Thomas Hales

Cannonballs and Honeycombs

Pitt Mathematician Solves Century-Old Problems

Cannoneers stacking cannonballs and grocers mounding oranges seem to know it intuitively, but determining the best way to store solid round objects is a problem that has been vexing mathematicians for centuries.

Pitt’s Andrew Mellon Professor of Mathematics Thomas Hales recently solved that problem and was honored with the Chauvenet Prize from the Mathematical Association of America (MAA) at its annual January conference, held jointly with the American Mathematical Society (AMS).

Hales described his solution to the problem, called the Kepler Conjecture, in an article titled “Cannonballs and Honeycombs,” which was published in the April 2000 edition of the Notices of the AMS.

“Honeycombs” refers to Hales’ solution to another conundrum mathematicians have wrestled with: how to divide a flat surface into equal-sized cavities with a minimum of boundary for each cavity. The stacking arrangement, called face-centered cubic packing, occurs when spheres are placed in the spaces between balls in adjoining layers. In an open space, this type of packing can take on the shape of a pyramid — hence the familiar shape of cannonballs seen at pre-20th-century war memorials, and of oranges in produce aisles.

“The problem evaded solution for almost 400 years, until Thomas C. Hales, the author of this article, gave a difficult, computer-aided, yet ingenious proof,” the anonymous MAA nominator wrote of Hales’ piece. “It has humor, history, talks about real people, presents significant mathematics, and has handholds throughout the article so you can keep finding good things even if you choose not to follow all the details as you go.

“The writing is delightful: It connects us to famous scientists of the past and to nature, it talks about the resolution of a centuries-old conjecture, it points out philosophical issues about mathematics and rigor, and it describes intriguing, understandable open questions that have an interesting history, thereby situating us in the flow of history and the challenges of the future.”

According to Hales, the article grew from a lecture sponsored by the undergraduate math club at the University of Michigan, where he taught before coming to Pitt last year. “The club launched an aggressive advertising campaign weeks before the lecture. The campaign promised an event, but I had never delivered anything more than a talk. Rather than face the crowd’s scorn, I prepared like never before. I am delighted to be honored in this way for those efforts.”

In the article, Hales acknowledges the frustrating lag between what cannoneers and grocers have intuited for centuries and the ability of mathematicians to prove their intuition correct: “Not long after I announced a solution to the problem, calls came from the Ann Arbor farmers’ market. ‘We need you down here right away. We can stack the oranges, but we’re having trouble with the artichokes,’” Hales wrote. “To me, as a discrete geometer, there is a serious question behind the flippancy: Why is the gulf so large between intuition and proof?”

The adventurous or mathematically inclined can read Hales’ paper at the AMS Web site: www.ams.org/notices/200004/fea-hales.pdf.

The prize, which has been awarded since 1925, is named for William Chauvenet, who was a professor of mathematics at the United States Naval Academy. • JF