More than 2,300 years ago, Aristotle was wrong.
Now, in the past year, a flurry of academic activity is suddenly zooming in on an answer to a problem akin to wondering how many people can fit into a Volkswagen Beetle or a phone booth. Except here mathematicians have been thinking not about the packing of people, but of geometric solids known as tetrahedrons.
“It’s pretty remarkable how many papers have been written on this in the past year,” said Henry Cohn, a mathematician at Microsoft Research New England.
A tetrahedron is a simple construct — four sides, each a triangle. For the packing problem, researchers are looking at so-called regular tetrahedrons, where each side is an identical equilateral triangle. Players of Dungeons & Dragons recognize the triangular pyramid shape as that of some dice used in the game.
Aristotle mistakenly thought that identical regular tetrahedrons packed together perfectly, as identical cubes do, leaving no gaps in between and filling 100 percent of the available space. They do not, and 1,800 years passed before someone pointed out that he was wrong. Even after that, the packing of tetrahedrons garnered little interest. More centuries passed.
A similar conundrum for how to best pack identical spheres has a more storied history. There, the answer was obvious. They should be stacked like oranges at a supermarket (with a packing density of 74 percent), and that is what Johannes Kepler conjectured in 1611. But proving the obvious took almost four centuries until Thomas C. Hales, a mathematician at the University of Pittsburgh, succeeded in 1998 with the help of a computer.
With tetrahedrons, the best packing arrangement is not obvious, and after it was pointed out that tetrahedrons did not pack perfectly, it seemed that they did not pack very well at all. In 2006, two Princeton University researchers, Salvatore Torquato, a chemist, and John H. Conway, a mathematician, reported that the best packing they could find filled less than 72 percent of the space — packing more loosely than spheres. That ran counter to a mathematical conjecture that, among so-called convex objects (those without dimples, holes or hollows), spheres should have the loosest ideal packing.
The Princeton paper prompted Paul M. Chaikin, a professor of physics at New York University, to buy tetrahedral dice by the hundreds and have a high school student stuff them into fish bowls and other containers. “We immediately found you could do better than 72 percent,” said Dr. Chaikin, who had earlier worked with Dr. Torquato on the packing of squashed spheres, or ellipsoids. (It turned out that squashed spheres pack more densely than spheres.)
The Princeton paper also led Jeffrey C. Lagarias, a mathematics professor at the University of Michigan, to ask Elizabeth Chen, one of his graduate students, to look at tetrahedron packing. Ms. Chen recalled his telling her: “You’ve got to beat them. If you can beat them, it’ll be very good for you.”
Ms. Chen examined several hundred arrangements over the next few weeks, and, she said, “there happened to be several that stood out as very dense.” Her best packing easily eclipsed what Dr. Conway and Dr. Torquato had found, with a packing density of almost 78 percent, surpassing spheres.
“In fact, my adviser totally did not believe me,” Ms. Chen recalled.
After making physical models of tetrahedrons and demonstrating the packing patterns, she convinced Dr. Lagarias that her packings were as dense as she had said they were, and finally published her findings a year ago.
Meanwhile, Sharon C. Glotzer, a professor of chemical engineering also at the University of Michigan, was interested to see whether the tetrahedrons might line up as liquid crystals do. “We got into it, because we are trying to design new materials for the Air Force that have interesting optical properties,” she said.
Dr. Glotzer and her colleagues wrote a computer program that simulated the jostling of tetrahedrons and how they arranged themselves when pushed together. They found not liquid crystals but complex quasicrystal structures with patterns almost repeated yet not quite. “That is the most astonishing crazy thing,” Dr. Glotzer said.
Examining the quasicrystals, they did find a periodic structure that represented another leap in packing density: over 85 percent. Just as that finding was prepared for publication last month in the journal Nature, a group at Cornell, using a different search method, found yet another packing that was just as dense.
But while Dr. Glotzer’s structure was surprisingly complex — the repeat pattern consists of 82 tetrahedrons — the Cornell crystal was surprisingly simple, with just four. It is also puzzling to researchers why the tetrahedrons in Dr. Glotzer’s simulations tend to the complex quasicrystal structures if the best packing is actually a much simpler structure.
“That’s part of what’s so surprising about this,” said Dr. Cohn, of Microsoft Research. “Each of these packings feels very different.”
A few days before Christmas, Dr. Torquato and Yang Jiao, a graduate student, reported that they had tweaked the Cornell structure to bump up the packing density by a fraction, to 85.55 percent.
“I’d be shocked if what we have right now is the densest,” Dr. Torquato said in an interview last week. “It just happens to be the densest known right now.”
Dr. Torquato need not be shocked.
On Monday, Ms. Chen, the University of Michigan graduate student, posted a new preprint, which describes a family of packings that include the latest Cornell and Princeton structures. But it also includes a better packing. The calculation was verified by simulations from Dr. Glotzer’s group.
The new world record for packing density of tetrahedrons: 85.63 percent.
A version of this article appeared in print on January 5, 2010, on page D3 of the New York edition.