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 15: Turn on the Dynamical Model applet and let it run for a while. Describe what you see, using if need be the running counter of generations on the upper left corner. Do you notice any change in the divergence angle? You can stop the program (without quitting it) by clicking in its window. Click again to restart it.

Task 16: How do you see the three basic principles transpire in the model as it is running? For the first principle, you can play my daughter's favourite game: try guessing where the next primordium is born. Count 1 point each time you get it right...

Task 17: Speculate on what the three principles of Hofmeister tell us about the way plant develop. Do these assumptions seem realistic?

Task 18: Running the Dynamical Model applet program once again if necessary, stop it (by clicking the mouse on it) each time you recognize some new spiral pattern. Click on "draw". This allows you to connect the dots with the cursor (keep the mouse button down). To change the color (to draw another set of parastichies, click on "color". Reproduce these drawings in your notebook, writing the generation numbers (appearing in the upper left corner) at which you stopped to draw for each one. To restart the program, click on draw again, and then click once more anywhere on the body of the window. Running up to 200 generations, what different parastichy configurations did the program show you? It might take you 2 or three runs to get all the different configurations: they are not very obvious at first. You may use the Spiral applet in parallele with the Dynamical model to help you recognize some of the patterns.

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 > 2. Spiral Applet > 3. Dynamical Model Applet > 4. Cylindrical Spirals > 5. Cylindrical Spirals Applet > 6. The Farey Tree and the Golden Mean