Yes, it would be asymptotically, thanks for the
correction.
>Interesting. Sounds like something I've heard but
>forgotten. ;-)
>
>> 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.
>
>Ah, OK, I get it.
>
>> 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.
>
>So the question, then, is why are the leaves spaced by
>137.5 degrees, instead of some other number? (BTW, how
>tight is this, as I can see plants from where I am
>sitting for which it isn't true. Is it just a few plants?
>If so, which ones?)
>
Well, I guess I better put forward my standard disclaimer
before getting started. I have no expertise at all when
it comes to plants so I'm just passing along stuff that
I've read. Perhaps a plant expert is lurking about and
would like to comment.
OBTW, congratulations for taking the time to actually
look at a plant. Reminds me of a story I heard about
Aristotle. Apparently he had this theory that women
have fewer teeth than men. For some reason he seems
not to have bothered counting his wife's teeth ;-).
I looked at several plants myself walking to my
office and found the pattern in two of them, some
type of thorn bush and an ivy. Of course, I didn't
actually measure the angle, but it was highly
regular and clearly somewhere between 120 and 150
degrees.
In his book, Goodwin mentions three types of phyllotaxis
patterns, Distichous, Whorled and Spiral. Whether
there are more and these are just the most common
is not clear. I also got the idea that the angle was
universal for the spiral phyllotaxis pattern.
Scrounging around on the web I found the following
definition:
===============
II. PHYLLOTAXY:
The arrangement of leaves on a stem is called
phyllotaxy. Tomato phyllotaxy is termed
spiral because only one leaf is present at
each node and each successive leaf is
displaced approximately 137.5 degrees from the last.
Thus a line connecting successively older leaves
(leaves #1 to #5 in the figure) would make a spiral.
http://trc2.ucdavis.edu/CoursePages/PLB105/Students/Tomato/Stems/Branching.html
===============
The specific plant under discussion is the tomato, but the
wording suggests that the angle is part of the definition
of spiral phyllotaxy.
Goodwin mentions that a model for this growth patter was
developed by a certain Douady and Couder. Looking through
my files I found (for some odd reason!) that I actually
had a copy of their three papers "Phyllotaxis as a Dynamical
Self Organizing Process" (in three parts) which appeared in
J. Theoretical Biology (volume 178, 1996).
Apparently, this "problem" has a long history with a
total of 7 papers in the category "recent reviews".
Also, it seems that the basic rules of growth were
figured out by Hofmeister in 1868.
Also, from D&C it appears that there are exceptions
to the 137.5 divergence angle but that these are
"rare".
Now let's get back to the question, which you surmised
correctly as
>So the question, then, is why are the leaves spaced by
>137.5 degrees, instead of some other number?
Well, it's probably pretty clear by this time what I
consider as the best explanation for this angle.
I wasn't really too surprised when D&C mentioned
that there had been attempts at explaining the
angle in terms of selection pressures. I suppose this
is possible, but another question comes up right
away. Why the exceptions? Should we try to imagine
different selection pressures for these cases? Or
perhaps its due to historical contingencies, perhaps
the various angles are Gouldian spandrels.
The nice thing about the D&C model is that it also
predicts the angles observed for the exceptions and
provides a rational reason why the 137.5 solution
predominates.
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