Here are some not-too technical papers about the maths which justifies the occurrence of the Fibonacci numbers in nature:
A H Church On the relation of Phyllotaxis to Mechanical Laws, published by Williams and Norgat, London 1904.
E E Leppik, Phyllotaxis, anthotaxis and semataxis Acta Biotheoretica Vol 14, 1961, pages 1-28.
F J Richards Phyllotaxis: Its Quantitative Expression and Relation to growth in the Apex Phil. Trans. Series B Vol 235, 1951, pages 509-564.
D'Arcy W Thompson On Growth and Form Dover Press 1992.
This is the complete edition! (Click on the title-link for more information and to order it now.)
There is also an abridged version from Cambridge University press (more information and order it on line via the title-link.)
T A Davis, Fibonacci Numbers for Palm Foliar Spirals Acta Botanica Neelandica, Vol 19, 1970, pages 236-243.
T A Davis Why Fibonacci Sequence for Palm Leaf Spirals?, Fibonacci Quarterly, Vol 9, 1971, pages 237-244.
The Algorithmic Beauty of Plants by P Prusinkiewicz, and A Lindenmayer, published by Springer-Verlag (Second printing 1996) is an astounding book of wonderful images and patterns in plant shapes as well as algorithms for modelling and simulation by computer. (For more information and how to order it online use the title-link).
Related to this book is:
The Algorithmic Beauty of Sea Shells (Virtual Laboratory) in hardback by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler (more information and order it online via this title-link).
The Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art Sir Theodore A Cook, Dover books, 1979, ISBN 0 486 23701 X.
A Dover reprint of a classic 1914 book. (More information and you can order it online via the title-link.)
Also see H S M Coxeter's Introduction to Geometry, published by Wiley, in its Wiley Classics Library series, 1989, ISBN 0471504580, especially chapter 11 on Phyllotaxis. (More information and order it online via the title-link.)