Fibonacci Numbers and the Golden Ratio

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The sequence of Fibonacci numbers is given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …, in which each number is the sum of the two preceding numbers. This can be expressed as with and .

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Fibonacci (whose real name was Leonardo Pisano) found this sequence as the number of pairs of rabbits months after a single pair begins breeding, assuming that each pair of rabbits produces a pair of offspring when it is two months old.

As , the ratio of successive Fibonacci numbers approaches the limit , known as the "golden ratio". The ancient Greeks regarded as the most aesthetically pleasing proportion for the sides of a rectangle. In this Demonstration the black rectangle has sides proportional to . See how these approach the "golden rectangle" as you increase .

Commercial interests are apparently aware of the appeal of the golden ratio. Credit cards, for which you probably have begun getting offers, have almost exactly this shape.

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Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


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Details

S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 pp. 7–8.



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