Fitting Vogel's formula for phyllotaxis to an actual plant.

Question

A simple model for the arrangement of florets in a sunflower was given by Vogel:

$r = c\sqrt{n}$

$\theta = 137.508 n$

Where $r$ and $\theta$ are polar coordinates, $c$ is some constant and $n$ is the index number of the floret. In this image each floret is marked by a blue point:

I am trying to recreate it. The problem is:

• How do I determine the correct constants to generate the flower in the photo?

I do have information about the flower such as the position of the florets. I will do the calculations in Mathematica.

Update. I later managed to do this but the results weren't impressive, probably because the mode simply isn't that good.

Answer 1

Not knowing anything else about the problem domain, I'd take a look at the Nelder-Mead method. It's general and robust.

As far as what to vary, here's what I'd start with.

• The origin of your polar coordinate system. Where is the center of your spiral? Does it have to be located on one of the florets, or not?
• Your scale parameter, $c$. This determines, in general, how far apart the florets are. It would be best to relate this to the physical size of the flower, but expressing it in terms of image pixels would work, too.
• Your starting index. Maybe you'd get a better fit starting with $n=5$. Or maybe the model dictates that you start at $n=1$.
• Your offset angle. You will probably need to offset all of the angles by some constant angle $\theta_0$ to account for the fact that you took the picture at some orientation, rather than $1^{\circ}$ off.