# Solving the Riddle of Phyllotaxis - Why The Fibonacci Numbers And The Golden Ratio Occur On Plants

Irving Adler (1913-2012) was a mathematician and teacher, but will probably be remembered best from the 56 books that he authored (sometimes using the pen name Robert Adler) while he co-authored some 30 other books. Many of these books were about science and mathematics for children and youngsters or popularize the subject for adults.

This book is a collection of papers on phyllotaxis (the science of how leaves grow on the stem of plants) that he published in journals and in a book as a mathematician. Four of them appeared in the *Journal of Theoretical Biology* in the period 1974-1977, one is published in 1998 in *Journal of Algebra*. There is also a preface and a chapter from the book *Symmetry in plants* published in 1998 by World Scientific. The foreword is from the hand of his son Stephen L. Adler.

In all these papers Adler was looking for an answer to the problem why the Fibonacci numbers and the golden ratio occur when describing the way plants develop their leaves. The first paper is the longest (79 pages) and the most fundamental. It describes a mathematical model that is conform to the idea of *contact pressure*. This means that leaves will grow such that the geodesic distances are maximized. Left and right parastichy pairs (the spiral lines along the stem on which leaves emerge at specific distances) are described and define a lattice on a cylinder that plays an important role, but much more botanical terminology and parameters important in phyllotaxis are discussed. For example the fraction of a turn between two consecutive leaves is called *divergence*. The divergence and the *internode distance* of where leaves emerge are important parameters in the model and characterize the phyllotaxis. Continued fractions and (generalized) Fibonacci numbers occur and obviously in the limit also the golden section ratio. Also relations with sphere packing and space-filling curves are considered.

The fundaments and general introduction being given in the first paper, the other papers are shorter and give additional results and refinements The later papers are more like surveys or summaries of the theory. For example the paper in *J. Algebra* is about the role of continued fractions in phyllotaxis, but also gives a survey of the opinions and contributions of different authors to his phyllotaxic model. The papers are descriptive, i.e., they describe how the leaves are arranged, but they do not explain why or when they grow in these arrangements.

**Submitted by Adhemar Bultheel |

**21 / Nov / 2012