Golden Ratio

Date: 01/03/98 at 22:32:18
From: Paul Park
Subject: Golden Ratio

Dr. Math, do you have any topics that I can use in my term paper about 
the golden ratio? There are so many things about it that I don't know 
what to write about.  Thanks.

Date: 01/04/98 at 02:43:02
From: Doctor Pete
Subject: Re: Golden Ratio

Hi,

Here are some ideas:

Geometry:  

   If you have a regular pentagon ABCDE, with side length 1, what is 
   the length of AC? (To find out, draw AC and BE to intersect at F, 
   and compare the similar triangles ACF, BEF.)

Trigonometry:  

   Prove that Cos[Pi/5] = G/2, where G is the golden ratio 
   (1+Sqrt[5])/2.

Recursions:  Let F[n] be the n(th) Fibonacci number, i.e.,

   F[n] = F[n-1] + F[n-2]
   F[0] = 0, F[1] = 1.

   This is a recursive definition.  Prove that

   F[n] = (A^n - B^n)/Sqrt[5],

   where A > B are the two roots of the polynomial x^2 - x - 1 = 0, 
   and Sqrt[5] is the square root of 5. (Hint:  Show that A = G. Then 
   clearly G^2 = G+1, G^3 = G(G^2) = G(G+1) = G^2+G = (G+1)+G = 2G+1, 
   and so successively compute G^4, G^5, and so forth in terms of G.  
   You should see a pattern, and prove it by induction, which gives 
   G^n. Similarly, do this for B.)

Continued fractions:  Find the value of the continued fraction

              1
     1 + ------------
                1
         1 + --------
                  1
             1 + ----
                  1 ... .

   (Hint:  Call the above value p. Then p = 1 + 1/p. Why?)

Continued roots:  Find the value of

     Sqrt[1 + Sqrt[1 + Sqrt[1 + ...]]] .

   (Hint:  Call the above value q. Then q = Sqrt[1 + q].)

There are many more curiosities and unexpected places where the golden 
ratio pops up, but that should give you a nice overview.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/