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The Farey Tree and the Golden Mean
Symmetry in Plants
A TaskBased Tutorial on the
Mathematics of Phyllotaxis
Note: this was part of the "Discovering Math" class
taught at Smith College in the Spring of 99. The previous part of
the class included studying symmetry of planar patterns.
1. Observing Spiral Patterns
In the first part of this tutorial, we get
introduced to the patterns and to some of the mathematical models
that help analyzing them.
Task
1:
In looking at the different
pictures below, notice that they all have the common trait of exhibiting
each two distinct families of spirals. These spirals emanate from
the center and wind in opposite directions. In each picture, count
the number of spirals in each of the two opposite families. Note
that Figures 3 and 4 are mathematically generated. Can you explain
how this type relate to botanical patterns of the type seen in Figures
1 and 2 as well as to one another? Which of Figure 3 and 4 seems,
to your taste, to better represent botanical patterns?
Figure 1

Figure 2

Figure 3

Figure 4

Creating Your Own Spirals
In this section, you will need the sheets
of concentric circles provided by your instructor (or download
them: bigcircle.pdf,
tightcircles.pdf),
a sharp pencil, a good eraser, scissors and maybe a ruler.
Task
2: Using
the sheet bigcircle, place one dot on each of the concentric circles
so as to form one spiral of dots. What geometric rule do you have
to follow to make this spiral look nice and regular? Invent a practical
device to implement this rule in a neat way, using paper, scissors,
and maybe a ruler (nature is quite precise!).
Remark: As
you may have observed, there are actually several ways of creating
regular spirals by placing one dot per circle. The patterns of Figures
14 are obtained by recursively moving a point to the next circle
and then rotating it by a fixed angle on that circle. This angle
is called divergence angle in botany.
Task
3:
Until now in this class, we
have defined a symmetry of a figure as a rigid motion (translation,
rotation, reflection, and glide reflection) that preserves the figure.
Assume that, in Task 2, you have followed the rule described in
the previous Remark to draw a regular spiral. If you imagine your
spiral going forever inwards as well as outwards, what kind of new
motion preserves the spiral?
Task
4:
Make another spiral on another
copy of bigcircle, following the rules described in the remark,
striving to spread the points so that they fill the space the most
evenly you can (Hint: try playing with the divergence angle a
to tighten or loosen your spiral as you would the coiling
spring of an old clock). Refrain for now to connect the dots. Once
you have done that on bigcircle, do it in the two figures of tightcircles.
What artistic device could you use in order for your pattern to
look more plant like?
Task
5: Look
at your artwork from Task 4. Do you see any spirals other than the
one you intended to draw? Somehow your eyes may connect the dots
of your original spiral in unexpected ways. Do connect some of these
dots, enough to show the pattern you perceive. What kind of rules
do you follow when connecting the dots in order to form these extra
spirals? How many families of such spirals do you detect and how
many spirals do you see in each family?
You have seen 2 kinds of spirals in your artistic
endeavour. The first kind, that you deliberately drew, is called
the generative spiral. The dots in this spiral follow all
the successive generations of circles and generate spontaneously
the second kind of spirals, which often come in families. This second
kind of spirals, which you drew in Task 5, is called parastichy.
You saw and counted parastichies in Task 1.
Task
6:
Is it easy to see the
generative spirals in Figures 14 of Task 1? Why? What does this
tell you about how plants form their botanical entities?
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

