 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges — required a less orderly arrangement (see page 876).

“It became very clear that the central symmetry of the object really played a fundamental role in the arrangement that you need to get a dense packing,” explains Torquato.

Having shown by empirical evidence and mathematical analysis that an ordered lattice is the densest packing for solids with central symmetry, Torquato says the conjecture now awaits mathematical proof.

Nearly 400 years elapsed between Kepler's conjecture and its mathematical proof. This one is more challenging, Torquato says, “because these objects have sharp corners. They're not as nice as spheres.” Will it take 400 more years? Torquato laughs and says, “We're providing work for future generations.”