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).

    sunflower

  • 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=(√51)/2, the most irrational of the irrational numbers and using α=1ψ, the following model of the sunflower growth is obtained, as can be seen from the following animation in R.

    golden

 

  • 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=1ψ can be represented by the continued fraction [02,1,1,1,1,1,1,…].
  • Let g / 2π = 1ψ = ψ^2 = 1 / Ø^2 = 1 / (1+ Ø) = [02,1,1,1,1,1,1,…]
  • 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)
    fib_proof.png
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