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 r2 and then r3 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 rn. 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 00 and 3600?
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? 2nd 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