Sunday, February 15, 2004

Daubert and the Packing of Oblate Spheroids

In the year 1611, as part of his investigations into atomistic hypotheses, Johannes Kepler conjectured that the way to pack the greatest number of identical spheres into a given space was to stack them like oranges on grocer's shelves, in an arrangement known as face-centered cubic packing. Kepler calculated that the spheres in such an arrangement would occupy about 74% of the space in question. In 1840, famed mathematician Carl Friedrich Gauss offered a proof of Kepler's conjecture for spheres arrayed in a regular lattice. Only in 1998 was the conjecture proved by Thomas C. Hales to hold true for all possible arrangements, even irregular ones.

Left to themselves, randomly distributed spheres are less efficient space-fillers, occupying, on average, only about 64% of the space available. Do other convex objects occupy space more densely, when randomly distributed?

It now transpires that oblate spheroids do. According to press reports, physicist Paul Chaikin, working with his Princeton colleague, chemist Salvatore Torquato, has established that randomly distributed oblate spheroids will occupy up to 68% of the space into which they are packed. The results were announced in the February 13 issue of Science.

This research has wide implications, we are told, for inquiry into such deep issues as the structure of matter, as well as various potential practical applications -- e.g., the construction of less porous ceramics.

What interests us for the moment, however, is the methodological approach taken by Chaikin and Torquato to investigating the problem. Their approach involved vats of M&M's. Plain chocolate M&M's.

Chaikin's interest in spheroid packing was sparked when his students, wanting to poke fun at Chaikin for his habit of lunchtime chocolate consumption, smuggled a 55-gallon drum full of M&M's into his lab. It happens that M&M's are almost perfect spheroids, and are extremely uniform in size and shape. Seizing on this serendipity, Chaikin and Torquato commenced to pour M&M's into containers and count them. When questions arose about whether the M&M's at the center of the containers were perhaps spontaneously arraying themselves in nonrandom configurations, they ruled out that possibility by giving the containers an MRI. To assess their hypothesis that the oblate shape of the M&M's permitted more numerous points of contact between individual M&M's, facilitating denser packing, they had an assistant pour paint through the containers, remove the M&M's, and count the unpainted spots on them.

What is striking about this description is how easily it lends itself to either of two characterizations. In one, the scientists are ridiculed (probably by some attorney) for the unorthodox and even unprecedented techniques they used to reach their conclusions. In the other, they are extolled for following the best experimental traditions of empirical science, and for submitting their findings to a learned journal. On how many occasions, we wonder, does this same basic pair of alternative characterizations play itself out in courtrooms?

We do know one thing. You can now win the raffles where the prize goes to whoever comes closest to guessing the number of plain M&M's in the glass container. Estimate the volume of the container in cubic centimeters, multiply by .680 (the fraction of the container's volume actually occupied by randomly distributed chocolate spheroids), and divide by .636 (the volume of a plain M&M in cubic centimeters).
Fed. R. Evid. 702: If scientific, technical, or other specialized knowledge will assist the trier of fact to understand the evidence or to determine a fact in issue, a witness qualified as an expert by knowledge, skill, experience, training, or education, may testify thereto in the form of an opinion or otherwise, if (1) the testimony is based upon sufficient facts or data, (2) the testimony is the product of reliable principles and methods, and (3) the witness has applied the principles and methods reliably to the facts of the case.