I recommend the book "The Algorithmic Beauty of Plants" by Przemyslaw Prusinkiewicz and Aristid Lindenmayer (Springer-Verlag, 1990) if you want to explore the mathematical systems behind plants like this. It covers Lindenmayer systems (L-systems) in detail, of which phyllotaxis is only a part.

I happen to have a first edition, but it is these days available as a free PDF: http://algorithmicbotany.org/papers/abop/abop.pdf