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

##### December 20, 2010 By Phillip F. Schewe, Inside Science News Service

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.

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.

Explore further: Densest dice packing and computing with molecules

## Related Stories

#### Densest dice packing and computing with molecules

May 3, 2010

Physicists have made two new breakthroughs -- discovering a shape that packs more efficiently than any other known, and completing a high speed quantum calculation thousands of times faster than possible with conventional ...

#### Physicists show way to count sweets in a jar -- from inside the jar

July 29, 2009

(PhysOrg.com) -- How many sweets fit into a jar? This question depends on the shapes and sizes of the sweets, the size of the jar, and how it is filled. Surprisingly, this ancient question remains unanswered because of the ...

#### Entropy alone creates complex crystals from simple shapes, study shows

December 9, 2009

(PhysOrg.com) -- In a study that elevates the role of entropy in creating order, research led by the University of Michigan shows that certain pyramid shapes can spontaneously organize into complex quasicrystals.

#### NYU physicists make room for oddballs

August 3, 2009

(PhysOrg.com) -- Here's a question. How many gumballs of different sizes can fit in one of those containers at the mall so as to reward a well-spent quarter? It's hard to believe that most people never consider it even when ...

#### Mathematicians set world record in packing puzzle

August 12, 2009

(PhysOrg.com) -- Finding the best way to pack the greatest quantity of a specifically shaped object into a confined space may sound simple, yet it consistently has led to deep mathematical concepts and practical applications, ...

#### DNA can act like Velcro for nanoparticles

November 18, 2010

DNA can do more than direct how bodies our made -- it can also direct the composition of many kinds of materials, according to a new study from the U.S. Department of Energy’s Argonne National Laboratory.

## Recommended for you

#### Using a laser to wirelessly charge a smartphone safely across a room

February 20, 2018

Although mobile devices such as tablets and smartphones let us communicate, work and access information wirelessly, their batteries must still be charged by plugging them in to an outlet. But engineers at the University of ...

#### Reaching new heights in laser-accelerated ion energy

February 20, 2018

A laser-driven ion acceleration scheme, developed in research led at the University of Strathclyde, could lead to compact ion sources for established and innovative applications in science, medicine and industry.

#### Atomic structure of ultrasound material not what anyone expected

February 20, 2018

Lead magnesium niobate (PMN) is a prototypical "relaxor" material, used in a wide variety of applications, from ultrasound to sonar. Researchers have now used state-of-the-art microscopy techniques to see exactly how atoms ...

#### MEMS chips get metatlenses

February 20, 2018

Lens technologies have advanced across all scales, from digital cameras and high bandwidth in fiber optics to the LIGO lab instruments. Now, a new lens technology that could be produced using standard computer-chip technology ...

#### Using organoids to understand how the brain wrinkles

February 20, 2018

A team of researchers working at the Weizmann Institute of Science has found that organoids can be used to better understand how the human brain wrinkles as it develops. In their paper published in the journal Nature Physics, ...

#### Computers aid discovery of new, inexpensive material to make LEDs with high color quality

February 19, 2018

A team led by engineers at the University of California San Diego has used data mining and computational tools to discover a new phosphor material for white LEDs that is inexpensive and easy to make. Researchers built prototype ...

##### 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