Fall 2001

Cannonballs and Honeycomb: Pappus

R. Weaire was writing a book on sphere packings when I finished the proof of the Kepler conjecture, and we began to correspond. Under his influence, I turned to the planar version of the foam problem. This problem goes back over 2000 years. What is the most efficient partition of the plane into equal areas? The honeycomb conjecture asserts that the answer is the regular hexagonal honeycomb.

### Pappus

Around 36 BC, the Roman scholar Marcus Terentius Varro wrote a book on agriculture in which he discusses the hexagonal form of the bee's honeycomb. There were two competing theories of the hexagonal structure. One theory held that the hexagons better accommodated the bee's six feet. The other theory, supported by the mathematicians of the day, was that the structure was explained by the isoperimetric property of the hexagonal honeycomb. Varro writes, ``Does not the chamber in the comb have six angles ¼ The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space.''

This ancient proof has been lost, unless it was the proof presented a few centuries later by Pappus of Alexandria in the preface to his fifth book. The argument in Pappus is incomplete. In fact it involves nothing more than a comparison of three suggestive cases. It was known to the Pythagoreans that only three regular polygons tile the plane: the triangle, the square, and the hexagon. Pappus states that if the same quantity of material is used for the constructions of these figures, it is the hexagon that will be able to hold more honey. Pappus's reason for restricting his attention to the three regular polygons that tile are not mathematical (bees avoid dissimilar figures). He also excludes gaps between the cells of the honeycomb without mathematical argument. If the cells are not contiguous, ``foreign matter could enter the interstices between them and so defile the purity of their produce.''\footnote"*"% {T. Heath, A history of Greek mathematics, Vol II, Oxford, 1921, page 390.}

In 1943, L. Fejes T\'oth gave a proof of the honeycomb conjecture under the hypothesis that all the cells are convex polygons. He stated that the general conjecture had ``resisted all attempts at proving it.'' In 1999, I found the first general proof.

Theorem (Honeycomb conjecture). Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

The main ingredient of the proof is a new isoperimetric inequality, which has the regular hexagon as its unique minimum.

My expectations of mathematics have been shaped by the Kepler conjecture. I have come to expect every theorem to be a monumental effort. I was psychologically unprepared for the light 20-page proof of the honeycomb conjecture. 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.

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