Dying Mathematician Spends
Last Days on Area of Polygon

PRINCETON, N.J. -- If you had just a short time to live, what would you do with it? David Robbins, a mathematician, is solving a really tough geometry problem.

Dr. Robbins, 60 years old, was diagnosed in April with pancreatic cancer and was told he had less than two years to live, maybe much less. He reacted to the news by considering his options: He could stick to his normal work routine at a government research institute. He could search desperately for a cure for his disease, even though his doctors told him the cancer is inoperable. He could go home and wait to die.

Or he could finally get around to a math problem that has been bugging him for decades. The problem is in a category some mathematicians consider "recreational" math. Solving it would do nothing practical for mankind. But to Dr. Robbins, the choice was easy. "I wanted to finish it," he says.

Here's the problem: What is the area of a polygon if you know only the lengths of the sides?

If the polygon happens to be a rectangle, the answer is easy: Multiply the height by the width to get the area. With a formula devised by the ancient Greeks, a triangle is easy enough to figure, too. But if the polygon has five sides or more, the math suddenly becomes gnarly.

Dr. Robbins came up with formulas for pentagons and hexagons that he published to little notice in 1994. He now wants to find the answer for a heptagon.

For everyday purposes, finding the area of a seven-sided figure is easy: One just divides the figure into triangles, physically measures the sides of the triangles and then uses the ancient formula to calculate the area of each triangle. But for Dr. Robbins that would be cheating.

If time allows, he wants to discover a general formula that would give the answer not just for a seven-sided figure but for any polygon no matter how many sides it has. "You'd think it would have a solution," he says. "Lots of people said they did want to do it. They just never got around to it."

Despite chemotherapy, Dr. Robbins is still in fairly good shape. Every morning at 7:30, he leaves home and walks two miles to his office at the Center for Communications Research in a leafy part of Princeton. His tools are pencil, paper, a computer and an old-fashioned blackboard with chalk. He bats around formulas and ideas with Julie Roskies, a 35-year-old researcher who is also working on polygons full time.

Dr. Robbins's determination doesn't surprise his wife, Deborah, although she sometimes wishes she and her husband had time to take a few trips. "When you're facing the end of your life," she says, "the fact that he is so clear about what he wants to do -- I really have to respect that." He is the sort of person, she says, who gets up in the morning complaining he didn't sleep because he was pondering a problem all night.

To colleagues with whom Dr. Robbins has shared a love of math, his philosophy seems perfectly natural. "That's what mathematicians do," says his boss, David Goldschmidt, who runs the Princeton center.

His Princeton colleagues quickly organized a conference in his honor after his illness was diagnosed. According to Mrs. Robbins, Dr. Robbins at first tried to stop them, because he was worried he'd be too sick to appear. But she persuaded him to let plans go forward for June 29 and 30. At the gathering, he chose as the topic of his presentation an algebra problem that was first raised by the Rev. Charles Dodgson, a mathematician better known as Lewis Carroll, the author of "Alice's Adventures in Wonderland."

The work on the polygon problem can be frustrating. "Sometimes we'll sit and stare into space and say, 'Boy, we need an idea,' " says Dr. Roskies. But in recent weeks the pair have made significant progress. With the help of computer programs devised by Dr. Robbins, about half of the heptagon formula has come into view.

Why polygons? Mathematicians tend to disdain solving basic puzzles that are understandable to ordinary people who took math in school. "I happen to like the immediately intelligible," says Dr. Robbins, who reminds some people of Woody Allen. There is a certain resemblance, in drab sweaters and unfashionable glasses. Also, both men grew up in Brooklyn. But Dr. Robbins seems to have happier memories of growing up. His father, a real-estate developer, used to quiz him on math problems. When he was 12 or 13, a discussion with his father about triangles led Dr. Robbins to figure out the general solution for the area of triangles.

Later he learned the Greeks got there first. He also learned that the solution for finding the area of four-sided polygons had been found around 650 A.D. by Brahmagupta, an Indian mathematician. Brahmagupta and Dr. Robbins both focus on a special group of polygons called cyclic polygons whose points all fall on a single circle.

But Dr. Robbins was intrigued to learn that no one had gone beyond Brahmagupta. Prior to the computer age, the only mathematician known to have even tried is August Ferdinand Moebius, the German mathematician of Moebius strip fame. (The Moebius strip is a one-sided geometric surface formed by giving a 180-degree twist to a long strip of paper and then connecting the ends.) In a 31-page 1828 treatise, Moebius discussed the cyclic-pentagon problem but didn't solve it.

Dr. Robbins is something of an odd bird in the math world. Despite his Ph.D. from the Massachusetts Institute of Technology, he spent seven years teaching high school.

Friends lured him to the Center for Communications Research -- Princeton 23 years ago. It is a unit of the Institute for Defense Analyses, which provides technical and scientific analysis on issues of concern to the Defense Department. The center, according to its Web site, applies mathematics and computer-science research to "cryptology and related disciplines." Dr. Robbins won't discuss his work for the center.

Dr. Robbins says he enjoyed the practical side to his work as well as the freedom he was given to work in pure mathematics. Best of all was the group of kindred spirits he found at the center. For years he and colleagues would entertain themselves at lunchtime trying to solve questions from the Putnam exams, an annual college math competition. The custom died out a few years ago, he says. "People started complaining they didn't want to do problems at lunch."

Among mathematicians, Dr. Robbins's main achievement is something called the alternating-sign matrix conjecture, which has connections with fashionable subjects in math and physics. The polygon-area problem, says independent mathematician Michael Somos, is "very theoretical and very difficult and you're not going to get a prize if you solve it."

Greg Kuperberg, a math professor at the University of California, Davis, who once worked briefly with Dr. Robbins, is one of those who calls his polygon problem "recreational." In a phone interview, Dr. Kuperberg said polygons are like a fluffy human-interest newspaper article, whereas alternating-sign matrices are like a "weighty story on clandestine nuclear-weapons research."

But after he reviewed Dr. Robbins's pentagon proof, Dr. Kuperberg changed his mind a bit and e-mailed this remark: "Yes, this is considered recreational mathematics. But as he has done before, David elevates the problem. If he or someone else finds more time to work on this question, it could well lead to math research that goes beyond the merely recreational."

Dr. Robbins hopes he has enough time left. So long as his health holds up, he intends to commute to his office, where he pads about in an old pair of his father's slippers, and keep tackling the problem with Dr. Roskies. Dr. Robbins doesn't like to speculate about his cancer, but he says: "If I somehow turn too ill, Julie would be there to finish it."

Write to Peter Landers at peter.landers@wsj.com²

Updated July 29, 2003