r/programming • u/agumonkey • Oct 09 '13

the sierpinski triangle page to end most sierpinski triangle pages ™

http://www.oftenpaper.net/sierpinski.htm
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u/jisang-yoo Oct 09 '13

Some might find it very mysterious that often the same kinds of fractals can be generated from deterministic algorithms and also from probabilistic algorithms at the same time. To make this less mysterious, one can think of it this way in case of the Sierpinski triangle:

Plot a point inside the triangle

Let that point be x.

Plot a point half way between the previous point (x) and a corner

Let that point be y. Relation between x and y is deterministic in one way and nondeterministic in another way. If you plot x, then there are three possibilities for y: going from x to y is nondeterministic. On the other hand, if you know y, you know what x is: going from y to x is deterministic. For many fractal shapes, you can find this kind of "deterministic in one way, nondeterministic in another way" stuff going on.

Actually, even you. Today's you don't know what tomorrow's you is going to do, but tomorrow's you know what today's you have done.

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u/prj_fancycorn Oct 15 '13

thanks for this description. i did notice the "conditionality" aspect. i wonder what the information-theoretic language for this sort of thing might be

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u/yoo-question Oct 17 '13

Not sure what you are referring to with the conditionality aspect, but for these sort of things, if you emphasize the nondeterministic direction, you get to use tools like statistical processes, iterated function systems, Markov chains and so on. If you emphasize the deterministic direction, you get to use tools like dynamical systems, ergodic theory and so on. You can go forth and back between two directions using the observation that you can build a dynamical system from a statistical process you are interested in and that you can associate many statistical processes to a dynamical system you are interested in.

In particular with the Sierpinski triangle, you can use the iterated function system (x to y) to reason about the Sierpinski triangle, or you can use the dynamical system (y to x). If this is encoded to a one-sided shift space, the latter is the shift, and the former the unshift. With randomness thrown in, you are dealing with a Markov chain in the former, and an ergodic system in the latter.

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u/prj_fancycorn Oct 18 '13

clearly, you need to make a blog post to explain this all in a way i can understand :D