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Science
5 NUMBERS - PROGRAMME 3: THE GOLDEN RATIO
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Simon Singh investigates five very important numbers.
Monday to Friday 11-15 March 2002

The Golden Ratio

5 Numbers: The Golden Ratio


Leonardo Fibonacci was an Italian mathematician with a penchant for decimalization and rabbits! Having introduced the numbers 0 to 9 to Europe (like some medieval Big Bird from Sesame Street), he turned his attention to a different series of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55......

The Fibonacci sequence is generated by adding the previous two numbers in the list together to form the next and so on and so on...


Divide any number in the Fibonacci sequence by the one before it, for example 55/34, or 21/13, and the answer is always close to 1.61803. This is known as the Golden Ratio, and hence Fibonacci's Sequence is also called the Golden Sequence. Unlikely though it might seem, this series of numbers is the common factor linking rabbits, cauliflowers and snails.

Fibonacci used his sequence of numbers to investigate the population growth of his favourite furry lop-eared friend, the rabbit. He based his model on a maximum-security bunny heaven where rabbits cannot escape or die, and the problem he devised goes like this...

Suppose a newly born pair of rabbits (one male and one female) are put in a field. These rabbits take a month to become sexually mature, after which time they produce a new pair of baby rabbits or 'kits' (again, one male and one female). How many pairs will there be in subsequent years?

Think about it and if your answer is "enough for a very large pie", think some more.

 Number of pairs per month

Fibonacci Rabbits

As models of population growth go, it may not be the best, as it does not allow for rabbits to have more than two kits, to have kits of the same sex, or to take time off at Easter to deliver goodies. However, as a special series of numbers, the Fibonacci sequence has a hidden beauty all of it's own. Count the number of florets spiralling out from the centre of a cauliflower. Look closely and you will find two spirals running in opposite directions, and the number of florets in each are two consecutive Fibonacci numbers.

Like cauliflowers and rabbits, snails too are touched by nature's golden ratio. Draw a rectangle in the proportions of the golden ratio, then draw consecutively smaller 'golden rectangles' within it and join diagonal corners with an arc. The result is a perfect snail shell.

Snail Spiral


Fibonacci's Midas touch may have given mathematicians the blueprint for Mother Nature herself.

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