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 360-222.5=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 parastechy 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 parastechy numbers (1,2), (2,3), (3,4), .... corresponds to a path of fractions 1/2, 2/3, 3/4, .... 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! The computer screen in front of you has a ration length/height which is most probably the golden mean. The Golden mean F is given by the formula
F = (1+÷ 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 360-360 (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!Task 36: Check, using the quadratic formula that F is a solution of of
x2 -x-1=0
.Task 37: We will denote here by Fn the nth term of the Fibonacci sequence. Hence:
F1=1, F2=1, F3=2, F4=3, F5=5 , etc...
What is F8? F10? Explain why the Fibonacci sequence can be characterised by :F1=1, F2=1
Fn+2= Fn+1+Fn.
Now remark that if we set Fn=xn then Fn+2= Fn+1+Fn. becomes
xn+2=xn+1+xn
which, dividing by xn simplifies to:x2=x+1
or:x2-x-1=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, Fn/Fn-1 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