A: Why does the chaos game generate fractals?
Certain (but not all) fractals have a special type of symmetry. Informally, you could describe that these fractals repeat in a certain way. You can get a good intuitive feel of this if you look at the Sierpinski triangle which you displayed as one of the examples.
The chaos game makes use of a special map which projects the symmetry of the (outer) polygon onto its image. Sometimes the image gets projected onto a new nice symmetry and other times it gets projected onto something which has no symmetry. In a sense, the map and polygon need to align to allow for the fractal shapes to occur.
Typically, maps that have symmetries can generate fractals. A well known example are quadratic maps on the complex plane. Here is a picture of a fractal generated by a quadratic map:
To make mathematically precise why these symmetries occur is complicated. If you pick up a book on fractals in the complex plane you will immediately see why :o) (I recommend this book -> https://press.princeton.edu/titles/8117.html) But I hope my slightly vague answers help :)
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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)
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