Packing Tetrahedrons, and Closing In on a Perfect Fit

Another NY Times story, this time taking its starting point with Aristotle: “More than 2,300 years ago, Aristotle was wrong.” What he was wrong about was regular tetrahedrons, four-sided figures where each side is an equilateral triangle. (Dungeons & Dragons players will recognize the tetrahedron as the game’s 4-sided die.) Aristotle thought that tetrahedrons could be packed together perfectly, so that, for instance, you could fill a box with tetrahedrons with no unfilled space appearing anywhere in the box. 1800 years passed before somebody realized he was wrong, but even then nobody had a clue as to just how ‘well’ tetrahedrons could pack.

Fast-forward to the modern era, when tetrahedron packing underwent something of an arms race. John Conway and Salvatore Torquato from Princeton found a packing that they could mathematically prove would fill at least 72% of the space. Paul Chaikin at NYU then had high school students fill aquariums and other things with hundreds of the D&D dice, discovering that 72% was easy! You could always do better. But how much better? And was there a systematic way to do it (rather than dumping dice into an aquarium)?

This past year saw the arms race accelerate, with competing groups coming out with successively better packings: first Elizabeth Chen and Jeff Lagarias, from the mathematics department at Michigan, published a packing pattern that guaranteed 78% coverage. Then two distinct patterns emerged that got to 85%: an extremely complicated one from Sharon Glotzer, also from Michigan, but in the chemistry department (and coming at things from the point of view of developing new materials for the Air Force), and an extremely simple one by a group from Cornell University. Just weeks ago, Dr. Torquato and Yang Jiao, his graduate student, tweaked the Cornell pattern to get coverage of 85.55%. And finally, just this past Monday, Elizabeth Chen posted a pattern that reached 85.63%. Can you do better?

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