Packing it all in for the holidays: Scientists see how many polyhedrons can fit into a box

Dec 20, 2010 By Phillip F. Schewe
Green laser (top right) being sent into a stack of shiny Christmas ornaments. Credit: Courtesy Troy Shinbrot

Whether it is Christmas shoppers mailing last minute gifts, or a steelyard dispatching ball bearings to factories, everyone wants to maximize the number of objects that fit into a container. Naturally the packing fraction -- the fraction of the container actually filled -- depends on the shape of the object.

Scientists conducting a new experiment hoped to find an optimal packing , but instead found that the basic sphere is still the best shape for filling a given volume.

Arshad Kudrolli, a physics professor at Clarke University in Worcester, Mass., is an expert on how things pack.

"Trying to pack one too many presents in the trunk of my car after shopping, or 'creating' a parking spot for my car during the holiday season are certainly times I think of packing efficiency," Kudrolli said.

Kudrolli thinks about packing at other times, too, and explains why it is important to know how particles are loaded into containers: "A large number of 'solid' products are manufactured by compacting powders into molds. A drug pill or sandstone may be familiar examples, but even iron bearings, carbide and ceramic parts are cast from powders of various kinds. The powders themselves are generally not spherical and the final manufactured solid composed of particulates is held together by friction or interlocking."

Spheres pack very well. If balls are carefully stacked, cannonball style, they can fill 74 percent of the available space in a container, such as a cylinder. If they're thrown in randomly, spheres will occupy about 64 percent. This was the number to beat. Actually, a 2003 experiment conducted at Princeton University in N.J. showed that by carefully modifying the shape of M&M's candies just a bit, a 74 percent packing fraction could be achieved.

X-ray picture of tetrahedron dice sitting in a container. Credit: Arshad Kudrolli

Scientists believe that in addition to shape, between objects can also the reduce packing fraction. The more sides an object has, it was thought, the lower the fraction. So Kudrolli and his Clarke University colleague Jessica Baker set off to measure the packing properties of five famous shapes studied since the time of the ancient Greeks, the so-called Platonic solids. These solids, named for the fourth century B.C. philosopher Plato, are those made of polygons having equally-long sides. The solids are the tetrahedron, a four-sided pyramid-shaped solid; the six-sided cube; the eight-sided polyhedron; the 12-sided dodecahedron; and the 20-sided icosohedron.

Using commercially available dice already in the Platonic shapes, Baker and Kudrolli loaded hundreds and thousands of centimeter-sized dice into a container and then noted what fraction of the container's volume was occupied by the shapes.

They took the precaution of loading the dice in the container in various ways. They dropped the dice randomly, they added dice but shook the container by hand to let the dice settle more efficiently, they used a machine to shake the container, and they added liquid in order to let the dice settle while floating before draining off the liquid.

The machine shaking produced the best packing fraction. But the two scientists found to their surprise that the packing fractions of cubes and tetrahedrons, 64 percent, were no better than for spheres. For randomly loaded dice -- and not using any shaking to settle them -- the Platonic solids pack even more poorly. The fraction for cubes was about 54 percent while the fractions for the other shapes were not much better than 50 percent. Spheres thrown haphazardly fill up about 59 percent of space.

Troy Shinbrot of Rutgers University in N.J. is a scientist who studies how granular materials like sand and powders flow. He maintains that the flow rate for objects is just as important as packing density. Loading and packing are both critical, for example, when putting grain into a silo or a ship's hold.

Shinbrot found Kudrolli's new paper to be "aesthetically very pretty," and praised Kudrolli's research on the flow of particles. Most simulations of granular flow of things like pharmaceutical powders assume that the particles are spherical. Kudrolli's work, said Shinbrot, has helped to reveal the consequences of the particles not being true . These are particularly important when flows get jammed up. "Jamming certainly depends on packing details," Shinbrot said. "So packed round particles can flow more or less smoothly, whereas packed pyramidal particles almost certainly do not. Likewise packed plate-like particles, such as coins or flat candies, slip nicely along the plane of the plates, but not in the direction perpendicular to the plane."

The Platonic packing research was published in the December 15 issue of the journal Physical Review E.

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lexington
5 / 5 (1) Dec 20, 2010
If my box is a cube, filling it with cubes is more space efficient than filling it with sphere. Suck it math.

More seriously why say that "the sphere is the best" and then halfway through the article explain that an oblate spheroid is actually better? Sloppy writing at its best.
Raveon
5 / 5 (1) Dec 20, 2010
The writer failed to mention that they are talking about random packing of identical shapes, just dumping the pieces into a container, not hand packing where a cube could fill a square box 100% and so would a couple other shapes. Just a minor detail to leave out.
El_Nose
not rated yet Dec 22, 2010
Its a minor detail until you understand that detail is the root of the question -- the question only has perspective if it is a random packing problem... The end result if a definite answer could be found would be a computer algorithm to pack containers, one does not exist at present because the problem is mathematically to hard to optimize a general solution that can be solved within a time frame of useful significance. Basically it takes to long for a computer to solve the problem... this paper is more evidense that it may not be necessary to come up with a complete solution just a practical one