## Introduction

In the realm of Mathematics, there are many concepts that can be applied to multiple fields of mathematics. One of these concepts is the “Fibonacci sequence” that was developed by Leonardo Pisano during the 13th century. The Fibonacci number and sequence proposed by him gives a clear indication that mathematics can be connected to many things that may seem unrelated to it. The Fibonacci sequence has remarkable characteristics; it can be applied to various fields like discrete mathematics, number theory and geometry. By definition, the numbers in the Fibonacci sequence starts with either 0 and 1, or 1and1, and each subsequent number is the sum of the previous two numbers. The Fibonacci sequence is the best example of any recursive sequence. This sequence appears in numerous mathematical problems.

Fibonacci problem: Let us assume 2 rabbits (one male and one female) are born on 1st January. Assuming all the months have same number of days and rabbit starts producing young rabbits exactly after two months they were born i.e. after reaching two months age, each pair produces another pair (one female and one male), and then mixed pairs are produced every month thereafter and no rabbit dies in between. After one year, let us calculate the pairs of Rabbit produced?

## Fibonacci Solution: Solution provided is the Sequence given by Fibonacci:

0,1,1,2,3,5,8,13,21,34,55,89,114 Thus, after completion of one year, pairs of rabbits produced will be 144. Thus the Fibonacci sequence and Fibonacci numbers can be defined recursively as:

fn+2= fn + fn+1 where n>0 or n=0.

## Applications of Fibonacci sequence

The Fibonacci sequence can be related to various items in the nature like: growing buds on the plants and trees, the growth of branches on trees, the spiral hexagon observed on the pineapple, the pinecone’s rows, the seed heads, the petals growing on various flowers such as sunflower, the starfish, shells observed on snails and nautilus, the spiral galaxies and the chambers of vegetables and fruits like apple, lemon, Chile etc.

## Properties of Fibonacci sequence

Taking the Fibonacci numbers from odd indices, their sum is f2n i.e. from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n , take numbers at odd indices and sum them, their sum will be equal to number to f2n. For example: 1+2+5+13= f8=21 which is correct.

Taking the Fibonacci numbers from even indices, their sum is f2n+1 - 1 i.e. from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n , take numbers at even indices and sum them, their sum will be equal to number to f2n+1 – 1. For example: 1 +3 + 8 = 12=f7 - 1.

Golden Ratio: Sum of all (fn)/(fn-1) produces golden ratio. For example: 3/1+ 5/3 + 8/5 + 13/8 gives golden ratio ) / 2.

## Golden Ratio

Golden section and Golden ratio were first mentioned by Euclid in Elements in 300 B.C. Euclid proposed the problem for the division of a line into golden section. He proposed that when a unit segment is divided into two lengths, then the ratio of the smaller part to the larger part of the whole segment is equal to the ratio of the larger part to the whole length of the segment. For example: Let the whole length of the segment is x+y, where x is the larger part and y is the smaller part, then golden ratio can be defined as:

(x+y)/x=x/y= golden ratio represented by Greek letter phi ( or ). Its value is

.

Golden ratio is an irrational number. Golden ratio fascinated the Greeks a lot, as it appears frequently in the field of geometry. One of the best applications of the Golden ratio in the geometry can be, the proportion in the lines of the pentagram. Also in pentagon, when all the sides are equal, the diagonals when intersect divide each other in the golden ratio i.e. the ratio between the diagonal and a side is divine ratio. Many of the greatest architectural structures in the world are designed in such a way that they have golden ratio in their proportions. Many artists have used the golden ratio throughout the history for their works as found it really beneficial and pleasing.

## Origin of Fibonacci

According to Dunlap (2003), the Italian mathematician Leonardo Pisano was born into a merchant family in Pisa around 1175 A.D. His father, Bonacci, held a diplomatic post in the city of Bugia in the North Africa and so Fibonacci was educated by the Mohammedans of the Barbary. Fibonacci travelled the Mediterranean region with his father during his childhood, which introduced him to innovative mathematical ideas and concepts from numerous countries. His travel fuelled his love for mathematics as he returned to Italy in 1200 A.D. to publish his book, Liber Abaci (Book of the Calculations). The Arabic system of numbers was introduced by this book to Europe and thus established Fibonacci as one of the most reputed mathematicians of his time.

