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This page explains what a fractal is and how the Mandlebrot set is generated.

A fractal is an object that repeats itself infinitely. Some examples of this are the coast of a country, a snowflake, or a tree. Really, anything is a fractal to some extent. The term fractal really just means that an object's complexity never ends. Without going into theoretical physics, I think I can safely say that most objects have close to infinite complexity. If you take a bowling ball, which looks smooth, and magnify it, you will see the smoothness fade away. Soon it will get bumpy and jagged, zoom in farther and you will see atoms, then the nucleus, then protons, then quarks, etc. It would seem that no matter how far we 'zoom' in, the bowling ball just keeps giving us new things to look at. The same holds true for the Mandlebrot set. If you zoom in farther, you will keep seeing new images, each more fascinating that the previous. You may wonder then, how is the Mandlebrot set generated? When I first became interested in computer generated fractals, I was frustrated at the quick and shallow explanations that were given, so I will try to make this understandable. The Mandlebrot set is generated on a complex plane. This is similar to a normal graph with x and y coordinates, with one exception. The x, or horizontal axis, is the real number line. The y, or vertical axis, contains imaginary numbers. So, a point (which in this plane is a complex number) would be graphed like so: The point given is 4-2i, so on the graph, we go over 4 and down 2.

So, now that you understand that, let's go onto how to multiply complex numbers. For an example let's multiply (3-i)*(2+i).

So, (3-i)*(2+i)=7+i, now lets see what this has to do with the Mandlebrot set. The algorithm used with the Mandlebrot set is X² + c. c stands for the original complex number, and X is the product of the last iteration. If that seems muddled, here is a clearer example. C will stand for the complex number in the following equations, while Z(n) will stand for the n^th time the calculation has been made.

There, the distance from the origin (the absolute value) is greater than 2, this means that each further calculation will get bigger and bigger. In a manner of speaking, the value has 'escaped'. Now, it took 4 iterations (Z(4)) for it to escape, so we color the original point (-1 + .5i) the color 4. The actual color of 4 can change, but by the pictures on the top of this page, it is light brown (You can see the color changes distinctly by clicking on the image for a blowup). On the graph, it would look like this:

Now, imagine that this is done for every point on the graph, with each point sent through this formula, and each point given a color. That's what the pictures are. This is what the mandlebrot pictures could look like; with the set of axis on top:

In the above picture, each hash mark is roughly at 2 away from the center.

If you want to download some more pictures of fractals, click here

If you want to make your own fractals with a java program (they're small, but cool), click here

If you want to download a program that I wrote (Size: 98k), then click here. It will generate fractals much quicker than the java program. It runs under DOS, and is fairly quick if you have a Pentium-based computer.

Have any questions, complaints, or corrections? E-mail me!

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