(Sandipan Dey, 14 August 2016)
- In this article, a mathematical model for the growth of a sunflower (shown below) will be described (reference: the video lectures of Prof. Jeffrey R Chesnov from Coursera Course on 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 , with .
- With , the most irrational of the irrational numbers and using , 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 a rational number.
- The model contains 34 anti-clockwise and 21 clockwise spirals, which are Fibonacci numbers, since the golden angle can be represented by the continued fraction
- 2π = 1−ψ = ψ^2 = 1 / Ø^2 = 1 / (1+ Ø) = [0; 2,1,1,1,1,1,1,…].
- Then we can prove that .
- Proof by induction (on n)