What's the best that can happen when a scientist sits in on a course from another discipline? For Robert MacPherson, a mathematician based at Princeton in New Jersey, and materials scientist David Srolovitz, it led to a collaboration that hit a theoretical jackpot and has broad practical applications.

“Bob attended my graduate course at Princeton because he'd heard there were some great geometry problems in materials science,” says Srolovitz, now dean of Yeshiva University in New York. MacPherson's hunch proved correct when Srolovitz spoke about the von Neumann grain growth problem, which predicts how cells grow in two dimensions but can't describe how they grow in three.

Within months they were able to take some abstract concepts from geometric probability theory and measurement theory to evaluate the integral curvature in three dimensions (3D). “After we saw that the idea worked in 3D and reduced to the von Neumann 2D result, we realized the solution could be extended to all dimensions,” Srolovitz says (see page 1053).

But perhaps the most valuable result of their collaboration is the identification of a quantity they term the 'mean width', a 1D measurement that they believe will become the standard measurement of length or size for 3D objects.