#### He Made Sense of Chaos

Mitchell J. Feigenbaum, a pioneer in the field of mathematical physics known as chaos, died on June 30 in Manhattan. He was 74.

The cause was a heart attack, his stepson Sasha Dobrovolsky said.

Dr. Feigenbaum's intense, eclectic curiosity led him to questions far astray from the ones usually asked by theoretical physicists. How does one make the most accurate maps? What makes the moon look larger when it is closer to the horizon? What design of paper money would thwart photocopying?

His friends recalled meals, full of red wine and red meat (no vegetables), at which the conversation -- not just about physics but also about literature and classical music -- would stretch late into the night.

Dr. Feigenbaum's lifestyle and his Renaissance intellect were a poor fit to the demands of modern publish-or-perish academia. But by following his own path, he uncovered a pattern of chaos that is universal in math and in nature

At Los Alamos National Laboratory in New Mexico in the mid-1970s, Dr. Feigenbaum, using a programmable calculator, found what seemed at first a mathematical curiosity. A simple equation generated a sequence of numbers, which were initially trivial: the same number over and over. But as a parameter in the equation shifted, the output became more varied. First the numbers bounced back and forth between two values, then they cycled among four values, then eight, and so on, with the rate of the change quickening until the patterns lost all hint of repeating cycles.

The dynamics had, in the terminology of physics, passed into the realm of deterministic chaos. That is, each number of the sequence could be computed precisely, but the resulting pattern appeared to be complex and random.

Dr. Feigenbaum looked at another simple equation, and it exhibited the same behavior, known as period doubling. More startling, the number that characterized the rate of doubling was the same: As the periods multiplied, each doubling occurred about 4.669 times as quickly as the previous one.

This number is now known as the Feigenbaum constant. Dr. Feigenbaum was able to prove why it is a universal mathematical value, much as pi -- the ratio of the circumference of a circle to its diameter -- is the same for all circles. "There aren't too many fundamental constants," said Kenneth Brecher, an emeritus professor of astronomy at Boston University, who met Dr. Feigenbaum when both were graduate students at the Massachusetts Institute of Technology. "And he was the only living person that had one."

In 1979, a French scientist, Albert J. Libchaber, observed the same cascade of period doublings in the temperature fluctuations in the center of a convecting fluid. Dr. Feigenbaum's theory of the transition from order to chaos now described phenomena in the real world.

"It's observed in nature and not simply mathematical equations," said David Campbell, a physics and engineering professor at Boston University and a longtime friend of Dr. Feigenbaum's.

The same ideas have now also been applied to describe the rise and fall of fish populations, the dripping of a leaky faucet and the swings of financial markets.

Mitchell Jay Feigenbaum was born in Philadelphia on Dec. 19, 1944, and grew up in Brooklyn, the son of Abraham and Mildred Feigenbaum. His father was a chemist, his mother a schoolteacher. After graduating with a bachelor's degree in electrical engineering from the City College of New York, he pursued particle physics at M.I.T., completing his doctorate in 1970.

During a postdoctoral fellowship at Cornell, Dr. Feigenbaum was already allowing his focus to wander. Instead of churning out a stream of publications, as was expected of young scientists, he gained a reputation as a deep thinker without much to show for his thoughts.