Re: Detecting Design

Brian D Harper (
Sun, 26 Apr 1998 00:17:44 -0400

At 11:50 PM 4/23/98 -0400, Steven Schimmrich wrote:
> I have read (but am not sure I could rigorously prove) that this
>angle of 137.5 degrees is the angle between leaves that insures that
>the maximum area of each leaf is exposed to sunlight. Surely such an
>angle could arise through the process of natural selection and so what
>if the constant phi (the symbol usually used to denote the golden ratio)
>shows up - other constants like pi and e also pop up in surprising places.

There are several points I wanted to make with this
example. One thing I would like to emphasize is that
finding an explanation for this observation in no way
diminishes the beauty of it. Of course, this is
subjective. If someone disagrees that's fine.

The fact that one discovers the Golden Ratio in the
growth of plants does not in any way rule out the
three explanations that I had previously mentioned:
(1) developmental constraint, (2) historical contingency,
(3) natural selection, although it may cast a shadow
over the second alternative.

There are several things that would argue against
natural selection, some of which were already mentioned
by Bob Dehaan. I'm sure this point can be disputed,
but I really don't want to get into that right now.
I'll grant that NS is a possible explanation but argue
that developmental constraint is a much better and
more likely explanation.

I'm just going to give a very brief description of
the problem here, more details can be found in a
sequence of three papers by Douady and Couder: "Phyllotaxis
as a Dynamical Self Organizing Process" (in three parts)
which appeared in J. Theoretical Biology (volume 178, 1996).

This is also discussed in a less technical manner in
Goodwin's <How the Leopard Changed its Spots> and
in Webster and Goodwin's <Form and Transformation>.

First of all, the problem is easiest discussed in terms
of another one of those mathematical thingies which
seems to pop up every where, the Fibonacci series.
Actually, its not really the Fibonacci series itself
that's important but rather the simple growth law that
generates the series, namely any term in the series
is the sum of the previous two. One gets the ball rolling
by specifying the first two numbers. The most natural
way to start seems to be with 0 and 1, which gives the
Fibonacci series:

0 1 1 2 3 5 8 13 21 34 55 ...

Starting with 1 and 3 gives the Lucas series:

1 3 4 7 11 18 29 47 76 123 ...

This is another odd place where the Golden Ratio makes
an appearance. Successive ratios of pairs of any sequence
having this growth law will approach the Golden Ratio
in the limit. Actually, one gets pretty close very

1/2 = 0.5 2/3 = 0.667 3/5 = 0.6 5/8 = 0.625
8/13 = 0.615 13/21 = 0.619

1/3 = 0.333 3/4 = 0.75 4/7 = 0.571 7/11 = 0.636
11/18 = 0.611 18/29 = 0.621 29/47 = 0.617

The Golden Ratio is about 0.618.

Here I just want to give a hint as to why plant growth
might have something to do with these series. The details
are very interesting but would require a long post.
Basically, as a new shoot forms, its position is not
immediately set. It can and will move a little due to
interference with previous shoots. This can be 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. Also, there would be more interference
from the closest neighbor just as 13 (for example) is a
larger fraction of 21 than is 8.

Again, this is just a hint at why a series growth might
be related to plant growth. Douady and Couder have
developed a nonlinear dynamical morphogenesis model which
describes the growth of real plants very well. One thing
I neglected to mention earlier was that while the 137.5
degree divergence angle predominates in spiral phyllotaxis,
there are several exceptions. The Douady and Couder model
predicts the exceptional cases as well as predicting
that the 137.5 degree case will be the predominant pattern.
Interestingly, the exceptional cases are also tied to
the Fibonacci series and Golden Ratio. A very satisfactory
explanation which, IMHO, greatly enhances the beauty of
finding the Golden Ratio.

Morals of the story:

Suppose our hypothetical botanist, upon discovering the
Golden Ratio, threw up his hands and shouted "its design".
He might then miss out on discovering the explanation
described above. This is, of course, a common criticism
of design arguments. They can stifle further study.

But the same might be said of an adaptationist. They
might say "hey, this pattern optimizes the light
that the leaves receive, it must have been selected
for". Then they go watch TV or something.

Also, wrt Dembski's approach to detecting design through
pattern matching, we see here a pattern obtained from
pure mathematics finding expression in biology in
a marvelous and beautiful way. Surely this is as
significant as finding a sequence of prime numbers.
Yet this doesn't indicate design. (Depending on how
design is defined of course, as Howard keeps emphasizing).

Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University

"It is not certain that all is uncertain,
to the glory of skepticism." -- Pascal