Modeling the growth of a sunflower with golden angle and Fibonacci numbers in R

Modeling the growth of a sunflower with golden angle and Fibonacci numbers in R


 

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 2πα, with 0<α<1.

With ψ=(√5−1)/2, the most irrational of the irrational numbers and using 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 a rational number.

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+2), where g(n) is the n-th rational approximation of the golden angle and F(n) is the n-th Fibonacci number.
Proof by induction (on n)


Link: Modeling the growth of a sunflower with golden angle and Fibonacci numbers in R