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 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 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