Mentor:Chaos has been found in all branches of science; in fact most scientists now think that chaos is
the rule in nature -- not the exception! Mitchell Jay Feigenbaum was a mathematician and
physicist who was a pioneer of the chaos theory. He discovered that systems show consistent
patterns as they approach chaos.

Student: What do you mean when you say, "they show consistent patterns as they approach chaos?"

Mentor: Well, he discovered that if two numbers very close together in value were input into the same
formula, the numbers become very different from one another after several repetitions of the
formula. He found two constants, which became known as Feigenbaum numbers, to represent these
patterns in chaos. Feigenbaum also used these new fractal methods in his work with
computerized mapmaking. He was awarded the Wolf Prize in Physics in 1986 and in 2005 he was
awarded New York City Mayor's Award for Excellence in Science and Technology for his studies
in chaos theory.

Student: Who else had a large impact on the chaos theory?

Mentor: A man named Edward Norton Lorenz studied weather patterns and population. He built a
mathematical model to show the way that air moves around in the atmosphere and from this he
coined the term
butterfly effect.

Student: I've heard of that before! It means that small initial changes can result in much larger
changes.

Mentor: That is right! The term butterfly effect came from the idea that the motion caused by a
butterfly's wings can cause small changes in the atmosphere, which eventually could result in
the impact of a tornado such as where it occurs or how strong it is. Big changes can occur as
a result of one small event. There is also a movie about this phenomenon, titled
The Butterfly Effect, which was released in 2004.

Student: Yeah, I think I've heard about that movie. I think that I've seen the idea of the butterfly
effect in several television shows, too. Sometimes, someone will do something and then you
will see what happens as a result of that action and what would have happened if they had not
done the action.

Mentor: Yes, exactly! It all relates to large effects because of a minor change in some system.
Another person that dealt with chaos theory was Aristid Lindenmayer. He modeled plant growth
and development in order to create realistic plant images and animate their growth processes.
In 1968 he introduced "L-systems."

Student: What are "L-systems?"

Mentor: "L-systems" (short for Lindenmayer Systems) are sets of rules and symbols that model growth
processes. Lindenmayer used them to model plant growth. He also published a book,
The Algorithmic Beauty of Plants, in 1990.

Student: Who are some other people that studied chaos theory?

Mentor: Biologist Robert May studied animal populations and, more specifically, predator-prey
systems. He examined natural communities through mathematics. May published
Stability and Complexity in Model Ecosystems in 1973. You can learn more about his ideas related to population dynamics through the applet
Rabbits and Wolves.

Student: Gosh, chaos is everywhere!

Mentor: Yes, and that's not all. Hans Meinhardt studied patterns and pigmentation on seashells. He
wrote
The Algorithmic Beauty of Sea Shells in 2003. Also, James D. Murray studied the patterns of fur markings on mammals. He published
Experimental and Theoretical Advances in Biological Pattern Formation in 1993. The work that these men did is closely related to chaos theory and fractals.

Student: Wow, all of these people have created models to explain different patterns in nature. I had
no idea that there were so many different patterns that could be explained mathematically.

Mentor: For even more experience with the idea you should try the
Life and
Rabbits and Wolves activities!