Fractals images reveal complex dynamics

Some surreal sights are hanging on the walls of the Physics Library in Chamberlin Hall, phantasmagoric art that flows from a surprising union.

You can see what you want to see in these framed images of colored lines. You might think they resemble a tropical fish as reflected in an amusement park mirror, a T-Rex head, or a bird’s-eye view of a river valley in the Land of Who-Knows-Where.

Then again, you might not — that’s the eye-of-the-beholder beauty of fractals, which is what you call these other-worldly images. They were recently given to the Physics Library by Clint Sprott, professor of physics and fount of fractals.

Sprott wrote the computer program that generated the fractals, also known as strange attractors. In fact, he wrote a book on it: “Strange Attractors: Creating Patterns in Chaos.”

Fractals are the offspring of the marriage of art, science and technology. They are generated by a computer to graphically represent solutions to mathematical equations. And in Sprott’s case, not just any solutions, but “chaotic” solutions.

“It all has to do with chaos,” says Sprott. “Chaos can be defined as unpredictable behavior within a system governed by laws of nature — variety within structure.”

Chaos can be seen in the weather or clouds or a turbulent stream. For example, if two sticks are dropped side-by-side into that stream, they will quickly separate in unpredictable ways — aka chaos.

That “separation” can also be seen in the weather. “A tiny change in today’s weather — initial conditions — can cause a larger change tomorrow and an even larger change in the next day’s weather,” says Sprott.

Sprott designed his computer program to scan millions of equations to find a few thousand — maybe 1 percent of the total — that have chaotic solutions. That is, it searches for an equation that has two initial values or starting points that end up separating at an exponential rate, just as those sticks do in the stream.

“Equations that are chaotic have solutions that are unstable but bounded,” says Sprott. “The solution never settles down to a fixed value or even to a repeating pattern, but neither does it move off to infinity.” Controlled chaos, in short.

In this kind of equation, the initial values are drawn to a “strange attractor,” which is a complicated geometric object, or fractal. Because those values don’t shoot off into infinity, the lines in Sprott’s printed fractals don’t shoot off the paper.

French mathematician Henri Poincaré knew of the existence of fractals a century ago, but he faced a huge hurdle in producing them: no computers.

“Even 10 years ago, computers and printers capable of producing high-resolution fractals were very expensive,” says Sprott. “More recently, it took 24 hours to print a fractal, using just eight colors, but now it takes minutes, using 256 colors and producing 25 times better resolution.”

Fractal art wasn’t what Sprott set out to produce. It was a byproduct of his study of dynamical behavior exhibited by simple equations and of the probability that such behavior is chaotic.

But it’s been a bountiful byproduct. “It opens up a whole new world for me,” he says. “Fractals are a form of scientific visualization, which is a hot topic now that computers and printers are improving. If you can see something better, you can understand it better. For example, if we can visualize stock market variables better, then we can plot trends more accurately. The same goes for predicting weather.”

If you find Sprott’s fractals aesthetically appealing, it’s likely due to what they remind you of, at least unconsciously.

“There is probably a deep connection between fractals and nature,” says Sprott, “because they both are produced by dynamical systems that are unpredictable.” In other words, the fractals in the Physics Library are chaotic cousins of the wind-whipped froth on Lake Mendota.