Fibonacci And The Golden Ratio Mathematics Essay

Some aspects of mathematics can be dull and tedious from start to end, much of it however is intriguing and inspiring, when you truly see the beauty and the relevance. This is why I would like to bring to your attention the magic of the Fibonacci numbers. If you have ever looked at a sheet of paper and wondered Why do we use those dimensions? or looked at the leaf or an attractive plant and wondered Why can I never find a four leaved clover? then this may be of some interest. Many of these things are quite interconnected in a way you would not realise, and most of them are connected by the Fibonacci sequence.
If I return to one of my original questions Why can I never find a four leaved clover? it seems reasonable, that if you can find 3 leaved clover and 5 leaved clover, you would be able to find the more symmetrical 4 leaved clover. Why then is it so rare to find one?
If we look closely at other examples of nature, we can perhaps find the answer. If you were to search through your average garden, you would find the majority of flowers have 5 petals, many have 3 or 8 or more but if you look closely, you will always find more of certain numbers, compared to others. These numbers just so happen to be part of the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
Although, why does nature choose these numbers over others? In addition, the connection between the real world and this sequence does not just end there; it can be found almost everywhere we look: spirals on a snail shell, the core of an apple, geometry, art, architecture, the stock market and even the human body. So what makes it so useful? Why is it so special?

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My project intends to answer these questions and along the way discover new applications and more examples. I will be delving into the mathematical concepts behind the nature we see every day, the regular objects we rely on, the human body and the stock market. I shall also investigate aspects of the golden ratio and how the Fibonacci sequence is related to this.
The Fibonacci sequence is found by adding the previous term to the term before that. For example:
0, 1, 1, 2, ?
0 +1=1 1+1=2 1+2=3 and so on…..
Overall equation for next term: a_(n+1)= a_n+ a_(n-1)
This creates an infinite sequence of numbers and is known as a recursive sequence, as each number is a function of the previous two. Also, as the sequence progresses the ratio between each consecutive term seems to converge upon a single number.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
2/1=2 3/2=1.5 5/3=1.667 8/5=1.6 13/8=1.625 21/13=1.615…
Eventually, it converges to 1.618034… This number has a specific interest to many mathematicians and is known as the golden ratio. It is also useful when we consider where it is found.
If you were to take your hand and bend the index finger as full as possible, measuring the dimensions of the rectangle created, you would find what is known as a golden rectangle. The average height (of the intermediate phalange) would be around 3cm and the average length (of the proximal phalange) would be 5cm.
As we can see from left this creates a shape of ratio 5:3 or simply 1.667:1 (the golden ratio).
This is only one of the many examples of golden ratio in the body. There are many, many more some of which have been known for hundreds of years (see Da Vinci s Vitruvian man – right).
Also, the golden ratio is not just confined to the human body. Rather than cutting and apple from pole to pole, if you were to slice in a horizontal fashion, you would find a simple five pointed star. However, it is much more complex than meets the eye. If you were to take the distance AB as 1 unit, the distance AC would be 1.618, the golden ratio. But why does this happen, what make this ratio so efficient and so appealing, and why has nature adopted it?
History of the Sequence and Ratio
From the start of the Palaeozoic era, 400 million years ago, animals of divine proportions have been roaming the earth. The most notable is the nautilus shell (right) which follows a logarithmic spiral based on the golden ratio in rectangles.
The earliest written documentation of a special ratio belongs to the Rhind papyrus. A scroll about 6 metres long and 1/3 of a metre wide, it is one of the first mathematical handbooks. It was discovered by Scottish Egyptologist Henry Rhind in 1858 and is believed to have been written by Egyptian scribe, Ahmes in 1650 BC. He is believed to have copied it, from a document 200 years older, dating the first notation of the sequence to 1850 BC. However, the pyramids, built 1000 years previous, show many examples of the use of golden ratio, although many scholars believe it is merely coincidence created by the need for right angles.
Between the 6th and 3rd centuries, Greek philosophers, mathematicians and artists used and analysed the golden ratio. It is visible in pentagons and pentagrams throughout the period and was attributed to Pythagoras and his followers. It was used as part of his symbol (a pentagram with a pentagon within) and it was he, who first suspected the proportion was the basis of the human figure.
Plato also studied the ratio naming it most blinding of mathematical relations, the key to the physics of the cosmos. and from his lectures so did Eudoxus, whose work was used by Euclid in his book of elements II. Here he writes one of the first definitions A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” During his work he creates problems based on the ratio in pentagons, equilateral triangles and some of his prepositions show the ratio to be an irrational number.
