Well, there are even deeper relationships.
Fibonacci numbers are always integers. The generalization
for fractions are "Farey Numbers". Farey numbers are
ordinary fractions, but are endowed with a funny addition:
a/b + c/d == (a+c) / (b+d)
Note that the farey sequence has fibonacci number in
the numerator & denominator. E.G.
0/1 + 1/1 = 1/2
0/1 + 1/2 = 1/3
1/2 + 1/3 = 2/5
etc.
(although most farey's do not have fibonacci's in them -- e.g.
0/1 + 1/3 = 1/4, etc.)
For every rational p/q, there is a corresponding farey;
you can use the above to fill out the mapping from rational's
to farey's.
The mapping for farey number to reals is bizzarre:
its infinitely differentialble, its derivatives are all
zero at all rational numbers -- i.e. its infinitely flat
at at all rational numbers. But its not a straight line,
its a bumpy curve that is increasing ...
Farey's occur all over the place in fractal & chaos theory;
they're kind of an eye in the center of the storm.
The buds of the mandelbrot set get labelled with farey's,
the phase-locked regions of the forced rotor (phase
locked loop, a practical application of chaos, found
in modern stereo's and tv sets) occur with widths given
by farey numbers. Farey's have been observed in quantum
mechanical processes.
Oh, and since farey's are generalizations of fibonacci's,
they of course can be observed in nature, and the Golden
mean will also crop up from time to time.
People like Prusinkiewicz have done a wonderful job of
showing how fractal/chaotic processes, when expressed
in terms of grammers, give rise to biological forms. Since
farey's & fibonacci's occur in fractals/chaos, its not
surprising that they should also appear in biology.
Thus, I firmly beleive that nature can be firmly anchored
in physics and math, and does not witness the intervention
of the hand of God.
The real theological question is:
"Is mathematics an accident, or was math cleverly designed
by an omnipotent God?"
In other words, did God create algebra and geometry, in such
a way that all of physics and mathematics finds expression
in our real universe? Or is the fact that 2+2=4 independent
of God? Did God create Pi=3.14159... so that it would have
all those magical properties? Or is Pi a fixed absolute
of euclidean space, something that exists of its own right,
without having a creator? Did God create the mandelbrot set,
or does it exist simply because math exists?
Since all numbers, integer, rational, irrational, real,
countably and uncountably infinite, and the uncountably
infinite number of infinitessimals that populate the "spaces"
between real numbers can all be derived with a very simple
construction (John Conway's) from the empty set, maybe the
question is, "Did God create the empty set (and the rest of
the universe just "happened" as a consequence of that)?"
We will never know, since knowing would probably violate
Godel's theorm, which states that some thngs can be true
without being provably true ...
--linas
Oh, and have a happy new year!
It's been rumoured that Rick Becker said:
>
> Cross-snip:
>
> any feedback?
>
> Return-Path: <owner-evolution@udomo.calvin.edu>
> X-Sender: bharper@pop.service.ohio-state.edu
> Date: Tue, 30 Dec 1997 21:23:02 -0500
> To: evolution@calvin.edu
> From: Brian D Harper <bharper@postbox.acs.ohio-state.edu>
> Subject: RE: ABCD... Fibbonacci and gold
> Sender: owner-evolution@udomo.calvin.edu
> X-UIDL: a7773610dac96e190af997000c45b1cc
>
> At 09:37 AM 12/30/97 -0800, Greg wrote:
> >
> >Brian,
> >
> >[...]
> >
> >> <<sidelight: here's an extra credit problem for any
> >> adaptationists in the crowd. Successive pairs of the
> >> Fibonacci series define the phyllotaxy patterns of
> >> leaves on a plant. In the case of spiral phyllotaxis,
> >> successive leaves are located at angles that
> >> divide the meristem in proportions of the Golden
> >> Section. How would natural selection account for
> >> such a precise arrangement? Or is it design ;-) >>
> >
> >Obviously design ;-). But I'm not clear on what is
> >the case. What are these patterns, and what exactly
> >do the successive pairs of F. numbers determine?
> >What is the meristem and which angles are in the
> >proportion of the Golden Ratio? (Is this true for
> >all plants, or just some or what?)
> >
>
> Actually, the Fibonacci numbers and Golden Section
> are referring to the same phenomenom. I introduced
> the idea in terms of FNs in order to have a tie in
> with AIT. I also believe AIT gives a clue as to
> what I believe is the best explanation.
>
> By way of background for those who don't know about
> the Golden Ratio, consider the problem of dividing
> a rectangle into a square and another rectangle where
> the new rectangle has the same proportions as the
> original. Let the long side of the original be
> A and the short side B. The new rectangle will
> have longer side B, shorter side C. For the two
> rectangles to be similar we must have
>
> R = B/A = C/B
>
> where R is the Golden Ratio. We also must have
> A = B + C. Combining these we get
>
> R + R^2 = 1
>
> the positive solution of which is
>
> R = (SQRT(5)-1)/2 =~ 0.6180
>
> Now take ratios of successive pairs in the Fibonacci
> sequence:
>
> 1/2 = 0.5 2/3 = 0.667 3/5 = 0.6 5/8 = 0.625
> 8/13 = 0.6154 13/21 = 0.6190
>
> this converges after a few more terms to the Golden
> Ratio.
>
> 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.
>
> 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.
>
> Fades to the theme of twilight zone ...
>
>
>
>
>
> Come to think of it, I saw a little math show for
> kids with my daughter several years ago. The
> history of the Golden Section was discussed in
> some detail with many examples from ancient Greek
> architecture. They then showed a multitude of examples
> where the pattern emerges in biological forms. The
> leaf pattern was one example but there were several
> others that I can't recall now. Is the Golden Ratio
> an example of the Archetype that Richard Owen searched
> for? Or is the arrangement beneficial to the plant
> in some way so that one could imagine it being selected
> for some time in the past?
>
> 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
>
> Refractal Design, Inc. | voice: 508-777-5500 / fax: 508-777-6575
> 57 North Putnam St. | Internet: rbecker@refractal.com
> Danvers, MA 01923 | Home page: http://www.refractal.com/
> USA (Our area code is changing to 978)
>
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