Where did the universe come from? Atheistic origin science has no answer.

[Alex K]

I think xe's not looking for connections or an underlying principle, but exactly the opposite: a miracle - that it is so unlikely for the "exact" golden ratio to appear in nature that it is an smoking gun indicator of supernatural interference.

Which is self-refuting: the probability of the golden ratio appearing in nature is precisely the same as for any random number. So I guess I'm giving the guy more credit than that.

Well, yes, the probability to encounter any arbitrary real number as a ratio in nature is 0 (and that's before we deal with finite resolution and uncertainty etc.). If you want to make an argument you have to

- predict the number (or a small range of numbers) before ever measuring, or
- as a weaker alternative, find a deep underlying reason why precisely this number is special, which is so convincing that one can even accept it as evidence after the fact.

However, so far this is basically the definition of pre- and postdictions in theoretical physics or generally the natural sciences. If you want to draw theological conclusions, your source of information about this special number has to be of a inherently religious nature, and I don't see how that should be verified even in principle if it's not a true prediction.
Unless, of course, you want to go the presuppositionalist route and claim that any connection between maths and nature can only be present due to god, but you don't need the golden ratio for that as it doesn't add anything new.

The golden ratio is special in mathematics (and by extension physics) as the limit of the fibonacci sequence which occurs in some dynamical systems, and such, so it is a feature of the natural world, like pi. In my opinion this argues against any religious conclusion one might draw, because even if the ratio did appear significantly more often than random (which it probably doesn't), or did appear to a precision not explainable by chance (*), there is a candidate mathematical/physics reason why nature might produce it every once in a while, just like rotational symmetry produces pi by allowing gravity to form balls.

It would not be the least bit surprising if some geometrical features of seashells and molecules exhibit this ratio, since their physical production mechanism, being constrained by geometry of space and maybe governed by chaotic dynamics, might somehow invoke this mathematical constant. That means absolutely nothing except that physics approximately follows certain mathematical relations, but we knew that already. It doesn't get much more geometric than say the group structure of the standard model. Of course we find numbers in nature which corresponds to some simple mathematical principles. That's called science, not God.

(*) A good example of that is given by the OPs Earth-Moon-Example. Using the numbers xe provides, the hypthenuse is the golden ratio to three or four digit precision. These numbers are however not even within the range of plausible definitions of Earth and Moon radius (pole and equatorial radii). If you plug those in, you miss the true golden ratio very nearly - by less than a permille, but still a definitive miss. Now you have to compensate for the number of astronomical comparisons you can make, and the number of ways you can compare earth and moon (really, it needs to be sqrt(e^2+(e+m)^2)/e ?

Still, there might be an interesting mechanism explaining why collisions of astronomical objects tend to produce such ratios. Something interesting might be learned from this observation. Unfortunately, the religion our OP does what it does to so many religious people: it gives hir a prefabricated answer and suffocates curiosity before it can even ask the really interesting follow-up question from which something could be learned.