The brown square is a gnomon, so the larger rectangle (blue and brown together) should have side lengths in the same ratio as the smaller (blue) rectangle.

The blue rectangle has longer side length x and shorter side length 1, so the ratio of longer side to shorter side is x to 1, or x/1, or simply x.

The larger rectangle has longer side 1+x, and shorter side x,

so the ratio is is 1+x to x, or

Therefore .Cross multiplying, we get

 

Solving this equation by the quadratic formula, we find

 

The second of these solutions is negative, which doesn't make any sense (you canít have a square with negative length sides!) so the value of x must be the other solution,

So, if a rectangle has one side 1.618 times as long as the other (or, more precisely, times as long), then it has a square gnomon.

This number, , is referred toby the Greek letter Φ, phi.

We can use this to find a pattern in the powers of :

 

 

What is the pattern?

 

What is the corresponding pattern for ?

 

 

 

 

 

 

There is another connection between and the Fibonacci numbers:

 

Recall that Binetís formula to explicitly compute the Fibonacci sequence is

Notice that the first of the numbers raised to the power n is , and the other number is the other root to the quadratic equation .


 

http://emptyeasel.com/wp-content/uploads/2009/01/golden-parthenon.jpg

http://emptyeasel.com/wp-content/uploads/2009/01/golden-parthenon.jpg

 

 

 

http://www.oprah.com/oprahshow/Measuring-Facial-Perfection-The-Golden-Ratio

http://images.oprah.com/images/tows/200903/20090304/20090304-tows-golden-ratio-a-200x200.jpg