Associated Topics || Dr. Math Home || Search Dr. Math

### 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
```

```
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/
```
Associated Topics:
High School Fibonacci Sequence/Golden Ratio

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search