Fractals in computing

Fractals and the related idea of chaos have profoundly changed the way scientists think about and model the world. Around 1960, Benoit Mandelbrot noticed that sup-posedly random economic fluctuations were not distributed evenly but tended to form “clumps.” As he investigated other sources of data, he found that many other things exhibited this odd behavior. He also discovered that the patterns of distribution were “self-similar”—that is, if you magnified a portion of the pattern it looked like a miniature copy of the whole. Mandelbrot coined the term fractal (meaning frac-tured, or broken up) to describe such patterns. Eventually, a number of simple mathematical functions were found to exhibit such behavior in generating values.

Fractals offered a way to model many phenomena in nature that could not be handled by more conventional geometry. For example, a coastline that might be measured as 1,600 miles on a map might be many thousands of miles when measured on local maps, as the tiny inlets at every bay and beach are measured. Fractal functions could repli-cate this sort of endless generation of detail in nature.

Fractals showed that seemingly random or chaotic data could form a web of patterns. At the same time, Mandel-brot and others had discovered that the pattern radically depended on the precise starting conditions: A very slight difference at the start could generate completely different patterns. This “sensitive dependence on initial conditions” helped explain why many phenomena such as weather (as opposed to overall climate) resisted predictability.

Computing Applications

Many computer users are familiar with the colorful fractal patterns generated by some screen savers. There are hun-dreds of “families” of fractals (beginning with the famous Mandelbrot set) that can be color-coded and displayed in endless detail. But there are a number of more significant applications. Because of their ability to generate realistic textures at every level of detail, many computer games and simulations use fractals to generate terrain interactively. Fractals can also be used to compress large digital images into a much smaller equivalent by creating a mathemati-cal transformation that preserves (and can be used to re-create) the essential characteristics of the image. Military experts can use fractal analysis either to distinguish artifi-cial objects from surrounding terrain or camouflage, or to generate more realistic camouflage. Fractals and chaos the-ory are likely to produce many surprising discoveries in the future, in areas ranging from signal analysis and encryption to economic forecasting.