OK, I'll try to explain it without getting into too
many boring details. The main factor in the growth
pattern has to do with the interference of a new
shoot with those just formed. The main parameter
here is probably pretty obvious, namely the time
T between the appearance of successive shoots, which
seems to be constant during most of the growth process.
The shorter this time period, the greater the interference.
There are two other important parameters which are
not so obvious. These are the radial growth rate V of
a new shoot and a critical radius R at which there
can be no further reorganization of the new shoot.
Before this critical radius is reached, the new
shoot is still "plastic" and can move around a little
thus affecting the divergence angle.
It turns out though that things are much simpler than
this since the three parameters above can be combined
into a single non-dimensional parameter G = V*T/R
which determines the divergence angle.
At small G there are multiple solutions, I count 13
solutions as G approaches 0. These range between
about 60 and 160 degrees. At intermediate to
large values of G there is only one solution. So, as
G gets smaller, more and more solutions appear. However,
these new solutions appear discontinuously. There is
only one continuous solution which turns out to be almost
constant for most values of G. This is the 137.5 solution.
The key point is that for almost all plants one starts at
some interm. to large value of G with G decreasing rapidly
to some constant value. Once G reaches this constant value
the divergence angle will remain the same. This final
G is often in the range where there are multiple
solutions, however, the initial value of G is almost always
in the interm. to large range where there is only one
solution. Starting at this particular point there is
then only one continuous path for decreasing G. This
explains the near-universal 137.5 degree divergence
angle.
Why does this near-universal value happen to be
associated with the Fibonacci sequence? Is this
something mysterious? What I hadn't realized
previously was that all the solutions are related
to something like a Fibonacci sequence in that
they have the same generative rule. This rule
is that the present term in the sequence is the
sum of the previous two terms. To get the Fibonacci
sequence one begins with 0,1. The rest of the sequence
is then generated by the rule. For example, the
Lucas series starts instead with 1,3 giving:
1 3 4 7 11 18 ...
This sequence turns out to be related to one of
exceptional (other than 137.5) cases.
I'm not completely sure why the dynamics of growth
is related to this particular generative law for
sequences. I suspect it's something like this:
The patterns are caused by interference from
previous shoots, which is modeled by new shoots
appearing in such a way as to minimize the repulsive
forces from previous shoots. It seems that only
the previous two shoots interfere significantly
with a new shoot (analogous to a particular number
in the Fibonacci sequence being determined by the
previous two numbers). So it kind of makes sense
generally speaking.
This is an example of what is often called a developmental
constraint. A constraint in the sense of a physical constraint
on the developmental process itself and a constraint on the
the paths that evolution might follow. The divergence
angle in this type of helical growth cannot be just any
old thing, it is constrained to take on certain values
because of the generative growth dynamics. This is
controversial in that some orthodox ultra Darwinian types
consider organisms to be kind of like putty, able to be
shaped in just about any manner due to external selective
pressures. For some odd reason, this reminds me of this
really silly article written by Dawkins (I think its on
the web) explaining why it is that organisms don't have
wheels. It would be funny except that he's serious :).
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