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Kingreaper comments on Open Thread: July 2010, Part 2 - Less Wrong
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IMO, that would be emergent behaviour of mathematics, rather than of pi.
Pi isn't a system in itself as far as I can see.
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.