by Angela Spivey

Yes, numbers misbehave -- it's only natural.

his story begins with a pencil, a piece of paper, and a set of numbers. You might hear “set of numbers” and think orderly, prim. But not these. These numbers first began causing trouble around 1900.

“A group of French and British mathematicians were trying to approximate solutions to some equations,” says Jane Hawkins, professor of mathematics. “But some of their attempts at approximation led to chaotic outcomes. The solutions never formed a usuable pattern. And they simply couldn’t explain it.”

To picture chaotic behavior, think of a waterfall. It’s a given that the mass of water will eventually go over the fall. But how each individual particle will get there, in what pattern, is completely unknown until it happens.

So these numbers were doing something similar, only mathematically. “They had been picked up, looked at, and put down many times over a century,” Hawkins says. “This set had certain features that everyone could understand and certain features that were completely intractable.”

You’ve probably guessed that these aren’t the kind of numbers you find in your checkbook. Hawkins’ number “family,” as mathematicians call it, is a group of functions, which are expressions of relationships between sets of numbers. Her family is written this way: f(z) = a(z + 1/z + 2).

Both a and z are complex numbers (see More About the Math). For each value of a, a different function results. All these related functions make up the family. Hawkins also refers to it as a family of mappings, because each function is represented by a different point on the complex plane.

ack in 1992, Hawkins reviewed a book that discussed using computers to help mathematicians visualize the behavior of such numbers. The book also contained a theorem (a mathematical proposition that can be proved using accepted rules) that gave the mathematical conditions under which you should expect a function to show chaotic behavior.

Later, in 1995, at a math conference in California, Hawkins gave a preliminary talk about using this visually oriented math to figure out these numbers. Someone at the conference whom she respected told her he liked the idea. So on the plane home, Hawkins sat down with her pencil and this number family and started doing some algebra. “I knew that there were functions that had to give chaotic behavior, but I didn’t know where to look for them,” she says. So she solved for the values of a that would produce functions that met the criteria set by the theorem from the book.

“I was using some things that we use in junior high and high school math—just solving for some number,” she says. “This was an easy problem to solve. Just realizing that this was how to set up the problem was the hard part.” After solving by hand for a few values of a, “I decided that I should use this algorithm that I used with my pencil, but let the computer do it faster.”

Next: "A mathematician must prove it"
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