The Golden Ratio

The golden ratio is a single number, 1.618034. It shows up everywhere in this system because of two things it does that no other number does.

First: its square equals itself plus one. Multiply 1.618034 by itself and the result is exactly one more than where you started. Second: 1 divided by it equals itself minus one. Divide 1 by 1.618034 and the result is exactly one less than where you started. No other positive number folds back on itself like that in both directions.

The Fibonacci sequence walks straight toward this number. Take any two neighbors far enough along the sequence — 21 and 34, then 34 and 55, then 55 and 89 — and divide the larger by the smaller. Each division lands a little closer to 1.618034 than the last, and by the time you reach 89 divided by 55, the result is 1.618182. The sequence never quite arrives, but it keeps closing the gap.

Those same four numbers — 21, 34, 55, 89 — are not just an example here. They are among the minute-lengths the timeframe ladder in this system is built from: four of the ladder's eight rungs, on a 15-minute seed. The golden ratio isn't a decoration on top of the ladder; it's the rule the ladder's spacing follows.

phi = 1.618034
phi^2 - (phi + 1) = 0 (to 1e-12)
1/phi - (phi - 1) = 0 (to 1e-12)
fib(34/21) = 1.619048
fib(55/34) = 1.617647
fib(89/55) = 1.618182
fib(89/55) approaches phi
OK