Fibonacci sequence and golden ratio

Question

I have recently read about the numerous occurrences of the golden ratio and Fibonacci numbers in nature. I have read that it occurs in everything from shells to plants and that the rectangle that is most aesthetically pleasing has its sides in this ratio aswell. Are these just coincidences or is there a reason that it occurs so often?

Answer 1

Take ten random numbers in range $[0,1]$, such as ten measurements on the human body or on the Parthenon.

Like

$$0.699, 0.637, 0.407, 0.599, 0.024, 0.998, 0.425, 0.248, 0.460, 0.616.$$

Form all pairwise ratios and look for $\phi$:

$$1.000, 1.724, 0.011, 1.475, 1.067, 1.103, 1.416, 1.101, 0.702, 0.077\\ 0.580, 1.000, 0.007, 0.856, 0.619, 0.640, 0.821, 0.639, 0.407, 0.045\\ 87.766, 151.292, 1.000, 129.477, 93.606, 96.849, 124.241, 96.613, 61.643, 6.801\\ 0.678, 1.168, 0.008, 1.000, 0.723, 0.748, 0.960, 0.746, 0.476, 0.053\\ 0.938, \color{red}{1.616}, 0.011, 1.383, 1.000, 1.035, 1.327, 1.032, 0.659, 0.073\\ 0.906, 1.562, 0.010, 1.337, 0.967, 1.000, 1.283, 0.998, 0.636, 0.070\\ 0.706, 1.218, 0.008, 1.042, 0.753, 0.780, 1.000, 0.778, 0.496, 0.055\\ 0.908, 1.566, 0.010, 1.340, 0.969, 1.002, 1.286, 1.000, 0.638, 0.070\\ 1.424, 2.454, 0.016, 2.100, 1.519, 1.571, 2.015, 1.567, 1.000, 0.110\\ 12.905, 22.246, 0.147, 19.039, 13.764, 14.241, 18.269, 14.206, 9.064, 1.000 $$

It's there !