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 . 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 satisfies the quadratic equation x² - x - 1 = 0. A number of interesting identities follow from this. For example, 1/ = - 1 = 0.6180339... and ² = + 1 = 2.6180339.... Hence ³ = + ² = 1 + 2 and so ⁴= 2 + 2² = 2 + 3 and in general ⁿ = a_n + a_n+1 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 : 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 .

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

Since can be defined using the square root of a rational, one can construct lines whose lengths have ratio 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 has the continued fraction expansion:

If you take the approximations to 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, is closely related to the Fibonacci numbers. For example, the nth Fibonacci number f_n = (ⁿ - ^-n)/₅ and an even cuter formula is f_n = [ⁿ/₅] where [x] means the integer part of x.