The Koch Curve
The Koch curve is another famous basic fractal system. To construct it, we start with a line segment:
We then trisect the segment, form an equilateral triangle from the middle third of the segment, and then erase that middle third. The result looks like this:
This is one iteration. The source code can be found here.
Subsequent iterations will produce the following:
and so on.
Simple modifications can made to the IFS to make more complex-looking attractors. The Koch curve is a good system with which to show this.
By simply scaling the triangle and moving it to the left (source code here), we can create a modified Koch curve:
We can then continue to iterate this new Koch curve, and the results are rather dramatic:
The attractor looks like a forest of bare trees!
And this was created just by making a few simple changes to our
Koch curve model.
This shows the beauty of fractals - they can be as simple or as
complex as you want them to be.
As we have just seen, fractal systems can be used to create images that appear quite natural. And in fact, fractals are often used to simulate nature. We explore this in more detail in section 3.1.