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Fibonacci number has something to do with natural growth. Though the function is very straight forward, we see this in nature. Does nature follow the function or its the simplified model of the function that nature follows?

Where do we see Fibonacci number? In flower petals, tree branches etc.

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For a skeptical view of the matter, you may want to look at the amusing article Fibonacci Flim-Flam. Late in the article, there are nice pictures of flowers with the "wrong" number of petals. –  André Nicolas Dec 21 '11 at 7:52
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We see a lot of things in nature. When we see the Fibonacci numbers (or something roughly close to them) in nature we are exulted and write books about it, and when we see less sexy sequences of integers we ignore them. –  Gadi A Dec 21 '11 at 8:01
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Thanks @AndréNicolas. It seems to be false claim. –  Shiplu Dec 21 '11 at 8:09
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@Shiplu: I believe the claim has valid content, but has been overstated. –  André Nicolas Dec 21 '11 at 11:46
    
Phyllotaxis is extensively discussed in the literature. A very readable account occurs in Chapter 4 of the 2010 book by Ian Stewart, The Mathematics of Life. Indeed, many other solutions of the generalised Fibonacci numbers are observed in nature. –  Willie Wong Dec 21 '11 at 12:18
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I read in the comments: It seems to be false claim. This is far from being consensual. You have many publications claiming that this is true, including very recent ones, eg this one: The phyllotaxis of the model plant Arabidopsis thaliana follows a Fibonacci spiral, the most frequent phyllotactic pattern found in nature.

Many publications propose models to explain this phenomenon, eg this one (published in 2011).

This book chapter by S Douady who wrote several papers on the subject might interest you (the last page is missing but all the models are fully described).

PS: Nice video with lots of examples, no explaination or modeling.

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