The Miraculous Space Efficiency of Hexagons

Slate/Science   July 22, 2016

Hexagons and the Science of Packing

By Marc Chamberland

Excerpted from Single Digits: In Praise of Small Numbers by Marc Chamberland. Out now from Princeton University Press.

What do grocers and honeybees have in common? The obvious answer is that they are both adept at providing food for others. But there is a richer, more technical answer to this question: These two groups know how to efficiently pack their resources.

Honeycombs, made from the wax secreted by bees, are used to store honey, pollen, and larvae. For thousands of years, the honeycomb’s hexagonal structure has been noted and admired. It is wondered whether this entomological architecture inspired the interior ribbing and hidden chambers in the dome of the Pantheon in Rome. Today honeycomb structures have numerous engineering and scientific applications, including in the aerospace industry.

Why do honeycombs have a hexagonal structure? Pappus of Alexandria declared that bees “possessed a divine sense of symmetry,” and Charles Darwin described the honeycomb as a masterpiece of engineering that is “absolutely perfect in economizing labor and wax.” A mathematical rationale was given by the Polish polymath Jan Brożek (1585–1652): The hexagon tiles the plane with minimal boundary. Stated another way, Brożek conjectured that the optimal way to cover a large region with shapes of the same area while minimizing the boundary is to use the hexagonal structure. This problem resisted a solution for centuries but was finally positively settled by Thomas Hales in 1998.