The Golden Ratio is at the boundary of Science and Pseudo-science. A. M. Selvam has published quite a body of work on self-organized criticality and non-linear dynamics using the Golden "mean" and the Penrose tiling pattern to "predict" patterns in just about everything. Her work tends to make sense even though it shouldn't make sense. In my opinion, it does because there are simply more natural patterns that can be reduced to irrational numbers than to rational whole numbers. We tend to think in whole number where nature doesn't think. Not everything in nature falls into a Golden ratio pattern, there is no reason everything should, but there is some logic to why the Golden ratio at least appears to dominate along with Pi, e and various roots. Nature doesn't do base 10.
Pi, e and various roots are commonly used to describe things in nature that are more common or ordered like circles, spheres, decay curves because we live in a "not a box" universe. Phi, the Golden ratio, is a bit like the red haired stepchild of the irrational number gang though.
The probable reason is Phi doesn't exactly fit anything. It is a close mean to a lot of things but it is inexact. Natural arches on Earth vary around the Golden mean and if every material had the same strength the Golden mean would be irrelevant. The Golden mean nearly fits more things on Earth and in the Solar system than it would if we lived around some other star. Fate makes the Golden mean somewhat relevant to us Earthlings.
Fate is not a scientific term, but chance is. The probability or chance of some object forming that has some approximation of a Golden dimension is greater than "normal". If you look hard enough you can find more examples. If you want to dispute the significance of Phi you can find plenty of examples to call Phi Phans nuts.
If I were to say that Earth is not a perfect sphere, folks would agree but Pi is here to stay even though it is not a perfect fit in nature, because it is a closer fit to nature. When we need more "exact" measure Pi is the guy. But when there is more irregularity we get all flustered and forget that Phi is a better general fit for all the stuff that isn't as ordered as a sphere or a curve influenced by gravity or the inverse square law. We need that order and better "fits" between our perception of nature and our construction of math. Had we kept base 60 as our math concept, Phi would be Phine. With base 10 and metrics, our perception of the universe just gets more biased to a nonexistent "normal". US units of measure, inches, feet, furlongs, bushels and peaks actual fit what we perceive instead of forcing things to fit plus requires some thought which is a good thing.
So let's think about Phi, 1.6180339887.... That is close to Pi/2, 1.5707963268 +/- 3 percent. Earth's orbit isn't perfect, it is off by about 3 percent.
What about e? 2.718281828 , since Phi = (1+(5)^.5)/2, Phi plus 1= 2.6180339887.... or 3.8% more than e.
Phi is a pretty decent approximation for most everything but not exactly equal to anything. How can you not "see" Phi relationships in nature and how can you deny Phi is relevant to nature? It is right on the boundary of science and pseudoscience.
Math purists, who tend to be extremely anal, will devise hundreds of statistical methods to allow for uncertainty in nature that are built into fractals approximate with Phi. Over billion of years of random erosion, adaption and evolution, things are not perfect and never will be, Phi is just a good estimate of the mean degree of imperfection.
Now which is the pseudoscience?
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