Suppose that a path connecting two arbitrary points, A and B, is defined (as shown by the colored line segments below):

Let's see what happens if we now apply the path to itself (by replacing each of the individual line segments with a smaller copy of the entire original path):

But let's not stop here! Let's continue to replace each of these new smaller line segments with even smaller copies of the original path. And then, let's replace each of those...

If we continue this recursive process forever, we end up with a fractal called a Dragon Curve: