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Complexity Digest 2000.17 - 02
http://comdig.unam.mx/index.php?id_issue=2000.17#554
24-Apr-2000

Nature's Cycles In A 	Fractal State Of Mind, Rochester Univ/Science Daily

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 Jorn&eacute; 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
Jorn&eacute;, professor of chemical engineering. Jorn&eacute; 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 Jorn&eacute;. "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." Jorn&eacute; 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."

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, Jorn&eacute; 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 Jorn&eacute; 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
Jorn&eacute; 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.

Nature's
Cycles In A Fractal State Of
Mind, Science
Daily, Posted 4/21/2000

Press
Release
Rochester Univ