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 (pronounced FIE-gen-bawm), 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.

“We know systems can do this,” Dr. Campbell said. “We understand that, and so we know to look for it. When you find, let’s say, period doubling transitions in dung beetle reproduction, it’s not a huge surprise.”

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. “He was not very happy with the physics he was doing at that time,” said Predrag Cvitanovic, a theoretical physicist at the Georgia Institute of Technology, who was a graduate student at Cornell at the time.

“The main thing he did is he solved the New York Times crossword puzzle every morning,” Dr. Cvitanovic said. “It was just clear that he was very smart. He didn’t produce anything I was aware of.”

A second unproductive postdoctoral fellowship followed, at Virginia Polytechnic Institute and State University in Blacksburg. And then a shake-up at Los Alamos rescued Dr. Feigenbaum.

Peter A. Carruthers had been installed as director of the laboratory’s theoretical division and wanted to revitalize it. He dismissed some senior scientists and recruited promising younger scientists — including Dr. Feigenbaum, even though he did not possess much of a track record.

Dr. Feigenbaum moved to Los Alamos in 1974. A year after his arrival, he made his breakthrough on period doubling. Publishing his findings took a few years more.

“The referees rejected it, because he had submitted it to a normal math journal,” Dr. Brecher said. “They didn’t know what he was talking about.”

Dr. Feigenbaum’s ideas took hold in the 1980s. He moved to Cornell in 1982 as a physics professor and to Rockefeller University in 1986.

In 1984 he won a MacArthur Fellowship (the so-called genius grant of no-strings-attached money), and in 1986 he and Dr. Libchaber shared the Wolf Prize in Physics, which is often a precursor to a Nobel. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences.

Dr. Feigenbaum worked in other areas as well — cartography, for example, with the Hammond World Atlas Corporation. Projecting cartographic details from a round world onto a flat piece of paper necessarily requires distortion. Dr. Feigenbaum figured out what projection would be most accurate for a given region and converted the time-consuming manual mapmaking process into a quick digital one.

He also figured out an efficient method of placing labels on maps. Instead of relying on the whim of the mapmaker, Dr. Feigenbaum used physics. He imagined placing electrical charges on the words. Similar charges repel each other, and thus the words would shift to ideal spacing, subject to other mapmaking requirements; the name of a river, for example, needed to be next to the river without obscuring the river.

While serving on a National Research Council committee on the design of next-generation currency design in 1993, Dr. Feigenbaum developed a fractal-based geometric pattern that digital copiers could not reproduce without visible flaws.

In 1996, he helped found Numerix, a company that produced software to quickly calculate the risk and pricing of financial derivatives.

“He didn’t publish very much, but he did a lot of work,” said Nigel Goldenfeld, a University of Illinois physicist and another founder of Numerix. “He was motivated by a sense of beauty, and he saw the beauty in a lot of different problems.”

In addition to his stepson Sasha, Dr. Feigenbaum is survived by another stepson, Kiril Dobrovolsky; a brother, Edward; and a sister, Glenda Jeunelot. His marriages to Cornelia Dobrovolsky and Gunilla Ohman ended in divorce.

In the last decade, Dr. Feigenbaum was fascinated by anamorphic art — images that appear three-dimensional when reflected off a cylindrical mirror.

He wrote a 238-page manuscript, “Reflections on a Tube,” about how what one sees is influenced not only by the physics of optics but also by the peculiarities of the human vision system. Even though the book has been almost finished for a decade, Dr. Feigenbaum continued to tinker with it. Friends say they will now finally publish it.

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