To learn why simple formulas can create complex patterns, we will start with the example of a very simple formula:

new x = bx(1 – x)

Let’s try iterating it for different values of b. We will start with any number between 0 and 1, let’s say .234. For b=1.5 we get the following values:

.234, .269, .295, .312, .322, .327, .330, .332, .333, .333, .333, ...

After a while, the number stays in the same place, called a fixed point. Now let’s try b=3.2:

.234, .574, .783, .544, .794, .524, .798, .516, .799, .513, .799, .513, ...

Now, after a while, the number starts jumping back and forth between two different numbers. For b=3.45 we find that the number jumps around 4 different numbers, and for b=3.54 it jumps around 8 different numbers. As we increase b, the size of the number’s cycle doubles. This is called bifurcation. Now try this for b=4. You will find that the number doesn’t seem to settle anywhere. This phenomenon is called chaos. A very interesting diagram can be made by plotting the values that a number settles two against the value of b. What we get is the famous bifurcation diagram:

Now try magnifying the diagram...

It’s a fractal!!! Indeed, the diagram contains smaller versions of itself at all levels of magnification. Sometimes, it is referred to as the Feigenbaum Fractal. We have created a fractal out of a very simple formula. This shows the basic connection of fractals and chaos. Amazing, isn’t it?