
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 Spiral Applet
In the previous tasks, you have
learned to construct regular spirals by placing points at a constant
divergence angle a
from each other on concentric circles. The circles I provided
you have radii increasing at a constant ratio, say r. The closer
to 1 the ratio r is, the tighter the circles are together and the
bigger r, the farther apart the circles are. If the first circle
in the center has radius 1, the next one has radius r, the one after
r^{2} and then r^{3} etc… Meanwhile, the angle
of your points, with respect to the original one, are 0,
a, 2a, 3a etc… Such spirals are called logarithmic
spirals because the angle na is a
logarithmic function of the radius r^{n}. The
Spiral Applet a JAVA program written by Biliana Kaneva (a computer
science student at Smith College) is a way to visually study all
these logarithmic spirals at once (I you have a big screen, get
the
Big Spiral Applet).
Task
7: Go to the
Spiral Applet and freely explore the different kinds of configurations
that you see as you move the cursor inside the circle of the Parameter
Space, noting down any general phenomenon you may perceive
Task
8:
Clicking on the "generative
spiral" button, where does the generative spiral start, in all these
configurations? What is the effect of moving the cursor radially
from the center of the Parameter space? What is the effect of rotating
the cursor around that center? Can you see some relation between
these motions and the way the values of r and a
change? Can you see that, as you move the mouse over the
entire disk in the parameter space, you will cover all the possible
values of r between 1 and • and
a between 0^{0} and 360^{0}?
Task
9:
Turn off the generative spiral
for now and click on the "Closest Point" button. Two points in the
Configuration Space now appear colored in red and green respectively.
They are the 2 closest points to the rightmost point of the central
disk. Which is closest? 2^{nd} closest? Now click on the
"Circle" button. Where is this circle centered, and through which
special point does it pass? Record some of the changes in the configurations
that occur when you move the cursor through the Parameter Space.
In particular, discuss what happens when the cursor crosses a white
line in the Parameter Space.
Task
10:
Click on the "parastichies"
button. What is the relationship between the green/red parastichies
and the green/red closest points? Can you confirm, or reformulate
the rule you gave in Task 5 in the light of this?
Task
11:
Count the green and red parastichies
in some of the spiral configurations, noting down, in a drawing,
the rough location of the mouse cursor in the Parameter Space. Are
there configurations that seem to you more like flower than others?
Are there typical locations of the cursor in the parameter space
where these configurations occur? What is the set of points in the
Parameter Space that yield only one family of parastichies in the
Configuration Space?
Each of the patterns you have observed
has a certain number of green and red spirals. Botanists have classified
these patterns according to these parastichies numbers. Hence, if
a pattern has 23 red and 47 green spirals, it is said that its Parastichy
numbers are (23, 47) (A pattern with 23 green and 47 red would
still be (23, 47) : the smallest number always goes first). The
amazing phenomenon observed in most (about 92%) plants that have
these spiral patterns is that the Phyllotaxis type numbers are successors
in the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13,
21, 34, ….
Task
12:
Check that the plant examples
you looked at in Task 1 have as type numbers 2 successors of the
Fibonacci sequence. If your original count did not give you this,
count again: it is probable (although not certain!) that you made
a mistake.
Task
13: Back to the
Spiral Applet. In the Parameter Space, the regions where the
configurations have same Parastichy numbers are separated by certain
curves. Which are they? Indicate them on a copy of the
Parameter Space. Write, in each of a few of these regions, the
Parastichy numbers to which it corresponds. Which path should you
follow if you want the Parastichy numbers to increase according
to the Fibonacci sequence? Indicate a possible path on your paper
copy of the Parameter Space.
Task
14 (voluntary basis!):
What do you wish this program
showed that it does not?
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

