It seems that there is an algorithm to generate the sunflower seed pattern based on the golden ratio. From this article in the Irish Times:
A simple mathematical description of the geometry of sunflower seed patterns was devised by Helmut Vogel (1979). He defined the positions of the seeds, using polar coordinates (r, θ), by
r(n) = √n and θ(n) = n φ
where φ ≈ 137.5º is the golden angle. Thus, as n increases by one, the position rotates through the golden angle and the radius increases as the square root of n. All points are on a curve called the generative spiral (r = √θ), a form of Fermat spiral which winds ever-tighter as it curls outwards.
But that doesn't explain how the sunflower actually does it. Now someone has come up with a differential equation that claims to generate the pattern. From the same article:
The equation comes from a paper by Pennybacker and Newell which I don't have access to (and probably wouldn't understand if I did).
Is this for real? Does this differential equation really work? And if so, can anyone give a physical explanation of what the individual terms are doing? I don't mean "explain to me what the del operator does", I mean like "why do you subtract the cube of the hormone concentration..." etc. Any insights?