It is known that the spiral phyllotactic pattern is common in Nature, especially in Botany.

The example of spiral phyllotactic arrangement of emerging flowers on Gerbera hybrida (doi: 10.1093/jxb/erw489).

It consists of two group of clockwise and anticlockwise spirals, starting from the center. In most cases the number of those spirals are two consecutive Fibonacci numbers: $F_n, F_{n+1}$. Also there are patterns where number of spirals are doubled Fibonacci numbers, Lucas numbers, or Fibonacci ± 1.

The equations for generating some sort of a spiral phyllotaxis are:

$$ \phi = \pi(1+\sqrt{5}) \\ \forall n \in [0, N] \\ \theta = n \phi \\ r = \sqrt{n} \\ x = r \cos{\theta} \\ y = r \sin{\theta} $$

How to change that equations to take into account the predefined number of spirals? For example, I want to generate a phyllotactic pattern consisted of (11, 18) spirals: 11 clockwise and 18 anticlockwise. How to do that?


It looks that the number of visible spirals is depended on the radius of a pattern.


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