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 can’t 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 to by
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 Binet’s 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://www.oprah.com/oprahshow/Measuring-Facial-Perfection-The-Golden-Ratio