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 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 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 .