The first person to apply numbers and sequence to construct the golden ratio was Leonardo of Pisa (full name, Leonardo Pisano Bigollo, lived 1180-1250). He was the son of an Italian businessman from the city of Pisa and grew up within a trading colony in North Africa.
At the time, Italy and the majority of Europe was using the Roman numeral system of counting, this was quite complex and meant most calculations required an abacus. While growing up in Algeria he learned the Hindu-Arabic system of calculation (the familiar 0, 1, 2…). After returning to Pisa as a young man in the thirteenth century, he recognised the superiority of this new structure and began to spread it throughout Europe. He did this through his book the Liber Abaci (book of abacus) published in 1202 under the nickname, Fibonacci (a contraction of filius Bonacci, meaning son of Bonacci).
To explain the system he used the Fibonacci sequence in his famous immortal rabbits problem (see next section of more detail). This allowed him to explain addition, subtraction and division using the Hindu- Arabic system and in turn allowed him to popularise it through Western Europe. Due to this he was later known as the founder of western mathematics and the “greatest European mathematician of the middle ages”. He introduced concepts such as algebra, geometry, the common fraction and even the square root symbol. He also considered the possibility of negative numbers and related it to merchant problems which began with a debt.
There was very little significant work done upon the topic until 1509, when Luca Pacioli published De Divina Proportione with the help of illustrations by Leonardo Da Vinci, who later used this within his famous work the Vitruvian man . In 1611, German astronomer Johann Kepler discovered the numbers within his own work on planetary motion saying as 5 is to 8, so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost in relation to the rings around Saturn. It was later found that the ratio of mean distance between planets was in fact the golden ratio.
Over the next two centuries many scholars investigated the sequence, deriving formulas and functions. In 1830, A. Braun first applied the sequence to the arrangement of bracts on a pinecone. A decade later and J.P.M. Binet derived a formula for the value of any Fibonacci number without the need for the previous two.
nth number= 1/(v5) ((1+v5)/2)^n- 1/(v5) ((1-v5)/2)^n
In 1920, Oxford Botanist A.H Church discovered spirals on sunflower heads corresponded to the numbers in Fibonacci s rabbit problem (see next section). This discovery inspired botanists to look for Fibonacci numbers elsewhere, teams then began to realise that many phyllotactic ratio s are golden ratio s (see flower patterns and primorda). In the 1930 s, Joseph Schillinger consciously composed a piece of music using Fibonacci intervals and Ralph Elliot began predicting the stock market in Fibonacci periods. By the 1960 s, a lively interest had been aroused and to this day mathematicians around the world are investigating the uses and problems connected with the sequence.
The Immortal Rabbits Problem
To explain his mathematical theorems, Fibonacci liked to create problems to allow his audience to use the maths he tried to describe. The immortal rabbits problem is one such challenge. It was first described within his famous Liber abaci and was later adopted as an explanation for the Fibonacci sequence.
Imagine if you will a large enclosure and within it a pair of rabbits. The immortal rabbit problem asks if there is one pair to begin with, how many rabbits will there be after a certain length of time if:
Each rabbit is immortal
They stay within their pairs
They breed once per month and produce a pair each time
Each new pair takes 1 month to mature, and then breeds to form a new pair the next month
January, we start with 2 rabbits, these then take one month to breed…..
February, there is now one adult pair and a new born pair of immature rabbits….
March, the new born pair have now matured, and the adult pair have reproduced…
April, the new born pair from March have now developed, the first pair reproduce again and the second pair reproduce for the first time..
The pattern continues until…
Month Pairs of mature rabbits Pairs of immature rabbits Overall Number of Pairs
January 1 0 1
February 1 1 2
March 2 1 3
April 3 2 5
May 5 3 8
June 8 5 13
July 13 8 21
August 21 13 34
September 34 21 55
October 55 34 89
November 89 55 144
December 144 89 233
After a while, we begin to notice a pattern, the total number of rabbits in any given month is a Fibonacci number. This is because the total is formed from the number of immature rabbits (the same as the number of mature rabbits the last month) and the number of mature rabbits (the total from the previous month) i.e. a_(n+1)= a_n+ a_(n-1)
Another interesting note is the rate of growth in the population….
2/1 = 2 3/2= 1.5 5/3= 1.666 8/3= 1.625 ……. this continues until we reach a_(n+1)/a_n =1.618034.. i.e. the Golden Ratio.
