Consider the red rectangle in Figure 1. If you add to that rectangle a square of height equal to
the height of the rectangle, you get another rectangle which has has different ratios.

Figure 1. A *poorly* proportioned rectangle.

The greeks, keen on geometry, were interested in a rectangle such that, when
you added a square, maintained the same proprotions, as is shown in Figure 2.

Figure 2. A *nicely* proportioned rectangle.

To find out what the ratio of the sides is, let us represent the width
of the rectangle by 1 and the height by *φ*. Then, the width of the
second rectangle in Figure 2 is *φ* + 1. This is represented pictorially
in Figure 3.

Figure 3. Ratio of sides.

Thus, we want *φ*:1 = 1 + *φ*:*φ*, or

Multiply both sides by *φ* results in the equation:

*φ*^{2} = *φ* + 1
or

*φ*^{2} − *φ* − 1 = 0
Solving this for *φ* yeilds:

Because there are two possible values, we will

## Properties

The golden ratio φ has some very nice properties:

- φ
^{2} = φ + 1
- φ
^{3} = 2 φ + 1
- φ
^{4} = 3 φ + 2
- φ
^{5} = 5 φ + 3

By this point, you probably recognize the Fibonacci sequence : 1, 1, 2, 3, 5, 8, 13, ... . This
brings us to our next observation: if f_{1} = 1, f_{2} = 1, and we recursively
define f_{n} = f_{n − 1} + f_{n − 2} as
the sum of the two previous values, then we get that:

Similarly, we note that φ^{-1} = φ - 1.

- φ
^{-1} = φ − 1
- φ
^{-2} = 2 − φ
- φ
^{-3} = 2 φ − 3
- φ
^{-4} = 5 − 3 φ
- φ
^{-5} = 8 φ − 5

Therefore, it is always possible to calcuate powers of φ by using just
integer addition and multiplication.

## Other Properties

Just like we can use the Fibonacci sequence to find φ, we can use φ to
find entries in the Fibonnacci sequence:

Additionally, you have probably seen the Sybase logo, a variation of the spiral shown in Figure 4.

Figure 4. Spiral resulting from the golden mean.