Monday, June 4, 2012

0 1 1 2 3 5 8 13 21 34 . . .

The Fibonacci sequence is named after Leonardo of Pisa, aka Fibonacci, although the sequence had been described earlier in Indian mathematics in connection with Sanskrit prosody.
In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202), where Fibonacci considers the growth of an idealized and biologically unrealistic rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
At the end of the each month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month. This is the Fibonacci sequence, which is intimately connected with the golden ratio.
Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes. They also appear in biological settings,such as branching in trees, Phyllotaxis (the arrangement of leaves on a stem), the fruit spouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

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