🌻 Phyllotaxis

About

In botany, phyllotaxis (from Ancient Greek φύλλον (phúllon) ‘leaf’ and τάξις (táxis) ‘arrangement’)or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. — Wikipedia

Live demo

🧠 Idea inspired by this video

Theory & implementation

The spiraling flower pattern is easily generated by varying the values of merely two parameters called $n$ and $c$, where $n$ counts the number of seeds/dots and $c$ is the so-called scaling factor, i.e. it determines how densely the seeds/dots are packed.

By varying the value of $n$ from one to the final number of seeds, we obtain values for (the polar coordinates) $r$ and $\phi$ with:

$$\begin{cases} \phi &= n \cdot 137.5^\circ \\ r &= c \sqrt{n} \end{cases}$$

where $r$ measures the distance of the seed/dot to the core, and $\phi$ the angle at which it is placed. Note the presence of the special number $137.5$, which is the famous golden angle!

Finally, we only have to transform the $r$ and $\phi$ values to the $x, y$ coordinates on our computer screen:

$$\begin{cases} x &= r \cdot \cos(\phi) \\ y &= r \cdot \sin(\phi) \end{cases}$$