
The 'Chomskyan Era' (2000) - GmeSalazar
http://www.chomsky.info/books/architecture01.htm
======
s_q_b
> _" Or, take the mathematical series called the 'Fibonacci series'. It shows
> up all over the place in nature; nobody knows exactly why. If you take a
> sunflower and you look at the flower, it has spirals that go in different
> directions. The number of parts that appear in adjacent spirals are related
> to one another as successive terms in the Fibonacci series. You find that
> kind of thing all over nature; it is not well understood why. There is
> something about the physical world that forces certain kinds of structures
> to emerge under particular conditions."_

So this is why I find Chomsky's conclusions to be suspect when he speaks about
topics of which I have no understanding: he's often casually incorrect about
topics in other fields. We understand why the Golden Ratio shows up all over
the place, especially in biology: it doesn't.

In reality, it's the logarithmic spiral that's common, and it's a due to
logarithmic spirals being a necessary characteristic of certain structures
that exhibit self-similarity.

This quality was well known to Renaissance and Enlightenment era
mathematicians and physicists, as well as to the ancients as the "spira
mirabilis", or "marvelous spiral." But it seems to have lost favor in the
modern era to a mythos surrounding the Golden Ratio Phi and the Fibonacci
sequence.

For example, Jacob Bernoulli, the famed mathematician and no stranger to the
Golden Spiral and the Spira Mirabilis, requested the latter be placed above
his headstone, with the inscription "Eadem mutata resurgo," meaning "Although
changed, I shall arise again," a reference to the then well-understood ability
of logarithmic spirals to change scale while preserving shape.

Living organisms likely exploit this property of self-similarity for easy
scaling. It's likely helpful to sunflowers to maximize the area of solar
exposure, and snails to maximize living space. Once an organism evolves a
roughly spiral structure, logarithmic spiral patterns are easy local maxima
for the evolutionary algorithm to find, because it gives organisms the ability
to scale aspects of their biology without major structural changes. Non-living
systems exhibiting logarithmic spirals, such as certain galaxies, are a result
of various (different) physical forcing functions that cause self-similarity
to arise.

One way to debunk the pseudo-mystical notions surrounding Phi is to very
carefully measure the spirals themselves. What you'll actually find is not a
series of systems all approximating the same Golden Spiral, but rather very
different systems all exhibiting different logarithmic spirals, with the
shared characteristic of self-similarity.

~~~
zenogais
So far as I can tell your post is actually "casually wrong". The sunflower
seed example given by Chomsky seems to be mathematically well supported as a
representation of Phi/Fibonacci sequences [1][2]. There's no mystification
going on here, Chomsky is simply saying we don't fully understand the
morphogenetic systems that produce these results. You offered some speculation
as to why this might be, but so far no formalized results. Rene Thom did some
excellent early work in this area, and demonstrated that chaotic dynamics can
help to model some of these issues, but the topic is far from fully explored
[3].

[1]:
[http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html](http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html)

[2]: [http://www.maths.surrey.ac.uk/hosted-
sites/R.Knott/Fibonacci...](http://www.maths.surrey.ac.uk/hosted-
sites/R.Knott/Fibonacci/fibnat2.html)

[3]: [http://www.amazon.com/Structural-Stability-Morphogenesis-
Adv...](http://www.amazon.com/Structural-Stability-Morphogenesis-Advanced-
Classics/dp/0201406853)

~~~
Cowicide
Whoops, you must have written your response and submitted as I was typing
mine. Thank you for saying this far more succinctly and intelligently than I
could have (or tried to).

