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Recent Advances in Tetrahedron Packing

January 28, 2010

The old circus gag "How many clowns fit inside a Volkswagen Beetle?" has a mathematical equivalent -- "How many tetrahedrons fit inside a given volume?" More precisely, tetrahedron packers focus not on how many, but on what percent of a fixed volume they can fill with these four-sided, triangular pyramids.

This geometric puzzle dates back at least as far as Aristotle's mistaken conjecture that tetrahedrons pack together perfectly, i.e. leaving no gaps and filling 100 percent of a given volume. Since then, the packing of spheres has received more press. In 1611, Johannes Kepler surmised that the most efficient arrangement of spheres was the way fruit is stacked in the grocery.  This conjecture remained unproven until University of Pittsburgh mathematician Thomas C. Hales provided a formal proof in 1998.

A team of researchers headed by Sharon C. Glotzer, a professor of chemical engineering at the University of Michigan, tackled the tetrahedron problem by jostling tetrahedrons to see what structures they formed on their own. They found "complex quasicrystal structures with patterns almost repeated yet not quite," which resulted in a packing density of over 85 percent. Around the same time, a group at Cornell University found another packing pattern with a similar density, but a far simpler arrangement – the basic units involved only four tetrahedrons. Currently the densest tetrahedron packing on record is 85.63 percent, discovered by Elizabeth Chen, a University of Michigan graduate student.

Practical applications of the packing problem range from the search for the most efficient arrangement of merchandise for shipping to the design of Air Force materials. However, the most basic explanation for the interest generated by packing problems is the same principle that leads us to spend hours with trying to rearrange wooden blocks into a cube – human beings love a puzzle.

Source: New York Times Jan. 4, 2010.

.gif courtesy of Wikipedia.

This Math in the News summary was written by Hannah Ross, a senior Math/English double major at Kenyon College.

Start Date: 
Thursday, January 28, 2010