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From: skinner@saber.UUCP (Robert Skinner)
Newsgroups: net.math
Subject: Re: Pascal's Inverse Triangle
Message-ID: <1677@saber.UUCP>
Date: Mon, 8-Jul-85 13:19:33 EDT
Article-I.D.: saber.1677
Posted: Mon Jul  8 13:19:33 1985
Date-Received: Thu, 11-Jul-85 00:00:49 EDT
References: <2216@utcsstat.UUCP>
Organization: Saber Technology, San Jose, CA
Lines: 74

> Now, consider this, which I shall refer to as Pascal's Inverse triangle...
> 
>                                   1
>                                1     1
>                              1   1/2   1
>                            1   2/3  2/3  1
>                          1   3/5  3/4  3/5  1
> 
>                 this triangle also continues on for infinity.
>            
>       Each entry is the INVERSE of the sum of the two entries 
>        directly above, for example
> 
>                                      1/2 = 1/(1 + 1)
> 
>                               and    2/3 = 1/(1/2 + 1)
>  
> 
....
> 
> 
>                                         1
>                             X   =  -----------------
>                                     1   +   1
>                                           -----------
>                                             1  +  1
>                                                  ---------
>                                                    1  +  etc.
> 
>                 and this is the same as
>           
> 
>                                         1
>                            X  =  ---------------
>                                     1   +   X
> 
>                                 X * X  +  X  -  1 = 0
> 
>                         and this equations one positive root is
> 
>                                  (sqrt(5)-1)/2
> 
>            Has anyone come across this triangle before, and if so
>                have you anything of interest to share concerning it ?

*** REPLACE THIS LINE WITH YOUR MESSAGE ***

I've never seen the triangle you describe here, but the root you show
is also the Golden Ratio.  (Or the inverse: (sqrt(5)-1)/2=.618=1/1.618=
1/(sqrt(5)+1)/2).  The way I first heard it stated, the ancient Greeks thought
that the most pleasing ratio for a (Golden) rectangle were such that if a 
square were chopped off one end, it would yield another Golden rectangle.

In other words, x/1=1/(x-1) ==> x=1.618.

The Golden Ratio also pops up in Fibonacci numbers, the ratio of fn to fn-1
converges to 1.618, regardless of the values of f0 and f1.

Yet another odd place:  the five pointed star.  This is hard for me to 
describe, but suppose you draw a five pointed star like you did in grade 
school.  Assuming you draw a perfect star, the ratio of the length from the
body to the tip of one point, to the length of the rest of the same line
segment is .618=1/1.618.

It's nice to know the Golden Ratio lives in other places as well.
Anybody know of other places in math or geometry where the Golden Ratio occurs?

------------------------------------------------------------------------------

Name:	Robert Skinner
Mail:	Saber Technology, 2381 Bering Drive, San Jose, California 95131
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