April 18, 2000

Weather, battery life and even the way your lawn grows are all linked by four mathematical laws, according to a paper published in the April 3rd *Physical Review Letters*. Yonathan Shapir and Jacob Jorné of the University of Rochester have shown how natural cyclical events, such as seasonal weather, generate very specific patterns-the same patterns that govern the geometric images called fractals.

Fractals are mathematical designs that repeat their patterns on infinitely smaller scales: No matter how much you magnify a fractal, the same patterns appear. These patterns can be created over time. As sediment builds up on a surface, for instance, the tiny irregularities in the first layer become larger and more exaggerated in successive layers as they are laid down. Scientists have shown previously that many structures in nature, from lightning bolts to cauliflower heads, produce this fractal pattern, but the new findings are the first to demonstrate that the fractal patterns hold true for nature's next level of complexity, cycles.

"Often things are not formed by a single process, but by a combination of growth and recession," explains theoretical physicist Shapir. "What's amazing is that so many growth and recession cycles can be described by just a few fractal solutions."

Fractal solutions-equations with numbers that create fractal patterns-can help predict events that are based on natural cycles that build up and break down materials over and over, explains Jorné, professor of chemical engineering. Jorné and Shapir expect that fractal equations can help physicians estimate the spread of cells that grow and recede, such as a tumor in a chemotherapy patient. They also expect that the life span of car batteries can be predicted faster and more cheaply because engineers will be able to extrapolate the data from a few charge cycles to thousands. Even predicting such seemingly random things like how your lawn will spread may be possible by measuring rain and light cycles and matching them to the proper equation.

"This work shows that there are some basic laws underlying many of nature's cycles," says Jorné. "They may not be obvious, we may not see the connections at first, but underneath it all the same patterns are running."

Jayanth Banavar, head of physics at Penn State University and an expert on fractal phenomena, said, "This work is very exciting and opens entirely new avenues for future investigations. Besides its scientific interest, this work promises to have important technological ramifications." Jorné first approached Shapir with a simple question: Would natural cycles create fractal patterns?

"I had a hunch they would," says Shapir. It took him several months of mathematical tinkering, however, before he discovered the right approach. "The hardest nut to crack was how to make a certain, very complicated mathematical framework fit this experiment." That complex framework, known by the equally complex name "Renormalization Group Theory," helps reveal fractal-like properties in equations, and earned its developer the Nobel Prize in physics in 1982. "Once we understood how to apply it to cycles, everything fell into place in a matter of days."

Shapir and graduate student Subhadip Raychaudhuri used a computer to run cycle simulations. Tiny objects were randomly deposited on different types of surfaces. After each deposition, the researchers simulated a process, like water erosion or battery discharge, that removed some of the objects in an equally random way. After running the simulations tens of thousands of times, Shapir and Raychaudhuri found that no matter what the type of objects, forces or surfaces involved, each of the simulations could be described by fractal solutions. As each new layer of objects was laid down, its surface became more and more irregular, repeating the same basic shapes on larger and larger scales, just like a fractal.

With the simulation results in hand, Jorné and David G. Foster, a former graduate student and senior engineer at Eastman Kodak Co., designed an experiment that deposited atoms of silver onto an electrode for five minutes, followed by a reverse in charge to remove some of the silver for two and one-half minutes. The silver atoms accumulated in a fractal pattern just as predicted.

Shapir and Jorné already see practical applications for their findings. Often rechargeable batteries fail because each charge deposits material inside the battery, and each discharge charge removes some of that material. After several such charge cycles, the buildup can span the two leads inside the battery and short it out. Since the material does not accumulate in a uniform fashion, battery makers have had to test batteries by discharging and recharging them over and over until they fail. Shapir and Jorné think that with fractal equations, manufacturers can run through only a few charge accelerated cycles and calculate how long it will be until the battery fails without doing expensive, prolonged testing.

The research was funded by the National Science Foundation, the Office of Naval Research, and Kodak.

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