Fibonacci Sequence.

Formulation of Fibonacci Sequence

 Suppose you have a pair of rabbits. They are young.  After a month they become adults and mate and the following month they give birth to another pair of rabbits. So now in the third month, there are 2 pairs of rabbits. In the fourth month, the first pair gives birth to another pair of rabbits and the second pair becomes adults and mates. so, now we have three pairs of rabbits in the fourth month. Then in the fifth month, the first pair and the second pair give birth to another pair of rabbits and the third pair becomes adults and mates. So, in the fifth month, we have five pairs of rabbits.

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And this sequence of numbers becomes something like this

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

In the zeroth moth 0

In the first month 1

In the second month 0+1=1

In the third month 1+1=2

In the fourth month 2+1=3

In the fifth month 2+3=5

In the sixth month 3+5=8 and so on.

From the above relation, we can predict incoming numbers as we can see that a number in the sequence is equal to the sum of two consecutive preceding numbers.

This sequence was described in Indian mathematics as early as 200 BC in Pingala's work enumerating the possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

Golden Ratio

In mathematics, two quantities are in the golden ratio if their ratio is equal to the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b,

where the Greek letter phi ( or )  denotes the golden ratio. The constant   satisfies the quadratic equation  and is an irrational number with the value

1.618033988749....

 Fibonacci numbers are closely related to the golden ratio: Binet's formula expresses the nth Fibonacci number F_n in terms of n and the golden ratio and means that the ratio of two consecutive Fibonacci numbers tends toward the golden ratio () as n increases.

 (2+3)/3=1.666...

(3+5)/5=1.6

(5+8)/8=1.625

(8+13)/13=1.6154

 1.618033988749....

Fibonacci Rectangles

But it might not be so surprising. You can also give any specific number session your own name. So what is so important about these Fibonacci numbers? Fibonacci numbers appear unexpectedly often in mathematics, so much so that the entire magazine Fibonacci Quarterly is devoted to their study. Applications of Fibonacci numbers include computer algorithms and graphs called Fibonacci cubes used to interconnect parallel and distributed systems. They also occur in biological environments such as tree branching, leaf arrangement on a stem, pineapple fruit sprouts, artichoke flowering, curly fern, and cone bract arrangement.

If we take squares with sides of length  F_n and join them in the following manner the spiral formed is known as the golden spiral.

Now, this is what makes mathematics so special when an ordinary thought turns out to be a fundamental part of the universe. An ordinary thought and it turns out to be something very special. An Italian mathematician Luca Pacioli termed this golden ratio "The Divine Ratio". The golden ratio is so important and essential component of the universe. We can find it in nearly every single aspect of this universe.