# The Golden Ratio

When in doubt, use the golden ratio. It is faith. Sacred geometry.



My favorite expression of it is the continued fraction. It is
astonishingly beautiful. Especially with the lense of Ergodic
theory.

Phi^2 = Phi + 1
Phi = 1 + 1/Phi

1 / Phi
= 1 / (1 + 1/Phi)
= 1 / [1 + 1/ (1 + 1/Phi) ]
= 1 / [1 + 1/ {1 + 1/ (1 + 1/Phi) } ]
= 1 / [1 + 1/ (1 + 1/ (1 + 1/ (1 + ...
= [0; 1, 1, 1, 1, ...]

a_n = F_n / F_{n+1} -> 1/Phi

Phi
= 1 + 1/Phi
= 1 + [1 / (1 + 1/ (1 + 1/ (1 + 1/ ...
= [1; 1, 1, 1, 1, ...]
= (1 + sqrt(5)) / 2 since Phi^2 - Phi - 1 = 0, Phi > 0
~ 1.61803398874989484820458683436563811772030917980576286213544862

1 / Phi
= [0; 1, 1, 1, 1, ...]
= Phi - 1
~ 0.61803398874989484820458683436563811772030917980576286213544862



When it comes to music, my favorite is Bartok's use of it.