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A shower thought turned into a Collatz visualization

I recently went on a nice long SCUBA diving trip with my wife and daughters. Lots of diving implies lots of showers, and lots of showers means lots of shower-thoughts! [1] An especially interesting one I had turned into a nice way to visualize some aspects of the Collatz Conjecture.

The output of the Collatz function at each step seems like it would be biased towards producing even numbers, making the bit-sequences more full of one binary digit than the other, and maybe obscuring any interesting features we could otherwise see. After making this plot I remembered a nice trick I read about as a teenager in James Gleick's fantastic book "Chaos" (I think the idea might have been attributed to Feigenbaum). I would guess the 2-adic approach probably makes their proofs nicer, but jumping straight to fractions is likely to be easier if you've never seen p-adic numbers before (or if, like me, it's been long enough that you've forgotten most of what you learned about them!

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