Cynthia Lanius

Fractal Properties

Iterative Formation

 
Fractals are often formed by what is called an iterative process. Here's what I mean.

To make a fractal: Take a familiar geometric figure (a triangle or line segment, for example) and operate on it so that the new figure is more "complicated" in a special way.

Then in the same way, operate on that resulting figure, and get an even more complicated figure.

Now operate on that resulting figure in the same way and get an even more complicated figure.

Do it again and again...and again. In fact, you have to think of doing it infinitely many times.

 
You can observe this iterative process in all the fractals that we make in this unit:

Not every iterative process produces a fractal. Take a line segment and chop off the end. What is the resulting figure? Just another line segment - not "complicated" at all, and not a fractal. You could continue the iterative process over and over, chopping off the end of the line segment, but it would just become a shorter and shorter line segment, not "complicated", not fractal.

Now here's a similar iterative operation that is fractal. Take a line segment (see below) and remove the middle third. What is the resulting figure? Hmmm. That's a more complicated figure. It's a line segment with a hole in it.

Repeat the process on that figure. In other words, remove the middle third of both of those sections. This produces an even more complicated figure. Now think of doing this infinitely many times. In fact, this is a famous fractal called Cantor Dust

Cantor Dust

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