φ calculated with Fibonacci
The Golden Ratio φ "phi" can be calculated with a recursive
definition. It is the limit of the quotient of the fibonacci sequence (f
n)
n∈N
defined by
f0 = f1 = 0, fn = fn-1
+ fn-2 for n≥2
φ = limn→∞
(fn/fn-1) After more than 60000 steps the Golden
Ratio appears to be:
Which has at least 9000 digits of exact precision (in fact it's full
precision). The sequence calculating it is converging to
(√5+1)/ 2 = (Sqrt(5)+1) / 2.
Which can be proven analytically to be the exact
value of φ. By using generating functions, it is possible to
prove a closed form for the fibonacci sequence (by using of the golden ratio)
fn = (φn - ψn) / √5,
with
φ = (1+√5) / 2
ψ = (1-√5) / 2