How did Knuth derive A?
See exercise 9 of section 6.4 of The Art of Computer Programming.

Any irrational $A$ would work, because $\{kA\}$ breaks up a largest gap of $\{A\}, \{2A\}, \ldots, \{(k-1)A\}$ (I use the notation $\{x\}$ for $x\mod 1$).

But if $A = \phi^{-1}$ or $A = \phi^{-2}$, it has a special property: these are the only values for which neither of the two newly created gaps is more than twice as long as the other.