# Iterated Function Systems

Iterate means to repeat an operation, generally using the last result of that operation as the input. An iterated funtion system then is a system where a function is repeated over and over again to itself. The patterns on this page were created using iterated function systems. When IFS's are implemented using computer programs, all kinds of fascinating designs can be made. To make an IFS you start out with figures such as a triangle. Next you make a function. This function will be: Whenever you see a:

you replace it with a:

So if you begin with a triangle, the first iteration would be:

Next you take the same function and repeat so you get:

after infinite iterations you get the Von Koch snowflake:

Here is another example. Whenever you see a:

you replace it with a:

So we start with a line, and the replacement figure after the first iteration is:

After two iterations, we get:

In each iteration the starting shape is replaced by an appropriately scaled, turned, rotated and, if neccessary, flipped over copy of the original figure. After the third iteration:

After the fifth iteration you get this figure by Mandelbrot:

When trying to measure the length or area of an Iterated Function System you will find it hard to get a result. The Von Koch replacement rule for every iteration increases the length by a factor of 4/3.

In the first of the two preceding images, you begin with a line, say that it's length is one. Then we can say it is made up of three segments with length 1/3. The second figure, which is after one iteration, is made up of four such segments. So its length is 4/3. After another iteration, the length will be (4/3)*(4/3) or 16/9. The length keeps increasing. After infinite iterations the length of the Von Koch arc would be infinite. Computing area can be tricky, too. Take the Sierpinski triangle. You start with a triangle:

The function is to cut out the center of any:

Like this:

After the 2nd iteration:

After more iterations:

The area begins at, say, one. After one iteration, one quarter of it has been removed, so the new area is 3/4. After the next, one quarter has again been removed, so the area is (3/4)*(3/4) or 9/16. As we keep iterating, the are keeps going down. After infinite iterations, there is nothing left. The area of the Sierpinski triangle is zero.