MT2002 Analysis

## The Golden Ratio

The number (1 + √5)/2 = 1.6180339... that we met in the last supplement is what the Ancient Greeks called the Golden Ratio *φ*. In 1509 the mathematician Luca Pacioli (1445 to 1517) wrote a whole book: *Divina proportione* (= The divine proportion) about this number.

It occurs in many different places in mathematics.

The number *φ* satisfies the quadratic equation *x*^{2} - *x* - 1 = 0. A number of interesting identities follow from this. For example, 1/*φ* = *φ* - 1 = 0.6180339... and *φ*^{2} = *φ* + 1 = 2.6180339.... Hence *φ*^{3} = *φ* + *φ*^{2} = 1 + 2*φ* and so *φ*^{4}= 2*φ* + 2*φ*^{2} = 2 + 3*φ* and in general *φ*^{n} = *a*_{n} + *a*_{n+1}*φ* where you might recognise the terms of the sequence (*a*_{n}). (See below.)

Given a rectangle with the property that when you remove a square from it, what is left has the same proportions as the original square, you may verify that this rectangle has the proportions *φ* : 1.

The Greeks used rectangles of this shape in many of their classical buidings since they believed such a shape was more aesthetically pleasing than any other rectangle.

The Ancient Greeks investigated the geometry of the *Pentangle*:

They found that the ratio *AB* : *AC* is the Golden Ratio *φ*.

From this a modern mathematician can deduce that 2 cos(36°) = *φ*.

Since *φ* can be defined using the square root of a rational, one can construct lines whose lengths have ratio *φ* using ruler and compass constructions. Hence one may construct an angle of 36° by such a method and so construct a regular pentagon by ruler and compass constructions.

You saw on the last page that *φ* has the continued fraction expansion:

*φ* =

If you take the approximations to *φ* given by cutting off this continued fraction after 1, 2, 3, ... terms you get the sequence (1 , 2 , ^{3}/_{2} , ^{5}/_{3} , ^{8}/_{5} , ^{13}/_{8} , ... ) of ratios of the terms of the Fibonacci sequence (which you might have spotted above).
As this last result suggests, *φ* is closely related to the Fibonacci numbers. For example, the *n*th Fibonacci number *f*_{n} = (*φ*^{n} - *φ*^{-n})/_{√5} and an even cuter formula is *f*_{n} = [*φ*^{n}/_{√5}] where [*x*] means the integer part of *x*.

This formula is named after the French mathematician Binet (even though de Moivre knew about it earlier).

JOC September 2001