φ (Golden Ratio) N decimals Generator

Phi, roughly 1.618034, is the ratio in which the larger part is to the smaller as the whole is to the larger. It solves the equation x squared equals x plus 1, which gives the closed form: 1 plus the square root of 5, divided by 2. From that equation comes the most-cited property: the reciprocal of phi is phi minus 1, that is 0.618034 — the same decimal places.

Choose the places and receive up to a thousand digits, computed from the square root of 5 with arbitrary precision and checked against published values. Worth knowing that phi is the limit of the ratio between consecutive Fibonacci terms, and that convergence is quick: dividing 89 by 55 already gets the first three places right.

On the applications attributed to phi, it helps to separate what holds up from what does not. In phyllotaxis the argument is solid: the golden angle distributes leaves and seeds without overlap, and phi is the hardest number to approximate by fractions, which prevents alignment. Its presence in the Parthenon, the Mona Lisa and human body proportions is, for the most part, coincidence within the error margin of any measurement.