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

• In this article, the 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 \(2\pi\alpha\), with \(0<\alpha<1\).

• With \(\psi=\frac{\sqrt{5}-1}{2}\), the most irrational of the irrational numbers and using \(\alpha=1-\psi\), the following model of the sunflower growth is obtained, as can be seen from the following animation.

• The model contains 34 anti-clockwise and 21 clockwise spirals, which are Fibonacci numbers, since the golden angle \(1-\psi\) that can be represented by the continued fraction \([0;2,\overline{1}]\) and \(\frac{g_n}{2\pi}=\frac{F_{n}}{F_{n+2}}\).