It has been many years since I have seen honeycomb in markets. Why it is no longer widely available, I don't know. Children today miss the opportunity to chew on a spoonful of comb, its rich sweetness slowly melting to leave only the wax. Surely bubble gum is a poor substitute.

But it is not just the taste of that honeycomb that modern youngsters miss. They rarely get to see the remarkable construction of those combs with their row upon row of hexagonal tubes.

Mathematicians and architects have been attracted to the engineering skills of honeybees for at least 2000 years. In about 70 A.D., for example, Pliny wrote of earlier men devoting lifetimes to the study of the geometry of honeycombs. Professionals in these fields will recognize some of the others who investigated the bees' architecture: Pappus, Kepler, Maclaurin, Wren, and a 20th century mathematician, L. Fejes Toth, who wrote a highly technical paper with the delightful title, "What the Bees Know and What They Do Not Know."

Let's see what is going on here.

Bees build comb for storage of honey to last them through the winter when the flowers they feed on are not available. The honeycomb is vertical with horizontal storage tubes, like a pile of unsharpened lead pencils carefully pressed together. (Social wasps and hornets build vertical tubes.) Honeycomb is two faced with different tubes on each side; thus an individual tube goes only half way through the comb. Two of the six sides of the tubes are always vertical and each tube slants slightly downward toward the middle of the comb, which helps prevent the honey from running out as the worker bees fill it.

Why did these honeybee engineers "choose" these six sided tubes? Why didn't they build cylinders or prisms with triangle or square or other cross sections? The answer is straightforward. Their hexagonal tubes use less wax for the volume of honey they hold. Each wall in the honeycomb serves two tubes which avoids the wasteful duplication of cylinders and most polygonal prisms. Only triangular or square tubes can also share all walls, but hexagonal tubes still use less wax for the amount of honey: 18% less than triangular tubes, 7% less than square tubes.

Even more remarkable is the way the tubes meet in the middle of the comb. If you removed the honey and the wax where the tubes meet and peered through the holes, you would see that the tubes on opposite sides are offset: the center of a tube on one side is at the corner of tubes from the opposite side. You can demonstrate this by constructing several comb units as suggested by the accompanying diagram.

And the wax that separates the opposite tubes is not a single flat wall. Instead each tube ends in three rhombuses that come to a point, like a pencil cut with only three knife strokes. (A rhombus is a four sided plane figure with all sides equal; it is like a tilted square.) The three end walls of one tube serve as single walls for three adjacent tubes from the opposite side of the comb.

In about 1720, Miraldi measured the corner angles of these end walls and found them to be about 70° and 110°. You can imagine the thrill mathematicians enjoyed when Koenig and Maclaurin used the then new techniques of the calculus to determine that these were the angles that would give the maximum volume for this configuration.

This result was scarcely tarnished when Toth showed in his 1964 paper that a more complicated set of four end walls would give slightly better results. His improvement was only about a third of a percent and the construction would be more difficult.

So what the bees don't know certainly won't hurt them.

Anthony L. Peressini, "The Design of Honeycombs," *Modules in
Undergraduate Mathematics and Its Applications Module 502* published by
the Consortium for Mathematics and Its Applications, 1980

William J. Roberts, "Honeycomb Geometry: Applied Mathematics in Nature,"
*Mathematics Teacher*.

David R. Siemens, Jr., "The Mathematics of the Honeycomb," *Mathematics
Teacher* 70 (April 1965): 334-337.

David R. Siemens, Jr., "Of Bees and Mathematicians," *Mathematics
Teacher* 60 (November 1967): 758-760.

D'Arcy W. Thompson, *On Growth and Form: The Revised Edition*
(Cambridge, England: Cambridge University Press, 1942), pp. 525-544. This
remarkable book is now available in a 1992 Dover reprint.

L. Fejes Toth, "What the Bees Know and What They Do Not Know," *Bulletin
of the American Mathematical Society* 70 (1964): 468-481.