*(Sandipan Dey, **14 August 2016)*

- In this article, a mathematical model for the
(shown below) will be described (reference: the video lectures of Prof. Jeffrey R Chesnov from Coursera Course on*growth of a sunflower***Fibonacci numbers**). - New florets are created close to center.

- Florets move radially out with constant speed as the sunflower grows.

- Each new floret is rotated through a constant angle before moving radially.

- Denote the
*rotation angle*by**2πα**, with**0<α<1**.

- With
**ψ=(√5−1)/2**, the*most*and using**irrational**of the**irrational numbers****α=1−ψ**, the following model of the*sunflower growth*is obtained, as can be seen from the following*animation in R*.

- In our model 2πα is chosen to be the
**golden angle**, since**α**is very difficult to be**approximated**by anumber.*rational*

- The model contains
*34 anti-clockwise*and*21 clockwise*spirals, which are**Fibonacci**numbers, since the**golden angle****α=****1−ψ**can be represented by the*continued fraction***[0; 2,1,1,1,1,1,1,…].**

- Let
**g /***2π = 1−ψ = ψ^2 = 1 / Ø^2 = 1 / (1+ Ø) = [0; 2,1,1,1,1,1,1,…]**.*

- Then we can prove that
*g(n)/2π =**F(n)/**F(**n+*, where*2)**g(n)*of the*n-th rational*

approximation**golden angle**and*F(n)**is the*.*n-th*Fibonacci number

- Proof by
**induction**(on**n**)

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