Transformations - Transformations and Projections in Computer Graphics - page 8

The Golden Ratio

Start with a straight segment of length l and divide it into two parts a and b such
that a + b = l and l/a = a/b .

l

a b
The ratio a/b is a constant called the Golden Ratio and is denoted φ .Itisoneofthe
important mathematical constants, like π and e , and was already known to the ancient
Greeks. It is believed that geometric figures can be made more pleasing to the human
eye if they involve this ratio. One example is the golden rectangle, whose sides are x
and xφ long. Many classical buildings and paintings involve this ratio. [Huntley 70] is
a lively introduction to the Golden Ratio. It illustrates properties such as

1+ 1+ 1+ √ 1+

1

φ = ··· and φ =1+ .

1
1+ ···

1+

The value of φ is easy to calculate. The basic ratio l/a = a/b = φ implies ( a + b ) /a =
a/b = φ ,which,inturn,means1+ b/a = φ or 1 + 1 /φ = φ , an equa ti on that can be
written φ 2 −φ− 1=0 . This equation is easy to solve, yielding φ =(1+ √ 5) / 2 ≈ 1 . 618 ... .

φ

1

1

1/ φ

(a) (b) (c)

Figure 1.4: The Golden Ratio.

The equation φ =1+1 /φ illustrates another unusual property of φ . Imagine the
golden rectangle with sides 1 × φ (Figure 1.4a). Such a rectangle can be divided into
a1 × 1 square and a smaller golden rectangle of dimensions 1 × 1 /φ .

The smaller

rectangle can now be divided into a 1 /φ 1 /φ square and an even smaller golden

rectangle (Figure 1.4b). When this process continues, the rectangles converge to a
point. Figure 1.4c shows how a logarithmic spiral can be drawn through corresponding
corners of the rectangles.

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