A cornerstone in phyllotaxis – insight into history

This chapter develops, intuitively and formally, a theorem of fundamental importance in phyllotaxis, and shows applications of this theorem. Other applications will be given in subsequent chapters. The theorem establishes the complete geometrical relation between two concepts presented in Chapter 1: the visible opposed parastichy pairs (m,n) and the divergence angle d of a system. From it I devised four algorithms to obtain easily one parameter from the other. The theorem expresses essential properties of the lattices used in most of the explanatory models and attempts to understand phyllotaxis. Chapter 3 presents an illustration of the theorem, which throws light on the Lestiboudois-Bolle theory of vascular phyllotaxis. This illustration opens the way to a third representation of phyllotaxis, the hierarchical representation that plays an important role in the interpretative model of phyllotaxis, the subject of Chapter 6. We will see in Chapters 5 and 7 that the theorem is involved in practical pattern assessment.

The lack of such a theorem has been a major reason for errors in mathematical and biological phyllotaxis, because of the confusion generated by the large number of families of spirals observed on a single plant. A family of parastichies can be generated simply by linking two points in the lattice representing the pattern on a plant, and the family partitions all the points of the lattice. From the very beginning of the investigations in phyllotaxis (Bonnet, 1754; Goethe, 1790), the multiplicity of spirals puzzled its observers.