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He thought about the eye’s ability to see consistent colors and forms in a universe that physicists knew to be a shifting quantum kaleidoscope.
WHERE CHAOS BEGINS, classical science stops.
To some physicists chaos is a science of process rather than state, of becoming rather than being.
There was always one small compromise, so small that working scientists usually forgot it was there, lurking in a corner of their philosophies like an unpaid bill. Measurements could never be perfect.
Serendipity merely led him to a place he had been all along.
The Butterfly Effect acquired a technical name: sensitive dependence on initial conditions.
Nonlinearity means that the act of playing the game has a way of changing the rules.
Analyzing the behavior of a nonlinear equation like the Navier-Stokes equation is like walking through a maze whose walls rearrange themselves with each step you take.
Professional scientists, given brief, uncertain glimpses of nature’s workings, are no less vulnerable to anguish and confusion when they come face to face with incongruity.
Often a revolution has an interdisciplinary character—its central discoveries often come from people straying outside the normal bounds of their specialties.
Work fell between disciplines—for example, too abstract for physicists yet too experimental for mathematicians.
But to accept the future, one must renounce much of the past.
New hopes, new styles, and, most important, a new way of seeing. Revolutions do not come piecemeal. One account of nature replaces another. Old problems are seen in a new light and other problems are recognized for the first time. Something takes place that resembles a whole industry retooling for new production.
People who conduct experiments learn quickly that they live in an imperfect world.
Unpredictability was only the attention-grabber.
A computer can address the problem by simulating it, rapidly calculating each cycle. But simulation brings its own problem: the tiny imprecision built into each calculation rapidly takes over, because this is a system with sensitive dependence on initial conditions. Before long, the signal disappears and all that remains is noise.
“You see this large-scale spot, happy as a clam amid the small-scale chaotic flow, and the chaotic flow is soaking up energy like a sponge,” he said. “You see these little tiny filamentary structures in a background sea of chaos.”
He knew to look for wild disorder, and he knew that islands of structure could appear within the disorder.
A year-by–year facsimile produces no more than a shadow of a system’s intricacies, but in many real applications the shadow gives all the information a scientist needs.
At the institute, Yorke enjoyed an unusual freedom to work on problems outside traditional domains, and he enjoyed frequent contact with experts in a wide range of disciplines.
In daily life, the Lorenzian quality of sensitive dependence on initial conditions lurks everywhere.
The solvable systems are the ones shown in textbooks. They behave. Confronted with a nonlinear system, scientists would have to substitute linear approximations or find some other uncertain backdoor approach.
Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.
Within the most disorderly reams of data lived an unexpected kind of order.
The insights of fractal geometry helped scientists who study the way things meld together, the way they branch apart, or the way they shatter.
Leibniz imagined that a drop of water contained a whole teeming universe, containing, in turn, water drops and new universes within.
Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects—in clouds, trees, mountain ranges, or snow crystals. The shapes of all these are dynamical processes jelled into physical forms, and particular combinations of order and disorder are typical for them.”
Even supercomputers are close to helpless in the face of irregular fluid motion.
As singular boundaries between two realms of existence, phase transitions tend to be highly nonlinear in their mathematics.
Where Newton was reductionist, Goethe was holistic. Newton broke light apart and found the most basic physical explanation for color. Goethe walked through flower gardens and studied paintings, looking for a grand, all-encompassing explanation.
All of renormalization theory depended on it. In an apparently unruly system, scaling meant that some quality was being preserved while everything else changed. Some regularity lay beneath the turbulent surface of the equation.
UNIVERSALITY MADE THE DIFFERENCE between beautiful and useful.
Universality offered the hope that by solving an easy problem physicists could solve much harder problems. The answers would be the same.
The notion of a scientific breakthrough so original and unexpected that it cannot be published seems a slightly tarnished myth. Modern science, with its vast flow of information and its impartial system of peer review, is not supposed to be a matter of taste.
When a discovery is made, when an idea is expressed, it is assumed to become the common property of the scientific world. Each discovery and each new insight builds on the last. Science rises like a building, brick by brick. Intellectual chronicles can be, for all practical purposes, linear.
But the history of ideas is not always so neat. As nonlinear science arose in odd corners of different disciplines, the flow of ideas failed to follow the standard logic of historians. The emergence of chaos as an entity unto itself was a story not only of new theories and new discoveries, but also of the belated understanding of old ideas.
The way ideas spread became as important as the way they originated.
“Somehow the wondrous promise of the earth is that there are things beautiful in it, things wondrous and alluring, and by virtue of your trade you want to understand them.”
His idea of a good experiment was like a mathematician’s idea of a good proof. Elegance counted as much as results.
Yet he had a feeling for the abstract, ill-defined, ghostly thing called flow. Flow was shape plus change, motion plus form.
“The flecked river Which kept flowing and never the same way twice, flowing Through many places, as if it stood still in one.”
These experimenters, the ones who pursued chaos most relentlessly, succeeded by refusing to accept any reality that could be frozen motionless.
“Sensitive chaos”—Das sensible Chaos—was Schwenk’s phrase for the relation between force and form.
This classicist, polyglot, mathematician, zoologist tried to see life whole,
He was enough of a mathematician to know that cataloguing shapes proved nothing. But he was enough of a poet to trust that neither accident nor purpose could explain the striking universality of forms he had assembled in his long years of gazing at nature.
Even to a man looking for hidden forms in messy data, tens and then hundreds of runs were necessary before the habits of this tiny cell started to come clear.
Theorists adapted Feigenbaum’s techniques and found other mathematical routes to chaos, cousins of period-doubling: such patterns as intermittency and quasiperiodicity. These, too, proved universal in theory and experiment.
In a computer experiment, when you generated your thousands or millions of data points, patterns made themselves more or less apparent. In a laboratory, as in the real world, useful information had to be distinguished from noise. In a computer experiment data flowed like wine from a magic chalice. In a laboratory experiment you had to fight for every drop.
Mitchell Feigenbaum
The Mandelbrot set is a collection of points.
