
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
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
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 180^{0}. 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

