Why the golden ratio? Because the continued fraction of the golden ratio is all 1's [0]. So it is uniquely hard to approximate with a rational number. The golden ratio is the bound on Hurwitz's theorem [1]. And avoiding a rational number is what you want for good hashing, because multiplying with a rational number doesn't mix your digits well.
[0] https://codegolf.stackexchange.com/questions/48589/generate-... [1] https://en.m.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_...
> So it is uniquely hard to approximate with a rational number.
Fun fact. The best rational approximations for golden ratio are sequential fibonnaci numbers. E.g. fibonacci(n+1)/fibonacci(n)
or for a more concrete example, 21/13