Attractors
- Take two moderately interesting functions that map the unit square to itself; for example, \[(x,y) \rightarrow \left(1-y^2, \sqrt{x}\right)\] and \[(x,y) \rightarrow \left(\frac{x+y}{2}, x\right)\].
- Take a random point in the unit square.
- Apply a random one of your two functions.
- Apply a random one of your two functions.
- Apply a random one of your two functions.
- ...
- Plot out all of the points you iterated through.
You might expect that you get a kinda random smear of points. And you'd be right that it's kinda random, but...
It takes a bit of care to find an interesting pair of functions: most simple pairs of functions tend to produce very uninteresting attractor-patterns, because there's a point, or a small set of discrete points, that are invariant under both functions, and so you very quickly end up trapped in that set. Like...
I forget where I learned about this feature of our universe. It was one of my first programming projects outside of Warcraft III's map-making engine.