Tonight I was reading some interesting publications about math in nature. I was drawn to the phenomena of Phyllotaxis. The beautiful and dense arrangement of leaves in nature is stunning to see.

The text below is cited from the free book “the algorithmic beauty of plants”, available on the website http://algorithmicbotany.org/papers.

— begin quote —

phyllotaxis

The regular arrangement of lateral organs (leaves on a stem, scales on a cone axis, florets in a composite flower head) is an important aspect of plant form, known as phyllotaxis.

In order to describe the pattern of florets (or seeds) in a sunflower head, a formula is proposed by Vogel:

\[
\begin{align}
\phi &= n * 137.5^{\circ} \\
r &= c \sqrt{n}
\end{align}
\]

  • n is the ordering number of a floret, counting outward from thecenter. This is the reverse of floret age in a real plant.
  • \(\phi\) is the angle between a reference direction and the position vector of the \(n\)-th floret in a polar coordinate system originating at the center of the capitulum. It follows that the divergence angle between the position vectors of any two successive florets is constant, \(\alpha = 137.5^{\circ}\).
  • \(r\) is the distance between the center of the capitulum and the center of the \(n\)-th floret, given a constant scaling parameter \(c\).

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Implementing these formulas and playing with colour parametrisation in just a few lines of code gives an incredible beautiful result. Click on the image below to see for yourself.

Phyllotaxis. Click on the image to start the simulation.
Phyllotaxis. Click on the image to start the simulation.