Date: Mon, 25 Nov 1996 19:28:35 -0800
From: Metin Aktay <maktay-AT-superonline.com>
Subject: PLC: Errata : Re:PLC Numb&Gold>>Fibonacci&Gold 


I apologise for the below corrections:
I had switched the answers two the two equations:


> 
>                         "provided a=b+c"
> 
> for otherwise it has infinitely possibilities for ratios. When one
> solves the two consraints in conjunction, one finds out that the
> equations hold true only for numbers which adhere to
> 
>                a/b = b/c = 0.618....

                 SHOULD BE = 1.618....


> 
> It can also be solved for the inverse of same ratio and then one finds
> 
>                b/a = c/b = 1.618...

		 SHOULD BE = 0.618....
> 
> So the Golden Section is THE SAME AS THE CONVERGENCE OF FIBONACCI






Metin Aktay wrote (with errors which are corrected above): 
> 
> The engineer in me would like to make some input (re the Pat Sloane
> post) on the FIBONACCI SERIES and the GOLDEN SECTION, showing that they
> are actually the same, and on the GOLDEN MEAN:
> 
> 1. FIBONACCI SERIES:
> 
> > The Fibonacci series begins like this.
> >
> >                       1, 1, 2, 3, 5, 8, 13, 21, 34, 55......
> >
> > Each number is the sum of the two previous numbers.  It was discovered during
> > the middle ages by Leonard of Pisa, and I betcha Joyce mentions it somewhere.
> >  It's apparently a very mathematically sophisticated series, and I've heard
> > of whole clubs of amateur mathematicians, mostly in Russia, who devote
> > themselves to studying the Fibonacci series.
> 
> I am always amazed with the Fibonacci series both for its examples in
> nature, and for its nature of converging on the same ratio no matter
> what the beginning conditions are. I want to share an expose of the
> latter aspect.
> 
> Fib series is began with 0 and 1, which gives the next term as 1, and
> then 2, and so on. When this is carried on for some terms it becomes
> eivident that the ratio of any term to the previous approaches a
> quantity 1.618033989...
> 
> One can begin with 0 and any other number, say 234, and end up with the
> same convergence, which is expected, since this series is simply the
> multiplication of each term of the basic series by 234.
> 
> What interests me is this: one can begin the series with any two
> numbers, and see that as terms are generated it still converges on the
> same ratio. Say one begins with 123 and 22, where the series is 123, 22,
> 145, 167, 312, .... and presto out comes the same ratio again.
> 
> I find this intriguing, the fact that a rule creates the same output
> independent of the initial input. It also rings bells of the type which
> ring at the Santafe Institute where they study Complexity Theory, which
> is explaining complex events with propagation of simple rules, be it
> abstracts like Mandelbrot fractals or down to earth such as rules for
> economic behaviour.
> 
> I would venture that one of you could utilise it to give a macro
> framework to dictums such as "to err is human" or "truth will out" and
> maybe find parallels of similar inescapability treated in Joyce/Dante re
> human behaviour. I do not mean generating mathematical formulas, but
> simply having some insight into why human character supercedes human
> context.
> 
> 2. GOLDEN SECTION
> 
> > I think
> > especially the Golden Mean or Golden Section, in which
> 
> >                       a/b = b/c
> 
> What is missing from above is
> 
>                         "provided a=b+c"
> 
> for otherwise it has infinitely possibilities for ratios. When one
> solves the two consraints in conjunction, one finds out that the
> equations hold true only for numbers which adhere to
> 
>                a/b = b/c = 0.618....
> 
> It can also be solved for the inverse of same ratio and then one finds
> 
>                b/a = c/b = 1.618...
> 
> So the Golden Section is THE SAME AS THE CONVERGENCE OF FIBONACCI
> 
> which is logical because what we are doing in solving for the Golden
> Section is finding the convergence solution for Fibonacci, where given
> three successive terms T1,T2,T3, we are saying that T3=T1+T2 and that
> T3/T2=T2/T1=1.618....
> 
> 3.GOLDEN MEAN:
> 
> I do not remember this term connoting anything numerical or
> mathematical. I remember it as connoting the way of life advised by
> Aristoteles,among other Greeks, which is to sample everything in
> moderation relative to each other and thus live a balanced life.
> 
> I thank you for your attention.
> 
> Metin Aktay
> 
> Businessman from Istanbul, Turkey
> Home : Ihsan Aksoy sok, EVA apt. No:7/2, Camlik Etiler
>        Istanbul 80600, Turkey
> e-mail     : maktay-AT-superonline.com
> Home Phone : +90 212 265 10 16
> Home Fax   : +90 212 257 73 74
> Work Phone : +90 212 212 60 30
> Work Fax   : +90 212 212 60 32
> Mobile     : +90 532 274 17 71
> 
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