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hat Are Fractals?

This is a simple question with a very complicated (and very long) answer. A technical answer, while accurate, doesn't help much because it uses other fractalspeak jargon that few people understand, so I won't even give that definition here.
    The simple answer is that a fractal is a shape that, when you look at a small part of it, has a similar (but not necessarily identical) appearance to the full shape. Take, for example, a rocky mountain. From a distance, you can see how rocky it is; up close, the surface is very similar. Little rocks have a similar bumpy surface to big rocks and to the overall mountain.
A Simple Example

full Julia
first zoom
second zoom
This concept of self-similarity can be a little hard to come to grips with, but it's a fundamental part of fractals. So take a look at this first image—from a Julia set, a very simple fractal type. I've highlighted a small box near the left side (it's a little faint). The portion of the image in that box is shown in the second image, below. That image also has a small box highlighted, and that area is shown in full in the third image.
    You can see in these images that smaller areas of the fractal shape look very much like the larger, full-size image. With Julia fractals, you can continue this enlarging ("zooming") process as often as you like, and you will still see the same sort of details and shapes at very tiny sizes that you see on the full-size image. This is what is meant by self-similarity.
    Now of course, with something so rigidly self-similar, there's not really much point in zooming in. After all, everything is the same; small detail looks like large detail. So while it's interesting that fractals are self-similar, if this is all there is to it, there isn't much point.
Not Quite So Similar

Fortunately for fractal enthusiasts, that isn't all there is to it. Many fractal types get wildly different as you zoom in. They're still self-similar, but they're not rigidly self-similar.
    This is what makes fractal exploration so intriguing. The features you see as you zoom are always changing—teasing you with a little bit of familiarity, and tantalizing you with new and unexpected twists. With just a single fractal shape, you can explore forever and never see everything it has to offer. The further you zoom, the more likely you are seeing something that nobody has ever seen before. And with modern computers, it's very easy to zoom and zoom and zoom. With just a few clicks you can have zoomed so far that the original fractal image is larger than the sun.
full Julia
first zoom
second zoom
View a Flash animation of this zoom sequenceView a Flash animation of this zoom sequence     Take, for example, the zoom sequence on the left. Starting with the upper left image, the center of each image was magnified tenfold. Whereas the Julia image looked virtually identical as it was zoomed, this Mandelbrot image shows new variations as the magnification increases. (Because of the small size of these images, it may not be obvious that each image is a magnification of the previous one, but they are.)
How It All Works

The basic technique of these fractals can actually be explained without resorting to confusing mathematical equations and jargon. It's rather simple, really.
    First, give every point on the screen a unique number. Now take that number and stick it into a formula; you'll get a result from the formula. Take that result and stick it back into the formula. Keep doing this and watch what happens to the numbers you get. Color each point based on what happens.
    That's it. Really—that's it. Now, with most formulas it probably won't do much of interest, but with the formulas used in fractal creation, some interesting things happen. Sometimes the numbers you get by feeding the results of a formula back into the formula (iterating) explode into enormous numbers, that just keep getting bigger and bigger. Those points get colored one way. Other times, the numbers "home in" on a number, getting closer and closer to it. They get colored a different way.
    The interesting thing—and the reason fractals work at all—is that sometimes, just a tiny little change in the number you start with can completely change what happens as you keep iterating the number. And the boundary between numbers that explode and numbers that home in is complicated and twisted—it's the shape of the fractal.
The Enormous Task at Hand

Calculating fractals this way involves a lot of work. A small fractal image—perhaps only 640x480—contains over 300,000 points. Each of those points may require running a number through the fractal formula more than 1,000 times. This means the formula has to be computed more than three hundred million times. And that's a mild example. Extreme images (such as poster-size fractals) can involve more than one trillion calculations.
    Fortunately for the impatient among us, modern computers are fast enough to do the job in a few minutes. Large fractals might take hours or days, but exploring fractals has never been easier.
Where To Go From Here

As I stated at the outset, fractals are a huge topic. All I've even talked about here are one particular type of fractals (escape-time fractals), but there are many other types as well. Unfortunately, the further you look into fractals, the more math you will need to know. There are very few fractal-related books or web pages that don't get into heavy mathematics. I've attempted to assemble some pages here that will get you started on the mathematics behind fractals in an accessible fashion, but there is no hiding the fact that it is math.

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