December 10, 2009
Breaking The Tetrahedra Packing Record
Two Kent State University professors are part of a team of researchers who recently uncovered a way to pack tetrahedra, considered to be the simplest shaped regular solids with its four triangular sides, more densely than ever before. Peter Palffy-Muhoray, professor of chemical physics and associate director of the Liquid Crystal Institute at Kent State, and Xiaoyu Zheng, assistant professor in Kent State's Department of Mathematical Sciences, along with four colleagues at the University of Michigan and one at Case Western Reserve University, have broken a world record for packing the most tetrahedra into a given volume.
Their findings will be featured in the Dec. 10 issue of Nature, one of the leading international scientific journals, in an article co-authored by the seven researchers. The article is titled "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra."The researchers were able to obtain the highest packing fraction of 85.03, meaning tetrahedra fill 85.03 percent of the volume of the container. This shattered the previous record of 78.2 percent set by two Princeton University researchers in August 2009.
"The question of how best to pack shapes into a volume is an age-old question," Palffy-Muhoray said. "Johannes Kepler asked how to pack spheres in the early 1600s, and it was only recently proven in 2005 that the best way is to stack them like cannonballs. It is easy to understand how cubes can entirely fill space with no voids, but the packing problem is still unsolved for the simple tetrahedron. Though it's a simple object, it can't fill space like cubes, so we wondered how hard tetrahedra would pack when you squeezed them together."
In the process, the Kent State professors and their colleagues discovered that quasicrystals formed when the tetrahedra were compressed.
"A crystal is a material structure which repeats periodically," Palffy-Muhoray explained. "A quasicrystal is similar, but it doesn't repeat itself exactly, despite its regularity. It's something quite new, having been discovered only 25 years ago. Not only did we show that tetrahedra can pack much denser than previously thought, but we also found the most remarkable result: that they form quasicrystals. It's amazing that the simplest solid "“ the tetrahedron "“ forms these intricately complicated, amazingly complex structures. This is the first example of particles forming quasicrystals with no interactions other than hard objects bumping into one another. Entropy, which is often associated with disorder and chaos, can in fact create order."
Sharon Glotzer, a professor in the University of Michigan's departments of Chemical Engineering and Materials Science and Engineering who conceived and designed the study together with Palffy-Muhoray, said, "This is the most complex structure we've ever seen arising from purely entropic interactions."
Previous approaches by other researchers began with the geometric constructions and compressing these. Glotzer, Palffy-Muhoray and collaborators started from a random initial structure and compressed it to allow natural evolution toward high-density states. Work began in 2006 with physical experiments being conducted at Kent State involving Palffy-Muhoray and Zheng.
"We carried out experiments here at Kent State using tetrahedral dice to see how densely we could pack them," Zheng said. "We also constructed various motifs. Then, the key computer simulations were carried out at the University of Michigan. By using a new, ingenious scheme, we were able to achieve 85.03 percent packing. When we first started this work, we did not expect the results that we achieved. This whole project has been very exciting."
The results of this research and the formation of quasicrystals offer some interesting and exciting possibilities on how it can be used in real-world applications. "This will enable the production of metamaterials, which are manmade materials that don't exist in nature, with interesting physical and optical properties," Palffy-Muhoray said. "Applications are far-ranging, including high-resolution imaging useful for microscopy in medicine and materials science. This new packing method could enable the production of new kinds of materials, useful for computer chips, building materials and fabrics."
Palffy-Muhoray, whose distinguished career includes being elected a Fellow of the American Physical Society in 2008, said this research has been very rewarding. "In trying to understand simple things, you occasionally stumble upon beautiful, complex phenomena," he said. "This does stand out as one of the most beautiful things we've discovered. It's very satisfying that this simple inquiry would lead to such remarkable results."
Funding support for this research was provided by an Air Force MURI (Multidisciplinary University Research Initiative) grant to Kent State University and by the National Science Foundation.
Image Caption: Peter Palffy-Muhoray and Xiaoyu Zheng, both professors at Kent State University, are part of a team of researchers who recently uncovered a way to pack tetrahedra more densely than ever before. Experiments and computer simulations, like the one shown here, helped the team to obtain the highest packing fraction of 85.03 and discover the formation of quasicrystals when the tetrahedra were compressed.
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