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Sunday, April 3, 2011

The Fibonacci sequence

I have heard from several people who believe that the Fibonacci sequence is a proof of god. 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 good evidence that 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 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.

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 fits a particular pattern found in nature, even if that pattern is 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. The examples from nature that you find in books often have considerable variations from the "golden ideal". 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.

Fibonacci Sequence in Nature?

Phylotaxis. The dictionary defines Phylotaxis as the history or course of the development of something. In biology it generally refers to how a living thing develops and changes over time. This is one part of nature where the fibonacci sequence and related sequences seem to show up uncommonly often, and it's legitimate to inquire why. The interesting cases are seedheads in plants such as sunflowers, and the bract patterns of pinecones and pineapples.
We have noted above that not all spirals in mathematics or in nature are golden spirals. Likewise, spirals can be produced by non-biological processes if the discrete elements which make up the spiral are laid down according to some simple rules. The problem for biologists is to find those rules. Merely asserting that "nature seems to prefer fibonacci numbers (most of the time, in certain particular cases) isn't an explanation

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