The appearance of Fibonacci numbers and the golden ratio everywhere in nature is mostly a persistent myth. Famous (but most presumably accidental) examples are the ratios between the phalanges of your fingers, or the position of the belly button in the human body. One example where there is an explanation for the golden ratio’s involvement is phyllotaxis, the arrangement of leaves on a plant stem. In sunflowers, pineapples, romanesco, aloe plants, pine cones, artichokes and numerous other flowers, the number of leaves or seed spirals frequently equals a Fibonacci number.
So we see this results in some radial spokes (as many as the denominator in α’s irreducible fraction). For a flower this is very unappealing, since this arrangements waste a lot of space. Instead a plant wants to maximize its exposure to sunlight, dew or carbon dioxide.
You notice this arrangements fill space more evenly but also stagnate into spiraling patterns, resembling the rational case. These spirals correspond with the best rational approximations for α. Centrally in the left example you clearly notice three spirals, because 1/π ≈ 1/3. After a while they break apart into 22 spirals: 1/π ≈ 7/22. One can show that the “best rational approximations” or convergents of an irrational number are precisely the fractions resulting from keeping only a limited number of terms in the continued fraction expansion.
So if a flower wants to distribute its seeds optimally, it needs an α which is “hard to approximate” with fractions, and the golden ratio is essentially the hardest one because its continued fraction consists only of 1’s. The convergents of the golden ratio have only Fibonacci numbers as denominators (in lower terms), which helps explain their ubiquitous occurrence in nature.
Finally, the parameter α appears to be rather sensitive: even a small deviation in the angle of rotation quickly spoils the delicate balance achieved by the golden ratio.