Fractal Interpolation
Interpolation is a term used in statistics and in several other branches of mathematics. It deals with trying to fit a graph to a model (usually a line, but not always). We can use fractals to illustrate this concept.
Suppose we start with a square:
And then we divide the square into fourths and remove the upper left and lower right fourths:
We can use this as our iteration, and repeat it a few times:
As you can see, the attractor of this simple IFS (Mathematica code found here) is a simple line. In fact, it is the line y = x.
This is the simplest example of fractal interpolation. We started with a graph (the square) and created an IFS to fit the model (the line y = x).
This principle can be extended by doing some contortions with the IFS, similar to what we did with the Koch curve in section 1.2. This allows us to model other graphs. And if we split our graph into two or more sections and used a different IFS on each section, we could interpolate more complex graphs (for example, think of our simple tree from section 3.1 but with braches only occurring on one side).
This higher-order fractal interpolation is often used by stock market analysts to predict stock prices. An interesting (albeit technical) discussion on this topic can be found at this Yale University page.