What is the Fibonacci series, and to what extent is it true?

Italian Mathematician Leonardo Pisano( born in 1175 and died around 1250) also known as Fibonacci is mostly famous for his Fibonacci sequence. His name actually got originated from a misreading on a manuscript of “filius Bonacci”(son of Bonaccio).

Now discussing on his exceptional work on Fibonacci Sequence.The name "Fibonacci sequence" was first applied by Theorist Edouard Lucas in the 19th century.

In the field of Mathematics, Fibonacci numbers denoted as F_n.The sequence states that that each number is the sum of the two preceding numbers starting from 0 followed by 1.

The general term of the sequence

\(F_n\)=\(F_{n−1}\)+\(F_{n−2}\) where \(F_0\)=

\(0\)and \(F_1\)=\(1\) \(∀\) \(n>1\)

Thus the sequence becomes

0,1,1,2,3,5,8,13,21,34, and so on.

Fibonacci discovered a very interesting concept of the rabbit population.Rabbits usually never die and they are able to reproduce at the end of its second month.

Now if a male and a female rabbit that is a newly born pair of rabbits are placed in a field then they will always produce a new pair at the end of each month starting from the second month.

This way the following observations were made.

1. By the end of first month there is only one pair. \((F_1\)=\(1\))
2. Now By the end of second month, a new pair is born thus amounting to 2 pairs \((F_2\)=\(2\))
3. By the end of third month a new pair is born from the original pair thus amounting to 3 pairs \((F_3\)=\(F_2\)+\(F_1\)=\(2+1=3\))
4. By the end of fourth month again a new pair is born from the original pair and another pair is born from the first female produced by the original female amounting to 5 pairs \((F_4\)=\(F_3\)+\(F_2\)=\(3+2=5\))

We can conclude from the above mentioned facts that by the end of n month, the number of pairs will be

\(F_n\)=\(F_{n−1}\)+\(F_{n−2}\), which is the Mathematical generalised expression of the Fibonacci Sequence.

This fact about rabits is a big example of it being true. It is true in the sense it does exist in nature.

Now lets discuss the relation of Fibonacci Sequence and the Golden Ration and few applications of it in nature

Two numbers are said to be in Golden ratio if they are in the ratio of the sum of the numbers to the larger number. Most of the things in nature occurs according to this ratio including the spiral arrangement of leaves . The value of Golden ratio 1.618.1.618.Fibonacci Sequence and Golden ratio are interlinked

the Fibonacci sequence is 0,1,1,2,3,5,8,13,21,34,.........55.........

Now going by the golden ratio dividing each number by previous number we get \(\frac{1}{1}\)=\(1\) , \(\frac{2}{1}\)=\(2\) , \(\frac{3}{2}\)=\(1.5\) , \(\frac{5}{3}\)= \(1.6666\), \(\frac{8}{5}\)= \(1.6\), \(\frac{13}{8}\) = \(1.625\) , \(\frac{21}{13}\)= \(1.61\) , \(\frac{34}{21}\)=\(1.619\) , \(\frac{55}{34}\)=\(1.617\). We notice the values are converging towards 1.618 which is the golden ratio.

Most of the things in nature follows the Golden Ratio and the Fibonacci sequence.

The spiraling pattern of the sunflower and Pine cones follows the Fibonacci sequence.The branches of tree are also arranged in Fibonacci sequence. Even the galaxies follows Fibonacci pattern.Another very interesting fact about the Fibonacci number is that number of petals on flower Daisy is always a Fibonacci number (21,34,55) being most common numbers).The number of petals of flowers are often arranged in Fibonacci numbers. Like lilies have 3, Chicory have 21,daisies have 13 and all these are Fibonacci numbers.

Physics. In optics the number of different beam paths when a ray of light shines at an angle through two different transparent plates of different refractive index and material, there are k reflections ,for \(k>1\) and k is the Fibonacci number.

As recorded,1597,was the last year that was a Fibonacci number and the next will be 2584.