4. Cylindrical Spirals Applet

In the previous section, you found a way to draw regular spirals on a cylinder. The sheets of regularly spaced lines you were provided closed up into cylinders with circles stacked up at regular intervals. You marked one point on each line, in such a way that, at each generation, you jumped one circle up and moved around the circle by a fixed amount, in analogy to Task 2 . The points you obtain in this fashion, when unrolling the cylinders are aligned in segments of lines with constant slope. Once you choose the point in the first generation (the 0th generation was printed for you), all the other ones follow logically. You obtained different configurations, with presumably different numbers of parastichies by choosing printouts with more or less tight circles, and by changing the amount (i.e. angle) you moved your point around each time.

The Cylinder Applet or Bigger Cylinder Applet enables you to see all the possible such cylindrical spirals interactively. Clicking the mouse on the left Parameter Space window, a blue point appears and follows the mouse as you drag it. This allows you to adjust the blue point on the right Configuration Space window. The computer program continues the configuration for you by showing the points in a spiral (if you could roll up that part of the screen!).

Task 25: Go to the Cylinder Applet, and turn the "circles" button on. What is the effect of moving the mouse vertically up and down in the Parameter Space? What is the effect of moving the mouse horizontally left and right in the Parameter Space? As you follow the perimeter of the Parameter space with the mouse, what figure does the blue point in the Configuration Space outlines? Draw this figure in a simple diagram.

Task 26: Turn off the "circles" button, and turn on the "partition" and "nearest point" button. What are the defining roles of each of the points labeled as King and Queen? Tell the tragic story of political feuds as you move the mouse in the Parameter Space. What kinds of change in the power structure do you observe and where (in the Parameter Space) do they occur?

Task 27: Turn on the "numbers" button, keeping the "nearest neighbour" and "partition" buttons on. Numbers appear instead of the King and Queen labels. What are these numbers? You may want to turn on the "circles" button to give yourself a hint. What kinds of change of numbers occur and where do they occur in the Parameter Space? Can you see a general, simple arithmetic law in the way these numbers change? 10 bonus points for your group if you find a rigorous geometric argument explaining this arithmetic law!

Task 28: Turn on the "parastichies" button, keeping the others (except perhaps for "circles") on. What parastichy passes through the King? the Queen? Note that, contrary to the spiral program, this one only draws one parastichy per family. You can easily trace the other ones mentally. What is the correspondence between the number of distinct parastichies, and the numbers assigned to the King and Queen? (Again, you must think of the Configuration Space as an unrolled cylinder to distinguish different parastichies) 10 bonus points for your group if you can give a rigorous geometric argument explaining the relation you found in the previous task between the numbers labeling the King and Queen and the numbers of parastichies.

Task 29: Keep on the partition, nearest points, numbers buttons on for this task and next. In analogy to task 13 , find out what is the region where the parastichy numbers are 1 and 2. The region where they are 3 and 4. More generally, try to label as many different regions possible with their different parastichy numbers. Label these on a paper copy of the partition.pdf. Try to find a path in the parameter space where the parastichy numbers follow the Fibonacci sequence and mark it on you partition sheet. You may definitely want to use the Bigger Cylinder Applet at this point.

Task 30: Look at the region marked by a green square boundary in the parameter space. Now turn the "zoom" keeping the other buttons on. What do you see in the Parameter Space? As in task 29, label as many region in the Zoomedpartition.pdf picture as you can. Find a path where the parastichy numbers follow the Fibonacci sequence. Can you find a rule to form this path?

Task 31: Turn off the "zoom" button, and turn on the "potential" button. The configurations favoured by our Dynamical Model are located on the curvy boundaries of the yellow regions. Follow down some of these curvy boundaries. How do the parastichy numbers change? You may want to zoom in at some point to follow some of these curves farther down.

Going down in the parameter space means going to denser configurations, i.e. slower growth. So the vertical axis in parameter space may be thought of as growth rate: fast when you're high, zero when you reach the bottom horizontal axis. (The horizontal axis gives the divergence angle)

Task 32: Show that if you start with a high enough growth rate and decrease it (as Hofmeister's third principle dictates), and you hugg the curvy boundaries, you have no choice but to follow the Fibonacci sequence! What other choices do you have if you start with a somewhat lower growth rate?

For the record: What have we shown so far? If our Dynamical model only produced spirals, then, as the growth rate decreases continuously from a sufficiently high value, the model is locked into a path of spiral patterns whose parastichy numbers follow the Fibonacci sequence! It turns out that my collaborator and I can show that indeed, as the growth rate is high enough, our model spontaneously converges to spirals. This completes our explanation of the predominence of the Fibonacci sequence in plant parastichy numbers.

Interestingly enough, the system also accounts for patterns that are less commonly seen: those, for instance whose parastichy numbers follow the Lucas sequence:

(of the plants which exhibit spiral patterns, it is estimated that 92% follow the Fibonacci sequence, 5% follow the Lucas sequence). These are obtained if the growth rate decreases too abruptly, or start at a lower value. Our model also creates, in certain conditions, patterns that are not even spirals. In these configurations the divergence angle instead of being constant varies periodically. It turns out that, checking the botanical litterature, we found records of such patterns, that had hitherto not been explained. One of our goals is to classify all the possible stable patterns our Dynamical Model exhibits and make as many ties as possible with known (or unknown) botanical configurations.

Whorled structures: Finally, we should point out that there is a type of configuration that this Dynamical Model will never exhibits. Whorled (and multijugate) configurations are configurations where several primordia (dots in our Dynamical model) are generated at the same time. You could obtain such configurations on the circle sheets by drawing several points evenly spread out on each circle. This would create several generating spirals. A beautiful example encountered in this class is the cactus that some of you pricked your fingers on: it has parastichy numbers (16, 26)=2*(8,13), indicating it has 2 generative spirals, each one forming a Fibonacci configuration! We are working on a model that allows simultaneous generation of primordia.

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