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The change rate in binary notation is fractal.

It would be interesting to see how this generalizes to other bases.

Base 3 has nearly 20,000 operators, of which 729 are commutative.

Yeah, I'm pretty sure as long as you have symmetry somewhere (e.g. a commutative operation), you'll get self-similar patterns.

That's more or less, because binary numbers are already fractal.

Ask yourself why you added the “pretty well-known” phrase, and consider xkcd 1053.