Actually, the Fibonacci numbers and Golden Section
are referring to the same phenomenom. I introduced
the idea in terms of FNs in order to have a tie in
with AIT. I also believe AIT gives a clue as to
what I believe is the best explanation.
By way of background for those who don't know about
the Golden Ratio, consider the problem of dividing
a rectangle into a square and another rectangle where
the new rectangle has the same proportions as the
original. Let the long side of the original be
A and the short side B. The new rectangle will
have longer side B, shorter side C. For the two
rectangles to be similar we must have
R = B/A = C/B
where R is the Golden Ratio. We also must have
A = B + C. Combining these we get
R + R^2 = 1
the positive solution of which is
R = (SQRT(5)-1)/2 =~ 0.6180
Now take ratios of successive pairs in the Fibonacci
sequence:
1/2 = 0.5 2/3 = 0.667 3/5 = 0.6 5/8 = 0.625
8/13 = 0.6154 13/21 = 0.6190
this converges after a few more terms to the Golden
Ratio.
Now let's apply the same idea to a circle of circumference
A. We want to divide the circumference into two parts
B and C in such a way that ratio C/B = B/A. Since
A = B + C we obviously get the Golden Ratio again.
The angle for the smaller arc segment is 137.5 degrees.
What does this have to do with plants? There are several
types of leaf patterns observed in plants. One is the
spiral pattern (ivy, lupin, potato). Imagine looking
down the stem of the plant from the top. Successive
leaves form a spiral pattern as you move up the stem
with the divergence angle being 137.5 degrees.
Fades to the theme of twilight zone ...
Come to think of it, I saw a little math show for
kids with my daughter several years ago. The
history of the Golden Section was discussed in
some detail with many examples from ancient Greek
architecture. They then showed a multitude of examples
where the pattern emerges in biological forms. The
leaf pattern was one example but there were several
others that I can't recall now. Is the Golden Ratio
an example of the Archetype that Richard Owen searched
for? Or is the arrangement beneficial to the plant
in some way so that one could imagine it being selected
for some time in the past?
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"... we have learned from much experience that all
philosophical intuitions about what nature is going
to do fail." -- Richard Feynman