Phi is the most irrational of the irrational numbers, meaning that successive rational approximations are just slightly better: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, etc. There are no rational approximations that "jump out" as being much better, so you won't find any unexpected patterns when using it.

The Golden Ratio (why it is so irrational) - Numberphile https://www.youtube.com/watch?v=sj8Sg8qnjOg