
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
6.
The Farey Tree and the Golden Mean
In this section, we try to explain the
following observed fact: the divergence angle in plants is overwhelmingly
often close to either 222.5 degrees or 360222.5=137.5 degrees.
You can check that in the leaves of many plants. We will see that
these angles are related to the golden mean (which we will discover
in this section) and the Fibonacci sequence. ( Remember that the
divergence angle is the angle a , which you use to draw spirals
in 1. Observing Spiral Patterns or
in 4. Cylindrical Spirals .) In the
previous section, you filled up the different regions in the parameter
space with pairs of parastichy numbers. You might have noticed that
there is a definite pattern in the way these numbers appear. It
turns out that, in a way that is not quite direct (for now) this
pattern is an expression of a construction that
mathematicians call the Farey Tree.
Task
33: Note
that each fraction in the Farey tree has two "parents" in generations
above (e.g., 3/5 has parents 1/2
and 2/3). What is the rule to find the "child" fraction knowing
its two parents? Find the fractions that should be where there are
question marks. Better still, find the sixth generation of fractions
in the Farey tree.
Task
34: The
Fibonacci path that you found in Task 32, with parastichy numbers
(1,2), (2,3), (3,5), .... corresponds to a path of fractions 1/2,
2/3, 3/5, .... in the Farey tree. Is there something remarkable
about this path? Use a calculator to compute the decimal values
of these fractions (e.g. .5, .6666, .7.5 , ?, ?......). Compute
and write down enough of these to find a remarkable pattern/fact.
Task
35:
The Golden mean is a number which
is very popular among architects, painters, musicians and .... plants!
(although how recent this fad is in the arts is subject to serious contention.) The computer screen in front of you may have a ration length/height which
is close to the golden mean. The Golden mean
F
is given by the formula
F
= (1 + sqrt(5))/2
Use
a calculator to compute this number. How does it relate to what you
observed in the previous task? Sometimes the Golden mean refers to
1/F.
What is that number? What
is the angle 360 (1/ F)?
What is the angle 360360 (1/ F)?
How are these angle related, geometrically? The same specie of plant
may choose one or the other. There is apparently a preference depending
on which hemisphere the plant grows!
How is all this related
to our cylindrical spirals? In Bigger
Cylinder Applet turn "partition", "nearest" and "numbers". Look
at the set of gray arcs in the parameter space. As you go down one
gray arc, it comes to a branching point, with two gray arcs parting
and a black one continuing the gray one you were on. To label this
branching point, go down the black arc that goes down from it. If
you extend this black arc all the way to the bottom (see the arrows
on the diagram below), you get to a point on the x
axis. This point is always the base of one of the roundish bowls (regions
with only one family of parastichies), and its coordinate is a fraction.
If you label this branching point with this fraction you get the Farey
tree! You can get the coordinate of this bottom point from the parastichy
information as well: As you get down on a gray branch where the parastichy
numbers are (2,3), you know the denominator of the fraction attached
to the branched point is going to be 2+3=5. To find the numerator,
one has to use a little number theory...
In
the next few tasks we'll try to unveil more of the relationship between
golden mean and the Fibonacci sequence.
Task
36:
Check, using the quadratic formula that
F is
a solution of of
x^{2} x1=0
. Task
37: We
will denote here by F_{n} the nth term of the Fibonacci
sequence. Hence:
F_{1}=1,
F_{2}=1, F_{3}=2, F_{4}=3, F_{5}=5
, etc..
What is F_{8}?
F_{10}? Explain why the Fibonacci sequence can be characterized
by :
F_{1}=1, F_{2}=1
F_{n+2}=
F_{n+1}+F_{n.}
Now remark
that if we set F_{n}=x^{n} then F_{n+2}=
F_{n+1}+F_{n.} becomes
x^{n+2}=x^{n+1}+x^{n}
which, dividing by x^{n}
simplifies to:
x^{2}=x+1
or:
x^{2}x1=0
The plot thickens!
Task
38: 25
bonus points for your group
if you follow the above lead
and show that, as n becomes very big, F_{n}/F_{n1}
approaches F arbitrarily
closely...
1.
Observing Spiral Patterns >
2. Spiral Applet > 3.
Dynamical Model Applet > 4. Cylindrical
Spirals > 5. Cylindrical Spiral Applet
> 6. The Farey Tree and the Golden Mean

