Spiral Phyllotaxy
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Suppose you are a botanist investigating possible
geometrical growth laws in plants. In particular, you
are studying a group of plants displaying a growth
pattern which has come to be referred to as spiral
phyllotaxy. But you don't know this obviously, since
you are the lucky scientist who is going to make this
discovery :). As you look down the stem of a plant
from the top you note that successive leaves form a
spiral pattern as you move up the stem with a constant
angle of divergence. Careful measurements reveal this
angle to be very nearly 137.5 degrees. As you study
more and more plants with this spiral pattern you
find this same constant divergence angle again and
again.
Well, this is not particularly surprising. Its not
really surprising that the divergence angle should
be a constant. This constant must be some number,
why not 137.5? As to why the same angle in so many
plants one imagines three possible explanations,
all perfectly reasonable: (1) some type of developmental
constraint, (2) historical contingency (frozen accident)
or (3) natural selection (this particular angle confers
some advantage and was thus selected for during evolution).
OK, fine. Several weeks later you are reading your
favorite "joy of math" book during one of your many
"time-outs" imposed by the Emperor, err, I mean the
Department Chair. You are fascinated to learn about
the Golden Rectangle and the mystical and magical
Golden Ratio. The ratio that Kepler referred to as
the "Divine Proportion" and a "precious jewel", one
of the two "great treasures" of geometry, the other
being the theorem of Pythagoras.
Now the thought occurs to you: What angle will
divide a circle into the divine proportion?
IOW, consider a circle of circumference A and
some angle that divides the circumference into two
parts B and C (A = B + C) in such a way that the
ratio C/B = B/A = R, the Golden Ratio. This is
a fairly simple problem and after a few moments
you discover, to your great horror :), that the
required angle is 137.5 degrees.
And so you have discovered that the divergence
angle during the growth of the plants you have
been studying divides a circle into the Divine
Proportion. Surely a more astounding result than
a sequence of prime numbers.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"It is not certain that all is uncertain,
to the glory of skepticism." -- Pascal