University Times

After completing his proof of the Kepler Conjecture, Thomas Hales turned his attention to a related problem of even greater antiquity: What is the most efficient partition of a plane into equal areas? The Honeycomb Conjecture asserts that the answer is the six-sided honeycomb.

Honeybees had always taken the conjecture as a given. But for more than 2,000 years, mathematicians failed to prove it — until, in 1999, Hales did so, in less than six months.

In 36 BC, in a book on agriculture, the Roman scholar Marcus Terentius Varro described two competing theories of the honeycomb's hexagonal structure. One theory held that hexagons best accommodated the bee's six legs. The other theory, supported by mathematicians of the day, was that a circular hexagon enclosed the greatest amount of space.

A few centuries later, Pappus of Alexandria presented an incomplete proof of the conjecture, based largely on the fact that only three regular polygons (the triangle, the square and the hexagon) fill out a plane, and the hexagon holds the most honey.

Some of Pappus's reasoning seemed to be based more on natural history than mathematics. For example, he dismissed the possibility of gaps between cells of the honeycomb on the grounds that unless the cells were contiguous, "foreign matter could enter the interstices between them and so defile the purity of their produce."

Mathematicians from Kepler to Kelvin studied six-sided prisms, but the Honeycomb Conjecture resisted all efforts to prove it until Hales came along.

Based on his struggle with Kepler's Conjecture, Hales had come to expect that every theorem would involve a monumental effort. "I was psychologically unprepared for the light 20-page proof of the Honeycomb Conjecture," Hales wrote. "It makes no significant use of computers and took less than six months to complete.

"In contrast with the years of forced labor that gave the proof of the Kepler Conjecture, I felt as if I had won a lottery."

— Bruce Steele