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RE: A: Why does the chaos game generate fractals?
Hi, first of all thanks for your answer!
I can't say I made complete sense of your explanation but that's due to my poor knowledge of mathematics.
By "map" I assume you mean some kind of mathematical transform operation?
The mystery here is that the algorithm introduces randomness, so it's hard for me to reconcile how this randomness doesn't end up in filling up the outer shape entirely over time.
For example, running the chaos game for a square with f = 1/2 results in all points within the square filling up with equal probability, while using a value f = 2/5 results in a fractal design.
Weird!
The following example might give you some intuition about
Consider the map f: [0,1) mod 1 -> [0,1) mod 1 given by f(x):= 2 x .
Take x_0=1/4. Then plot or compute all forward iterates of x_0 under f
Now take x_0=1/4 + 10^-6*sqrt(2) (or some other point close to 1/4). Then plot or compute all forward iterates of x_0 under f
(if you are going to use a computer you need to pay close attention to rounding errors)