Packing Bubbles into a Foam More Efficiently

September 14, 2009

Another counterexample to Kelvin's conjecture on minimal surfaces has surfaced: a foam structure made up of four different shapes of equal volume that fit together to minimize surface area. The discovery is making waves in the world of mathematics—and may have applications in the world of medicine.

The conjecture states that the minimal surface-area partition of space into cells of equal volume is a tiling by truncated octahedra with slightly curved faces. Ruggero Gabbrielli (University of Bath), however, came up with a partition with a lower surface area: a periodic foam containing quadrilateral, pentagonal, and hexagonal faces.

Gabbrielli's method for generating the foam uses a partial differential equation well-known in two-dimensional pattern formation. "The novelty is that I've applied it to a three-dimensional problem to model the shape of foams," Gabbrielli said.

"It's not just about bubbles," he added. "Three-dimensional patterns spontaneously arise in many systems nature designed."

The findings appear in the article "A New Counter-Example to Kelvin's Conjecture on Minimal Surfaces" (Philosophical Magazine Letters, Aug. 8, 2009).

Source: University of Bath, Sept. 2, 2009.