Here is a fun equation

4^phi = 3*Pi, to an error of .003.

Wierd? Not sure, but I think it means irrationality might be a spherical definition.  It means that 2Phi = log2(3pi), or the precision of measuring 3Pi-1, assuming Gaussian noise of one is about Phi. That is, when measuring 3Pi-1, one notch on your ruler gets you about Phi bits of precision when notches sample at Nyquist-Shannon rates.

Now if you were to reverse the equation, assume 3Pi-1 was gaussian noise and compute log2 precision you get:

log2(1 + 1/(3pi-1)) = Phi/10, to within .0001.  Wierd? With Gaussian noise, only the one or the other is valid. The Shannon equation, as written, assume perfect orthogonality between signal and noise..

Anyway, spreadsheets are fun.