Put on your designer's eye, and consider the following rectangles.  Which is most pleasing to the eye?
 

Rectangle 1	Rectangle 2	Rectangle 3
Too flat! Ratio of length to height: ·· = ··	Too blocky! Ratio of length to height:  ·· = ··	Pretty nice! Ratio of length to height: ·· = ··

Notice that it is the ratio of the long side to the short side that determines the shape of the rectangle:

Now let's approach this in a more mathematical way.

Here's a rectangle with the sides labeled s = short side, l = long side:

Since we previously introduced the ratio of the long side to the short side (l/s) as the item of interest, let's rewrite our above equation by "flipping" both sides:

If we divide numerator and denominator of the right-hand side by s, we get:

and replacing l/s  (the crucial ratio, long to short) by x, we get:

Our method: use the quadratic formula:

Let’s do it.  What is the first thing you have to do?

• do these numbers look familiar? ··
• what does the Fibonacci sequence have to do with the solution to x² = x + 1 ?
• we have seemingly unrelated things being related mathematically
• it happens all the time!
• more about this below

• let's consider the positive solution of x² - x - 1 = 0 : 
• remember how it came about: it is the rectangle ratio long/short that satifies the "geometric mean" property of rectangles introduced earlier
• what is its actual value? do it on your calculator...  ··
• it is an irrational number: it actually goes on and on forever and never repeats in a pattern
• but rounded to 3 decimal places, it is:  ··
• remember that we discovered a "pleasing" ratio of long/short which was about 1.6?
• this very special "pleasing ratio" has a very special name:

• also referred to by the ancient Greeks as the Divine Proportion

Any rectangle for which the ratio of the long side to the short side is the Golden Ratio (approx 1.618) is called, appropriately enough, a golden rectangle.  All golden rectangles have the same "pleasing" shape,

For example, suppose we have a rectangle whose short side is 50 ft.  How long must the long side be for it to be a golden rectangle?  Since long/short must be 1.618 we have the equation

long/50 = 1.618

• we’ll use the Greek letter F (phi, pronounced "fee") to represent the Golden Ratio:

F = 

• we have: F² = F + 1      Why?  ··

F³ = FF² = F² + F = (F + 1) + F = 2F+ 1
F⁴ = FF³ = 2F² + F = 2(F + 1) + F  = 3F+ 2
F⁵ = FF⁴ = 3F² + 2F = 3(F + 1) + 2F =5F+3

• look at the boldface numbers above
• can you guess what F⁶ will be? ··
• YES! Can you guess a GENERAL FORMULA:

• a remarkable relationship between the Golden Ratio and the Fibonacci sequence!

Tell all your friends! They will be so excited! They will look at you funny!

Consider again the Fibonacci sequence:

1  1  2  3  5  8  13 . . . 89  144  233

Do some computing:
 

• as n gets bigger and bigger
• F_n/F_n-1 gets closer and closer . . .
• to what?   ··