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The Golden Mean

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:

φ2 = φ + 1

or

φ2φ − 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:

  • φ2 = φ + 1
  • φ3 = 2 φ + 1
  • φ4 = 3 φ + 2
  • φ5 = 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 f1 = 1, f2 = 1, and we recursively define fn = fn − 1 + fn − 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.