Tuesday, March 27, 2007

Fibonacci Numbers

What are they and what do they show us?

The Fibonacci numbers form a number sequence that is named after Leonardo Fibonacci who was an Italian mathematician and arguably the most talented of the Middle Ages. Fibonacci did not however invent the sequence but did use it as an example in his most prominent work. Fibonacci posed this problem:


“How many pairs of rabbits will be produced in a year,
beginning with a single pair, if in every month each
pair bears a new pair which becomes productive
from he second month on?”






Image courtesy of Dr. Ron Knott


1) After the first month the two rabbits have mated and produced 1 pair.
2) After the second month the female produced another pair, making 2 pairs of rabbits.
3) After the third month the original female produces a second pair, making 3 pairs total.
4) After the fourth month the original female produces yet another pair and the female born two months ago produces, now sexual mature produces her first pair as well, making 5 pairs. total.



The use of breeding rabbits is a simple example to illustrate the existence of this sequence. By starting with 0 and 1 the next number in the sequence can be determined by adding the last two. For example, to determine how many rabbits would exist on the 13th month you would simple add the last two numbers in the sequence 89 &144 to find that 233 is the next Fibonacci number.
The diagram below (Fibonacci squares) represents the proportionality between the 1st month and all sequential months leading up to the 8th month:


Image courtesy of mathacademy.com

This representation may seem arbitrary to its application in the natural world, but when superimposed over the image of a nautilus shell we can finally see the existence of Fibonacci numbers in nature:




Image courtesy of mathacademy.com

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