Fractal techniques can produce scenes which mimic nature in remarkable ways. Because natural objects are often made up of smaller pieces, each of which is roughly similar to the whole, natural looking scenes can be produced by recursively applying a simple set of rules.
The applet below renders a simple wire frame drawing of a hillside. It uses the random midpoint displacement method to generate the three dimensional points of the mesh. Starting with a basic square, it repeatedly splits each edge in half, raising or lowering the midpoint an amount proportional to its distance from the end points of the edge. Clicking the mouse over the applet will cause a new scene to be produced.
The Koch Curve is an example of a fractal created by a replacement rule. Such fractals begin with a simple image. In the case of the Koch curve and its variants, this image is a straight line. This image is then changed to something else, based on the replacement rule. The new image contains elements which correspond to elements in the original image. The replacement rule is then applied to these portions of the image, creating a more detailed picture. This process continues indefinitely, creating infinite detail from a simple picture and a replacement rule.
If you don't have a Java enabled browser, the two images below show the Koch curve and one of its variants. If you do have a Java enabled browser, clicking on the image below will cause the replacement rule to be applied to whatever image is currently displayed. The applets have different replacement rules, causing them to create two variants of the Koch curve.
Perhaps the most amazing thing about fractals is that totally random processes can lead to totally deterministic results. The Chaos game is an example of such a processes. To play the Chaos game, pick a set of transformations, and a single arbitrary point. Plot that point, and pick one member of the set of transformations at random. Apply that transformation to the coordinates of that point to obtain a new point. Plot that point and start again. Because each new point is determined at random you never know what point will be plotted next. However, if you play long enough with the same set of transformations you will always gets the same final image. You can control the final image by carefully selecting the transformations in your set.
The applets below show two variations on the chaos game. One plots the Spierinski gasket, the other displays the Barnsley fern. Clicking on one of the applets after it is done displaying will cause it to erase and start again.
One of the most famous fractals is the Mandelbrot set. Like all fractals the Mandelbrot set contain infinite detail. Unlike many man made fractals, this detail varies the closer you look. Each closeup of the set is similar to the previous image, but has its own unique twists and turns. These twists and turns continue indefinitely, provided that the set is computed with adequte precision.
The applet below allows you to explore the Mandelbrot set. The applet's drawing area is divided into nine regions, like a tic tac toe board. Clicking on one of these regions expands it to fill the entire region.