A search of the internet, or your local library, will convince you that the Fibonacci series has attracted a lunatic fringe of Fibonacci fanatics who look for mysticism in numbers and in nature. 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 φ in composing music. (He liked number games, but there's no good evidence that he ever deliberately used φ in a musical 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 even 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.