The book provided a solution to the problem involving the progeny of a single pair of rabbits which became the basis of the Fibonacci sequence (series). The Fibonacci sequence is thus defined by the formula: fn= fn-1+fn-2 where n>3 or n=3. Each Fibonacci number is obtained by adding two previous Fibonacci numbers together.

## Other names for the Golden ratio

Golden Ratio is also known as Golden mean, Divine proportion, Golden section, extreme and mean ratio, golden section and golden number. The golden ratio is often used when taking ratios of distances in geometric figures like decagon, pentagram, pentagon and dodecahedron. The golden ratio is denoted by , or sometimes . For example there is a rectangle with sides in the ratio 1: x. can be defined as the unique number x such that when the original rectangle is partitioned into a new rectangle and a square with the new rectangle having the sides in the ratio 1: x. Such a rectangle is called golden rectangle. So, it can be defined as:

/1=1/-1 giving using the quadratic equation and taking positive sign, the value of comes put be: . .

## Human Body, Fibonacci sequence and the Golden Ratio

Human body illustrates the Divine proportion. For example, when we look at the proportions of our index finger, it is observed that each section of the index finger is larger than the proceeding one by Fibonacci numbers 2, 3, 5 and 8 also producing the Fibonacci ratio of 1.618

Figure 1 Proportions of the index finger

## The ratio of the forearms to the hand is also phi.

Figure 2 Ratio of Forearms to hand

## Golden Ratio in Nature

One of the most interesting features of both the Golden ratio and Fibonacci series is that they appear frequently in nature, Golden Ratio is one of the fundamental characteristics of the universe that are followed by many items in nature. For example: Flower petals arrangements.

The number of petals in any flower constantly follows the Fibonacci sequence. For example lily which has three petals follows the Fibonacci sequence. Buttercups have five leaves. Each petal in flower is placed at 1.618034 per turn (out of 360-degree circle) which allows best exposure to sunlight and other factors.

Figure 3 Petals arrangement

The Fibonacci sequence is also found in the spiraling pattern produced by leaves on the plants’ stems. Animal bodies also exhibit Fibonacci sequence like on moving from navel to the floor and top of the head to the navel is the Divine ratio.

Figure 4 Animal Body

## Conclusion

Ultimately, the Fibonacci sequence applies to not only various parts of mathematics but also to many different aspects of nature and environment. It can be concluded that there is an abundance of the golden ratio, Fibonacci sequence, Fibonacci numbers and spirals in nature. They are found everywhere. Since, Fibonacci sequence can be found in numerous areas both inside and outside of the realm of the mathematics, research done on Fibonacci sequence and Golden ratio helps to understand its applications in areas like nature, human and animal bodies and art etc.

The existence of this Fibonacci sequence is not coincidental; it has been properly established and maintained. The mathematical properties of the Fibonacci numbers can be explored even more in today’s mathematical curriculum. Research on Fibonacci numbers helps in exploring the existence of Fibonacci sequence in the aesthetic nature of God. The Fibonacci sequence and numbers are simply the example of God’s power and authority over mankind.

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## References

Charran, S. R. (2011). The Equation of Life. The United States of America: LifeCode Series.

Dunlap, R. (2003). The Golden Ratio and Fibonacci Numbers. Singapore: World scientific publishing.

gizmodo.com. (2015). 15 Uncanny Examples of the Golden Ratio in Nature. Retrieved from io9.gizmodo.com: http://io9.gizmodo.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature

Leonesio, J. M. (2007). Fascinating Characteristics and Applications of the Fibonacci Sequence. Liberty University.

Posamentier, A. S., & Lehmann, I. (2012). The Glorious Golden Ratio. New York: Prometheus Books.