Back in the early 1960s, a mathematician-turned-meteorologist Edward Lorenz tried to model a weather system using some simple mathematical equations and a primitive computer. He was pleasantly surprised by the reasonably complex results generated by such simple setup and the fact that they never repeated themselves, just like a real weather system would behave.
One day he decided to repeat a particularly interesting part of his weather model and reset all the variables to the state they had been in before the period he wanted to rerun, then set it going again and went to get a cup of coffee.
When he returned he was surprised to find his weather system was doing something completely different than what it had done before. At first he thought he might have typed in the data wrong, but that was not the case, except he did type in some numbers that were a few decimal places shorter than the original ones, for example, 4.526 instead 4.526173. Small as the difference seemed, it eventually caused the system to diverge and behave totally differently from the original.
Lorenz published his findings in his paper "Deterministic Nonperiodic Flow" in 1963, which subsequently gave rise to a new field of study called "chaos mathematics" that studies the behavior of dynamical systems that are highly sensitive to initial conditions and whose outcomes are utterly unpredictable.
Lorenz popularized the idea through use of the phrase "the butterfly effect." If a single butterfly in Brazil decided to flap its wings, he explained, that could ultimately decide whether a tornado formed in Texas.
Or like those "Back to the Future" movies show, one minor twist of events in the past could result in extreme make-over of things today.
We humankind had successfully rocketed ourselves to the moon hundreds of thousands of miles away and back, and cracked the "God particle" out of the tiny atoms with ultra-fast super collider, but the truth of the matter is 99.9% of the day-to-day world we live in is still chaotic in nature. There is no pure straight line or perfectly round circle or truly symmetrical pyramid in real world--they only exist in our imagination. One accident on the highway, a transformer blowout in the power grid, or a glitch in computer software, reminds us how swiftly our peace and prosperity can be taken away and we returned to the chaotic state we thought we had long managed away!
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Another mathematician, Benoit Mandelbrot, another computer, and another simple equation, 16 years later. Mandelbrot fed the answer of his simple equation--no more than a multiplication and an addition--back to the computer and had it repeat the calculations millions of times. He then painted the results on his computer display, with a pixel of black representing the answer heading toward zero and pixels of various colors representing the speed at which the answer raced off toward infinity.
The result, after the equation had been applied to the whole screen, was an appealing black splodge in the center of the screen with colored, crinkly edge.
The edges of the shape were wrinkled and unpredictable, and sometimes bulged out to form another near-circle. Zooming in to them, it revealed more and more detail. There were swirls that looked like elephant trunks, and branching shapes that looked like ferns or leaves. The patterns kept coming as you looked closer, there were even miniature versions of the initial shape hidden deep within itself. But at no point did the patterns repeat themselves exactly, they were always entirely new.
A video rendition of the explanations above can be seen here:
https://youtu.be/PD2XgQOyCCk
Mandelbrot coined the word "fractal" to describe what he had discovered. A fractal, he said, was a shape that revealed details at all scales.
Fractals can be found everywhere in the natural world. They are in the branching of trees, the shape of leaves, snowflakes, ice crystals, and the shape of human lungs. They describe the distribution of blood vessels, and the path of a flowing river.
They seem to be the common language spoken among the created--a tiny antenna built according to the fractal rule can receive a wide range of wireless signals much better than a large, cumbersome antenna that is not--and the building principle used by the Creator: the branching pattern of one single tree in a rain forest is similar to the distribution of large and small trees in the whole region.
Another mathematician, Benoit Mandelbrot, another computer, and another simple equation, 16 years later. Mandelbrot fed the answer of his simple equation--no more than a multiplication and an addition--back to the computer and had it repeat the calculations millions of times. He then painted the results on his computer display, with a pixel of black representing the answer heading toward zero and pixels of various colors representing the speed at which the answer raced off toward infinity.
The result, after the equation had been applied to the whole screen, was an appealing black splodge in the center of the screen with colored, crinkly edge.
The edges of the shape were wrinkled and unpredictable, and sometimes bulged out to form another near-circle. Zooming in to them, it revealed more and more detail. There were swirls that looked like elephant trunks, and branching shapes that looked like ferns or leaves. The patterns kept coming as you looked closer, there were even miniature versions of the initial shape hidden deep within itself. But at no point did the patterns repeat themselves exactly, they were always entirely new.
A video rendition of the explanations above can be seen here:
https://youtu.be/PD2XgQOyCCk
Mandelbrot coined the word "fractal" to describe what he had discovered. A fractal, he said, was a shape that revealed details at all scales.
Fractals can be found everywhere in the natural world. They are in the branching of trees, the shape of leaves, snowflakes, ice crystals, and the shape of human lungs. They describe the distribution of blood vessels, and the path of a flowing river.
They seem to be the common language spoken among the created--a tiny antenna built according to the fractal rule can receive a wide range of wireless signals much better than a large, cumbersome antenna that is not--and the building principle used by the Creator: the branching pattern of one single tree in a rain forest is similar to the distribution of large and small trees in the whole region.
As the poem says: "To see the world in a grain of sand."
It may also help resolve the long puzzling contradiction between the second law of thermodynamics that says in an isolated system entropy (disorder) would increase, and the ever more complex and orderly development of organisms evolution theory says and biologists observe. With the arrival of chaos mathematics and fractal discoveries, biologists now have a key that helps them study the way order spontaneously arises in nature.
And for artists and designers, sophisticated art and creative design can be generated through adoption and manipulation of fractals. There are order and beauty hidden behind seeming chaos, after all.
Happy Thanksgiving!
It may also help resolve the long puzzling contradiction between the second law of thermodynamics that says in an isolated system entropy (disorder) would increase, and the ever more complex and orderly development of organisms evolution theory says and biologists observe. With the arrival of chaos mathematics and fractal discoveries, biologists now have a key that helps them study the way order spontaneously arises in nature.
And for artists and designers, sophisticated art and creative design can be generated through adoption and manipulation of fractals. There are order and beauty hidden behind seeming chaos, after all.
Happy Thanksgiving!
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