Observing Spiral Patterns > 2. Spiral
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6. The Farey Tree and the Golden Mean
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.
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
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
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
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
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
Interestingly enough, the system also accounts for patterns that
are less commonly seen: those, for instance whose parastichy numbers
follow the Lucas sequence:
1, 3, 4, 7, 11, 18...
(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