June 7, 2001

# Bees and Beijing in One Hour?

An Open Letter to Mr. John Gould

Dear Mr. Gould,

The math world has a new and special interest in your column on "What's true and what isn't about bees" (The Christian Science Monitor, May 18). In 1999, Professor Thomas Hales of the University of Michigan finally proved that a honeycomb of regular hexagons is the most efficient way to partition the plane into unit areas (using the least average amount of material; see Math Chat of June 17, 1999). The earliest extant claim of this economy appeared in 36 BC in a final missive of advice from the dying Marcus Terentius Varro to his wife about taking care of their farm. Varro actually gave two possible reasons for the six-sided honeycombs:

(1) "Does not the chamber in the comb have six angles, the same number as the bee has feet?"

(2) "The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space."

The second, though premature, has two thousand years later proved accurate. Varro's comments were not always so accurate, as when he wrote that the bees

". . . follow their own king wherever he goes . . ."

Apparently it was not discovered until the seventeenth century that the "king" was a queen.

The bees actually have a more complicated, three-dimensional problem involving how the ends of the hexagonal cells are shaped to interlock with the ends of the cells on the other side. L. Fejes-To'th, in a famous 1964 article on "What the Bees Know and What They Do Not Know" (Bulletin of the AMS 70), revealed the sad truth that the material in the bees' idealized three-dimensional structure can be reduced by a fraction of one percent.

Old Challenge. In fifty years, what will be the fastest commercial transportation from New York to Beijing?

Answer. Timur Dogan and Joseph DeVincentis suggest a one-hour trip on a space shuttle. Genele Rhoads reports that Paul Moller has already invented a personal skycar, which "uses a rotary engine to travel 350mph and gets great mileage" (see www.moller.com). Joe Shipman adds that for express packages there will be a super-high-speed evacuated tunnel powered by electromagnetism.

New Challenge. If the party affiliation of each of the 100 US Senators were determined by flipping a fair coin (heads Republican, tails Democrat), what would be the chance of a 50-50 split? Is it surprising that a 50-50 split actually occurred?

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a \$1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).