1D, 2D, circle, back

I don’t know how one gets these ideas.  The Box-Muller algorithm—which quickly fashions a normal distribution out of a few dice rolls—comes from the following unintuitive chain of logic.

  1. Take a problem in a simple domain (1-D).
  2. Put it into a more complicated domain (2-D).
  3. Change from grid coordinates to Navy coordinates (rect → polar).
  4. Do the problem there, then push it back.

solving in different domains

It’s normal to move problems to other domains to solve them: f⁻¹( g( f(●))).  Some might say that’s the whole point of math.  But this is so unintuitive, to make something more convoluted before making it simpler.

Using the Fourier transform, for example, you take a wave (like music) and turn it into a frequency (like notes).  Well both music and notes are understandable domains — sometimes you want one, sometimes you want the other, so you just transition back and forth between the domains as suits.  You use the equalizer to adjust things in the second domain, and then you go back to the music domain to listen to the transformed song.

Not this time, though.  You go into a seemingly unrelated domain, solve the problem there, and then come back.  Very weird.  Or as some people like to say, “clever”.

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