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Ole! Imagine a bronco bucking furiously around a corral without trying to jump out of it. Would you believe a mathematical equation could behave in a similar way? This fractal was generated by a computer program written by Clint Sprott, UW-Madison professor of physics. Fractals represent solutions to mathematical equations.
Sprott's program scans millions of equations to find a few thousand — maybe 1 percent of the total — that have "chaotic" solutions. For Sprott, chaos is not disorder, but rather "unpredictable behavior within a system governed by laws of nature — variety within structure." Mathematical laws govern equations -- think of them as the corral fence and gate. Sprott's computer program seeks out equations that have two initial values or starting points that end up separating at an exponential rate.
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. The solution -- our bucking bronco -- never settles down to a fixed value or even to a repeating pattern, but neither does it move off to infinity. Equations with chaotic solutions are unstable but bounded.
Whoa! Click here to see more fractals by Clint Sprott.