Fifty-five years ago the late ber-genius John von Neumann proved that the area of any two-dimensional region subject to surface tension—such as a bubble—changes in proportion to the number of its sides. (Five or fewer, it shrinks; seven or more, it grows; six, it maintains its area.) Since then, physicists have struggled to apply von Neumann's result to the analogous case of microscopic crystal grains growing in three dimensions, according to materials scientist David J. Srolovitz of Yeshiva University.

Recently Srolovitz and Robert D. MacPherson of the Institute for Advanced Study in Princeton, N.J., derived such an equation for tension-induced volume change in three (or more) dimensions by adopting an abstract quantity called mean width. Analogous to surface area or volume, mean width is a nonintuitive measure of a region's size in units of length, no matter what shape it has. If physicists can apply the result to shifting clusters of crystal grains or bubbles, they may be able to engineer stronger materials—or minimize the foam on a glass of beer. “This formula,” Srolovitz says, “basically tells you how every single bubble in the head of beer is going to change.” Size up the April 26 Nature for more.