Iterated Function Systems

In this section we will learn a different way to create geometric fractals. Instead of repeated removal as in the Sierpinski fractals, we will now explore the process of Repeated Substitution. These types of fractals are called Iterated Function Systems, or IFS Fractals.



The sequence above shows the development of the Dragon Curve fractal, which is formed by repeated substitution. The image on the left, labeled '0' is called the generator. The first iteration is formed by replacing each half of the dragon curve with a smaller copy of the same shape, rotated to fit. This process keeps repeating, where each iteration has twice as many copies of the original generator.

Click the box above to step through the iterations of the Dragon Curve. (Reload the page to start over.)

Question:
How many line segments are in the 4th order shape? [ ]

Next we'll explore a famous IFS fractal known as the Koch Curve. Here the generator is simply a line with a triangular bump in it, shown below.

Change the iteration number in the control panel at the top right of the applet. At iteration 2, each of the 4 line segments in the generator is replaced by a copy of the generator. In Chapter 9, when we explore fractal dimensions, we will revisit this curve and learn how the length increases with each iteration, until it eventually becomes infinitely long!


    


We can create Iterated Function System fractals with more complicated generators, as in the two examples above. Click on each of them to see them evolve. In the next lessons, we will learn about Lindenmayer Systems, which provide us with a language to define the substitutions at each iteration. Also known as L-Systems, they allow us to use the process of iteration to replicate the forms of many common plants.


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