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How Fractals Work


Fractal Terminology
Fractal art may initially appear random and disjointed, but closer inspection reveals a repeating structure.
Fractal art may initially appear random and disjointed, but closer inspection reveals a repeating structure.
Courtesy Frances Griffin

Before we get into any more detail, we need to cover some basic terminology that will help you understand the unique qualities that fractals posess.

All fractals show a degree of what's called self-similarity. This means that as you look closer and closer into the details of a fractal, you can see a replica of the whole. A fern is a classic example. Look at the entire frond. See the branches coming out from the main stem? Each of those branches looks similar to the entire frond. They are self-similar to the original, just at a smaller scale.

These self-similar patterns are the result of a simple equation, or mathematical statement. Fractals are created by repeating this equation through a feedback loop in a process called iteration, where the results of one iteration form the input value for the next. For example, if you look at the interior of a nautilus shell, you'll see that each chamber of the shell is basically a carbon copy of the preceding chamber, just smaller as you trace them from the exterior to the interior.

Fractals are also recursive, regardless of scale. Ever go into a store's dressing room and find yourself surrounded by mirrors? For better or worse, you're looking at an infinitely recursive image of yourself.

Finally, a note about geometry. Most of us grew up being taught that length, width and height are the three dimensions, and that's that. Fractal geometry throws this concept a curve by creating irregular shapes in fractal dimension; the fractal dimension of a shape is a way of measuring that shape's complexity.

Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through infinite detail. Simple, right? Don't worry, we'll go over all the pieces soon enough.


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