Sunday, January 2, 2011

Fibonacci Fibonacci

Leonardo Fibonacci discovered the Fibonacci, which converges on phi

In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.

Starting with 0 and 1, each new number in the series is simply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

The ratio of each successive pair of numbers in the series approximates phi (1.618 . . .), as 5 divided by 3 is 1.666... and 8 divided by 5 is 1.60.

The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi.  After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . . .


n mathematics, the Fibonacci numbers are the numbers in the following integer sequence:
 (Sequence A000045 in OEIS)
By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.
In mathematical terms, the recurrence relation defines the sequence Fn of Fibonacci numbers

With seed values

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonacci, "son of Bonaccio"). Fibonacci 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence was independently described in Indian mathematics and it is disputed which came first.
Fibonacci numbers are used in the analysis of financial markets, in strategies such as Fibonacci retrenchment, and are used in computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure. The simple recursion of Fibonacci numbers has also inspired a family of recursive graphs called Fibonacci cubes for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruit spouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pinecone.

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