# Bees Do It Best

Honey may be sweet, but honeybees' engineering is even sweeter. A mathematician has proved that a comb's hexagonal lattice allows bees to store the most honey for the least amount of beeswax. The proof, presented last month at the Turán Workshop in Mathematics, Convex and Discrete Geometry in Budapest, ends centuries of speculation--and confirms the intuition of human engineers, who use honeycombed materials to produce light but strong panels for cars, planes, and spacecraft.

Since at least the first century, researchers have theorized that a hexagonal lattice is the most economical design for a honeycomb's single layer of equal-sized cells. Until this summer, however, no one could mathematically confirm the hunch.

Now, Thomas Hales of the University of Michigan, Ann Arbor, has shown that a hexagonal honeycomb has walls with the shortest total length, per unit area, of any design that divides a plane into equal-sized cells. The proof, which Hales began working on last year, builds on his earlier work showing that equal-volume bubbles in a wet foam will form a regular lattice. But no one knew if that result applied to a honeycomb, which Hales says is more like a dry foam in which the bubbles squeeze one another's shape, leading to tradeoffs in volume. In the most efficient dry foam, the cells might have different shapes. Hales, however, showed that a beehive's hexagonal compartments optimize the gains and losses when the effect of each cell on its neighbors is taken into account, giving the most efficient overall arrangement.

While other mathematicians had made progress on the problem, Hales is the first to propose a method which correctly accounts for the costs and benefits of making the sides of a cell curved or using more than 6 sides in a cell, says John Sullivan of the University of Illinois at Urbana-Champaign. "Hales's bright idea was that no single cell can do better than a hexagon if appropriately penalized," he says. And he and other geometers are pleased that the proof, unlike similar solutions, does not require elaborate computer calculations. "There should be an easy reason for a pattern this simple," Sullivan says, "and I think Hales has found it."

Posted in Math