They are naturally occurring examples of fractals.

Before fractals were understood, cartographers were perplexed by the difficulty in measuring the length of coastlines accurately. The shape of an island as seen on a world map appears to be a simple variation of a circle. As your view magnifies, bays and peninsulas begin to appear, then inlets and coves. The coastline appears more jagged the closer you get. The effect is that as your measuring stick gets smaller the distance you cover gets longer. Imagine a giant and an inch worm traveling the edge of Austrailia as closely as they can—the inchworm has a lot farther to go!

The increased complexity of what appeared as a simple line begins, with all its zigs and zags, to fill up space. What results is more than a one-dimensional line yet less than a two-dimensional plane. This is a glimpse of what it means to have fractional dimension, a fundamental characteristic of fractals.

In the 1970's the Polish mathematician who coined the term "fractal", Benoit Mandelbrot, created one of the most recognizable illustrations in mathematics. His creation, now known as the Mandelbrot set, is a connected set of points in the complex plane. This set is determined by evaluating each point, one at a time. First, feed the point into a simple equation, then take the result and feed it back into the original equation, then take that result and feed it back into the original equation. This is repeated over and over again until a pattern is seen to emerge: either the result is diverging from or it is staying close to the initial point that was tested. If the result stays close, the point is said to be in the Mandelbrot set and is typically colored black. If a result diverges, it is given a color which varies in hue depending upon how many times it has to go through the equation before it's clear that it will (so to speak) not be coming back.

The light show Fractal Journey follows a route of magnification through the Mandelbrot set, introducing us to another fundamental property of fractals. As we zoom in on the boundary between set and not-set, we begin to pick out a seemingly infinite number of near-exact copies of the original Mandelbrot set embedded in points that appeared to be outside the set. This is known as infinite self-similarity and is seen throughout the natural world, as well as in this pure realm of mathematics.

Note how an individual finger of a fern leaf resembles the whole leaf; how a jumble of rocks at the foot of a mountain resembles the mountain; how dust clouds resemble hurricanes; how twigs are like branches; how atoms resemble solar systems.

Explore fractal art galleries, or get software to make your own fractals.