96 // What is the Golden Number?

Hint: it will be useful to know that the nth harmonic number

1 1 1 1

Hn ¼ 1 þ þ þ þ Á Á Á þ

2 3 4 n is approximately equal to

log n þ g

where g is Euler's constant, which is roughly 0.577 215 664 9.

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Answer on page 276

What is the Golden Number? The ancient Greek geometers discovered a useful idea which they called `division in extreme and mean ratio'. By this they meant a line AB being cut at a point P, so that the ratios AP : AB and PB : AP are the same. Euclid used this construction in his work on regular pentagons, and I'll shortly explain why. But first, since nowadays we have the luxury of replacing ratios by numbers, let's turn the geometric recipe into algebra. Take PB to be of length 1, and let AP ¼ x, so that AB ¼ 1 þ x. Then the required condition is

1þx x

¼

x 1 so that x2 À x À 1 ¼ 0. The solutions of this quadratic equation are

p

1þ 5

f¼ ¼ 1:618 034 . . .

2 and

p

1À 5

1Àf¼ ¼ À0:618 034 . . .

2 Here the symbol f is the Greek letter phi. The number f, known as the golden number, has the pleasant property that its reciprocal is

p

1 À1 þ 5

¼ ¼ 0:618 034 . . . ¼ f À 1

f 2 What is the Golden Number? // 97

The golden number, in its geometric form as `division in extreme and mean ratio', was the starting point for the Greek geometry of regular pentagons and anything associated with these, such as the dodecahedron and the icosahedron. The connection is this: if you draw a pentagon with sides equal to 1, then the long diagonals have length f:

How f appears in a

regular pentagon.

The golden ratio is often associated with aesthetics; in particular, the `most beautiful' rectangle is said to be one whose sides are in the ratio f : 1. The actual evidence for such statements is weak. Moreover, various methods of presenting numerical data exaggerate the role of the golden ratio, so that it is possible to `deduce' the presence of the golden ratio in data that bear no relation to it. Similarly, claims that famous ancient buildings such as the Great Pyramid of Khufu or the Parthenon were designed using the golden ratio are probably unfounded. As with all numerology, you can find whatever you are looking for if you try hard enough. (Thus `Parthenon' has 8 letters, `Khufu' has 5, and 8=5 ¼ 1:6 ­ very close to f.* )

Another common fallacy is to suppose that the golden ratio occurs in the spiral shell of a nautilus. This beautiful shell is ­ to great accuracy ­ a type of spiral called a logarithmic spiral. Here each successive turn bears a fixed ratio to the previous one. There is a spiral of this kind for which this ratio equals the golden ratio. But the ratio observed in the nautilus is not the golden ratio.

* Well, actually `Parthenon' has 9 letters, but for a moment I had

you there. And 1.8 is a lot closer to f than many alleged instances

of this number.

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