Dripping with Geometry What is fractal? Fractals are geometric shapes that are very complex, infinitely de- tailed and quite often stunningly beautiful. A portion of a fractal will have just as much detail as the whole fractal, and some sections are similar to large ones. Fractals are related to chaos be- cause they are complex systems with definite properties. They can be found almost everywhere in nature, from crystals and all clouds to coastlines and galaxy clusters. Fractual uses. Fractals provide the means to depict enormous detail in irregular objects, such as natural landscapes and in developing architectural and engineering designs. The real fractal dimensions of a metal’s surface can suggest it’s particular characteristics. Scientists use fractals to describe and predict events in astronomy, economics, meteorology and ecology. Source: Physicist adds mathematical perspective to Jackson Pollock’s artistic equation. By: Jennifer Ouellette. NEW YORK TIMES NEWS SERVICE I
n 1949, when was “the greatest living painter in the United States,” the resulting outcry voiced nearly a half century of popular frustration with abstract art. Others said that a trained chimpanzee could do just as well. Yet, Pollock’s reputation has far outlived his detractors. A retrospective of his work sever years ago at the Museum of Modern Art in New York City drew lines around the entire block, and an award-winning film of his life and art was released at the end of 2000. Apparently, “Jack the Dipper” captured some aesthetic dimension beyond the scope of his famous critics. That logic, says physicist and art historian Richard Taylor, lies not in art but in mathematics------or more specifically, in chaos theory and its offspring, fractal geometry. Fractals may seem haphazard at first glance, yet each one is composed of a single geometric pattern repeated thousands of times at different magnification, like Russians dolls nested within one another. They are quite often the visible remains of chaotic systems----systems that obey internal rules of organization but are so sensitive to slight changes that their long-term behavior is difficult to predict. If a hurricane is. a chaotic system, then the wreckage strewn in its path is its fractal pattern. Some fractal patterns exist only in mathematical theory, but others provide useful models for the irregular yet patterned shapes found in nature----the branching of rivers and trees, for instance. Mathematicians tend to rank fractal dimensions on a series of scales between 0 and 3. One-dimensional fractals (such as a segment line) typically rand between 0.1 and 0.9, two-dimensional fractals (such as a shadow thrown by a cloud) between 1.1 and 1.9, and three-dimensional fractals (such as a mountain) between 2.1 and 2.9. Most natural objects, when analyzed in two-dimensions, rank between 1.2 and 1.6. Physicist Richard Taylor was on sabbatical in England six years ago when he realized that the same analysis could be applied to Pollock’s work. In the course of pursuing a master’s degree in art history, Taylor visited galleries and pored over books of paintings. At one point in his research, he began to notice that the drips and splotches on Pollock’s canvases seemed to create repeating patterns at different size scales----just like fractals. Months later, back in his lab at the University of New South Wales in Sydney, Australia, Taylor put his insight to the ultimate test. First, he took high-resolution photographs of 20 canvases dating from 1943 to 1952. (Pollock had moved away from the drip painting method in 1953) Then he scanned the photographs into his computer and divided the images into an electronic mesh of small boxes. Finally, he used the computer to assess and compare nearly 5,000,000 drip patterns at various locations and magnifications in each painting—from the length of a full canvas to less than a tenth of an inch. The fractal dimensions of Pollock’s earlier paintings, Taylor finally concluded, corresponds closely to those found in nature. A 1948 painting titled Number 14, for example, has a fractal dimension of 1.45, similar to that of most coastlines. A skeptic might suggest that the effect is coincidental. But, Pollock clearly knew what he was after: The later the painting, the richer and more complex its patterns, and also, the higher its fractal dimension. “Blue Poles,” one of Pollock’s last drip paintings, now valued at more than $30 million, was painted over a period of six months and boasts the highest fractal dimension of any Pollock painting Taylor tested.: a 1.72. Pollock was apparently testing the limits of what the human eye would find aesthetically pleasing. Pleasing fractals To really find out if Pollock’s fractals account for his lasting appeal, Taylor next invented a device he calls the Pollockizer. It consists of a container of paint hanging from a string like a pendulum, which can be kicked into motion by electronic coils near its top. Now, when the container moves, a nozzle at the bottom flings paint on a piece of paper on the ground directly beneath it. By tuning the size and frequency of the kick, Taylor could make the Pollockizer’s motion chaotic or regular, there by creating both fractal and non fractal patterns. When Taylor surveyed 120 people at random to see which patterns they preferred, 113 (all but 7) chose the fractal patterns. Now, two recent studies in perceptual psychology had also found that people most clearly prefer fractal dimensions similar to those found in nature. But, the studies disagree on the exact value of that dimension: In one study,
subjects preferred a dimension of .3; in the other, 1.8. Last year, after
relocating to the University of Oregon, Taylor collaborated with perceptual
psychologists in Australia and England to see if they might be able to resolve
the discrepancy. The teams began by dividing fractal patterns into three
categories: natural, computer-generated, and man-made----the last category
consisting of cropped sections of Pollock’s famous drip paintings. Next they
asked 50 subjects to evaluate about 40 patterns each, with each test subject
having to choose between just two patterns at a time. The results, published
in The same predisposition seems to be at work in other mediums as well. Studies have found that people prefer patterns that are neither too regular, like the test bars on a television channel, nor to random, like a snowy screen. They prefer the subtle variations on a recurring theme in, say Beethoven concerto, to the monotony of repeated scales or the cacophony of someone pounding on a keyboard. According to James Wise, an adjunct professor of environmental science at Washington State University and one of Taylor’s collaborators, those preferences may date back to our earliest ancestors. On the African savanna, they could tell if the grass was ruffled by the wind or by the stalking lion by tuning in to variations in fractal dimensions. But in settings with high fractal dimensions (a densely branching rain forest, for instance) early humans would have ben vulnerable----and thus more uneasy. “Perhaps our appreciation of lower-dimension fractal patterns isn’t so much about beauty,” Taylor says, “but more about our survival instincts.” Architects, writers and especially musicians may instinctively appeal to their audiences by mimicking the fractal patterns found naturally in nature. In Pollock’s case, at least, the inspiration seem clear. He began his very first series of drip paintings, with their tightly meshed surfaces, soon after moving from Manhattan to a farmhouse on Long Island, N.Y., in late 1945. The paintings are worth more than their price tags, Taylor says. “If someone asked, ‘Can I have nature put onto a piece of canvas?’ the best example there had ever been of that in his 1948"s ‘Number 14.,” Return to the words of wisdom, science index..
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