Fibonacci Foolishness.
A search of the internet, or your local library, will convince you that the Fibonacci series has attracted the lunatic fringe who look for mysticism in numbers. You will find fantastic claims:
The "golden rectangle" is the "most beautiful" rectangle, and was deliberately used by artists in arranging picture elements within their paintings. (You'd think that they'd always use golden rectangle frames, but they didn't.)
The patterns based on the Fibonacci numbers, the golden ratio and the golden rectangle are those most pleasing to human perception.
Mozart used f in composing music. (He liked number games, but there's no evidence he ever deliberately used f in a composition.)
The Fibonacci sequence is seen in nature, in the arrangement of leaves on a stem of plants, in the pattern of sunflower seeds, spirals of snail's shells, in the number of petals of flowers, in the periods of planets of the solar system, and in stock market cycles. So pervasive is the sequence in nature (according to these folks) that one begins to suspect that the series has the remarkable ability to be "fit" to most anything!
Nature's processes are "governed" by the golden ratio. Some sources even say that nature's processes are "explained" by this ratio.
Of course much of this is patently nonsense. Mathematics doesn't "explain" anything in nature, but mathematical models are very powerful for describing patterns and laws found in nature.
I think it's safe to say that the Fibonacci sequence, golden mean, and golden rectangle have never, not even once, directly led to the discovery of a fundamental law of nature. When we see a neat numeric or geometric pattern in nature, we realize we must dig deeper to find the underlying reason why these patterns arise.
The "golden spiral" is a fascinating curve. But it is just one member of a larger family of curves/spirals collectively known as "logarithmic spirals", and there are still other spirals found in nature, such as the "Archimedian spiral." It's not difficult to find one of these curves that fit patterns found in nature, even if those patterns are only in the eye of the beholder.
But the dirty little secret of all of this is that when such a fit is found, it is seldom exact, and considerable variations from the "golden ideal" are seen in nature. Sometimes curves claimed to fit the golden spiral actually are better fit by some other spiral. The fact that a curve "fits" physical data gives no clue to the underlying physical processes that produce such a curve in nature. We must dig deeper to find those processes.