For a more comprehensive explanation of this behavior, look into the subject of Diophantine approximation; root 2, phi and other quadratic irrationals are badly

For a more comprehensive explanation of this behavior, look into the subject of Diophantine approximation; root 2, phi and other quadratic irrationals are badly approximable numbers (phi being the king of those), and their ability to "evade" rationals for a long time means their multiples are free to scatter around the interval more flexibly than better approximable ones. Diophantine approximation formalizes this heuristic; a good introduction is Khinchin's classic Continued Fractions.