Fractals: Jagged Symmetry

October 1, 2009
By arrangement: Computer-generated fractals include Julia set

By arrangement: Computer-generated fractals include Julia set. Solkoll

From the curl of a fern frond to the coastline of Britain, the shapes we see in nature are strikingly different from the squares and circles of traditional geometry. Jagged and seemingly random, many nevertheless possess a surprising symmetry: Zoom in on a picture of something very large—that coastline of Britain, say—and you'll begin to notice that the same types of geometric patterns visible in a satellite photo keep recurring as the magnification increases, right down to the pebbles on the beach. In the terminology of mathematicians, the coastline of Britain is "self-similar," repeating the same geometric motifs over and over on every scale.

More than thirty years ago, Benoit Mandelbrot coined the term "fractal" to describe these surprising geometries, which include everything from foliage patterns to animal camouflage. This may seem surprising, until we recognize that most fractals (including computer-generated fractals like the Mandelbrot and Julia sets) are produced by the infinite repetition of some procedural rule, what mathematicians call recursion. Because most features of the natural world are built by the same processes repeating themselves over and over (think of a branching tree), it's no wonder that so many of the patterns we see around us end up being fractal. Even chaotic systems exhibit hidden patterns of order that, when plotted graphically, turn out to be fractals.

The most famous fractals, however, are entirely artificial, the creations of movie special-effects artists. Because fractals so closely mimic real plant life and terrain, and because computers so easily perform the computations needed to generate them, fractals have been used to build everything from the second Death Star in Return of the Jedi to the dinosaurs of Jurassic Park.