An irrational number is a path that never arrives. Left or right at every fork, forever. Its continued fraction is the list of instructions: how many steps before each turn.
The golden ratio turns at every step. Right, left, right, left. No momentum. No coasting. Its approximations improve by the smallest possible amount, always. No windfall. Just work.
Pi takes corridors.
Three rights. Seven lefts. Fifteen rights. One left. Then two hundred ninety-two rights without turning — a hallway so long that the fraction at its end, three hundred fifty-five over one hundred thirteen, is accurate to the millionth.
The straightness is the luck. A number lives at the end of a corridor the way a coin lands in a fountain. Not aimed. Just near.
The golden ratio has no corridors. Every step reverses. And because the path is maximally winding, it is the hardest number to catch with any fraction.
Hurwitz, 1891: Every irrational can be approximated within one over root-five q-squared. The bound cannot be improved because the golden ratio fills it.
The number that takes no shortcuts sets the speed limit for all the others.