However, there is a fair amount of literature debunking the prevalence of the golden ratio. Very notable is an essay linked on Reddit, Fibonacci Flim-Flam by Donald Simanek. It's enough to make one abandon the idea altogether. (And, it reinforces my primary reason not to get tattoos: in a single lifetime, no concept or image is so enduring that it retains its meaningfulness forever--not even my best attempt).
This speaks to the danger of having too much mystical admiration for any one thing in nature, whether that be the mysterious origins of living tissue, the Big Bang, or the self-organizing properties of matter. Attach too much meaning to something, and then when it is swept out from underneath you, what do you have left to stand on?
This is part of the pathos of the debate between modern atheism and belief. The atheists correctly tear down the idols, and the believers, whose belief was founded so thoroughly on those idols (their allegedly bulletproof arguments), then lose their faith. Neither the believer nor the atheist had ever considered an appropriately transcendent understanding of God.
Now to be sure, my enthusiasm for nautilus shells was a careful enthusiasm. I was less rapt by the exact numbers than I was by the more abstract loveliness of nature--nature is not an equilibrium of halves, but an asymmetrical dynamo. Equality and balance are not synonyms; they are actually opposed. True balance involves complementarity and difference; equality causes inertia and stagnation. The nautilus expresses some of my earlier conviction in a symbolic form.
But Simanek's article nevertheless creates a vacuum of meaning. Maybe it is all just random after all.
But Simanek himself explains that this is not the case.
The reason f shows up in nature has to do with constraints of geometry upon the way organisms grow in size. Irrational numbers (those that cannot be expressed as a ratio of integers) are often revealed in this process. The well-known irrationals are Ö2, f, e, p and any multiples or products of them. To make matters more interesting, these are related. For example, phi is f = (Ö5 - 1)/2. And the Euler relation, eip = -1 relates e, i and p where i = Ö(-1). The natural processes that display irrationals are not governed or caused by f in order to achieve some desired purpose or result, but rather they are constrained by the geometry of the universe and the limitations imposed by that geometry on growth processes.His point is not that nature is chaotic; quite the contrary, his point is that nature is supremely simple. The processes that develop into beautiful spirals--which follow logarithmic patterns if not Fibonacci patterns--are based on the internally consistent properties of the bodies in question. Their matter is self-organizing, and this process undergoes countless influences with varying degrees of impact, producing different results. From this point of view, natural objects are not "reaching towards" an ideal; their apparent patterns and self-replicating structure can be explained by purely material, immediate causes.
The human mind likes parsimony. We are satisfied when diverse phenomena can be explained by the least number of possible rules. Fibonacci mysticism and mundane scientific observation both offer parsimony, but only the latter offers an explanation for it. True, if all logarithmic spirals had golden proportions, our hearts would be all aflutter. But the diversity of observed ratios is neither disheartening nor a point for atheists. Christianity must be comfortable with the messiness of nature.
The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait. (GK Chesterton)Thus, we should should not be discouraged because nautilus shells, vines, don't often reflect the golden ratio.
This issue then becomes a platform to begin the conversation about the validity of teleogical thought. Teleology is thinking about nature in terms of its proper "ends", e.g., "This egg is a chicken egg because, left to its own devices, it will become a chicken."
Teleology has been rejected as true knowledge ever since Descartes. For science, things are explained by what immediately precedes them--not what they will become. But this discussion is for another time.