>Return-Path: <owner-evolution@udomo.calvin.edu>
>From: linas@linas.org
>Subject: Re: real life application
>To: fractal-art@aros.net
>Date: Tue, 30 Dec 1997 23:10:04 -0600 (CST)
>Cc: bharper@pop.service.ohio-state.edu, evolution@calvin.edu
>X-Hahahaha: hehehe
>Sender: owner-evolution@udomo.calvin.edu
>X-UIDL: 0e37f748f0ffb920a0a7d4d2f52d177d
>
>
>Hi,
>
>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)
>>
>> -------------------------------------------------------------
>> Thanks for using Fractal-Art, The Fractal Art Discussion List
>> Post Message: fractal-art@aros.net
>> Get Commands: majordomo@aros.net "help"
>> Administrator: noring@netcom.com
>> Unsubscribe: majordomo@aros.net "unsubscribe fractal-art"
>>
>
>
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)