1. Observing Spiral Patterns

The Dynamical Model Applet

In the previous section, you explored all the Logarithmic spirals that mathematics can create. You have seen that, of these spirals, only very specific ones are usually exhibited by plants. Two fundamental questions arise:

• Why do plants favor spiral configurations at all?
• Why, among all possible logarithmic spirals, plants favor those with parastichy numbers successors in the Fibonacci sequence?

Even though this phenomenon has been observed for hundreds of years and studied by many botanists, mathematicians and crystalographers, only recently has there been a begining of an answer for these two questions. My collaborators Pau Atela, Scott Hotton and I have a partial answer to these questions. The goal of this tutorial is to give you an idea of this answer.

In our research we have been looking at a model by the french physicists Stephane Douady and Yves Couder. Incidently, Douady came to the field of Phyllotaxis one day when, coming back from the market with a cauliflower, he was intrigued by the fractal nature of this plant. But, after some time looking at it, his attention turned to the magic of parastichies on each of the little flowerets. He and Couder came up with a simple model for the formation of these spiral patterns, which they implemented both physically and on the computer. This model, based on assumptions made by the botanist Hofmeister, spontaneously generates the Fibonacci spiral patterns. The Dynamical Model applet is our version of Douady and Couder's model. The three basic principles of Hofmeister on which this model is based are the following:

• A new dot is formed periodically in the place around the central disk where it is least crowded by the others dots.
• Once they form, the dots move radially away from the center.
• As time increases, the rate at which new dots move away decreases

The dots represent the center of microscopic bulges of cells (called primordia in botany) that occur at the growing tip (Called apical meristem in botany) of the plant. These bulges eventually differentiate to become the leaves, petals, sepals, flowerets or scales of the plant.

##### Task 19:Running the Dynamical Model applet again, click on the "show graph" button. The top graph shows how the divergence angle a varies with time, starting at 1800. The bottom graph shows how the "growth rate", which is just our old friend r from the previous sections, varies. Letting run the program to 200, describe what happens to the divergence angle a and to the growth rate r. How does this correspond to your statements in Tasks 15 and 16?

1. Observing Spiral Patterns