Flower patterns and primorda
As we have seen in the introduction, nature has applied the Fibonacci sequence and golden ratio from the number of petals on a flower, to the core of an apple and the spirals of a sunflower. On the face of it, this seems to be a fortunate and appealing coincidence, but since the 1920 s botanist have searched and found more and more of these coincidences . This leads us to believe that perhaps, they have a much deeper and more interesting meaning for the life of your average plant. Maybe these numbers and ratios were chosen for a reason.
Even from Egyptian times it was noted that most flowers had 5 petals, the rest by majority also have Fibonacci numbers of petals.
Also, if you examine the many plant stems you will find the regular pattern or 1, 2, 3, 5, 8 stems at standard heights.
Another interesting phenomenon, and one which may reveal the mystery of why plants behave so regularly in conjunction with the Fibonacci sequence, are the spirals shown by plants.
Look carefully at the picture of the pineapple left. As you move from the top right to the bottom left you may begin to see a set of spirals, curving round the pineapple in a diagonal fashion. Upon closer inspection you may also find a similar on from top left to bottom right and less obvious, from top to bottom. If we count the number of spirals we (fortunately for this topic) seem to find only Fibonacci numbers. In fact in a study of over 2000 pineapples not a single on differed from the pattern.
The same principle applies to the pinecone. Upon close inspection, you will find two different spirals, one vertically and another horizontally, all of which come in Fibonacci numbers. A separate study to that of the pineapples showed that this was the case 99% of the time.
The sunflower however, has its own unique spiral display. Starting from the centre and continuing in a clockwise fashion to the outside, the number of spirals again adds to a Fibonacci number. Although, if you look in the opposite (anticlockwise) direction you will find yet another spiral and adding the number of these gives the consecutive Fibonacci number. The majority of the time this is the case, however from time to time there are variations; with larger sunflowers the number of spirals can be double Fibonacci numbers (i.e. 2, 4, 6, 10, 16, 26….). These spirals may be interesting and attractive to look at, but hold much more value than just aesthetics; they allow us to show just why Fibonacci numbers are so widely used in nature and give us an insight into how nature uses maths at its very core.
To understand the maths behind the growth of plants we must look deep into the way it grows. As the plant grows taller the interesting components (i.e. petals, sepals, stamens, leaves) all grow from small clumps of tissue called primorda. As these begin to grow they aim to have the largest distance between leaves as possible, this means they have the maximum amount of space and light to grow, ultimately making the plant stronger and more likely to survive. This distance has been decided through evolution to allow the maximum about of light to hit the plant and it turns out this maximum point of efficiently is related to the golden ratio.
It just so happens that the Golden angle is the angle one golden ratio away from the starting position.
360 1.618.. 582.5
i.e. 582.5 -360 = 222.5 away clockwise (or 137.5 anticlockwise).
As they grow at their angles the leaves have enough light and space to grow. However, when the 6th leaf begin to grow the angle means it is only 32.5 from the first, this leaves it in the shade meaning it is less likely to grow and develop; this is the reason many plants use the number 5 in some areas (i.e. in the number of petals) as the 6th would have less room and is less likely to grow.
Sometimes called the phyllotactic ratio, the connection between this and efficiency in plants does not just end there. If we take ourselves back to the sunflower and its spirals we can see that this also has connections to the same ratio. As it begins to grown from the centre outwards each primorda (and therefore each seed head) grows on golden angle away from the previous.
As the ratios between consecutive Fibonacci numbers are approximations to the golden ratio (and therefore used to create approximations to the golden angle) we begin to see them within the spirals. This is the main reason Fibonacci numbers are present in so many places; they form the best approximations of the golden ratio. Although, the actual number of spirals that arises depends upon the size of the seed head and slight variations in the rate at which the primorda migrate away from the tip of the growing shoot.
As we saw from the rotations in plant leaves above, the golden angle is used to give the most space and therefore the most light. In the seed head however this is not a problem so why has evolution adapted to use it?
The answer to this was first discovered by Professor H. Vogel in 1979. He noticed that using the golden angle allowed the seed head to pack together with hardly any missing space. This meant it was very efficient as more seeds could fit in a small area and also much stronger. In turn it meant there would be more seeds and better chance of offspring.
This was later supported by French physicists Yves Couder and Stephan Douady, who found the choice of angle the natural consequence of the dynamics of growing a plant shoot . They stated that each primorda gets pushed into the largest available space, so they pack more efficiently, making the golden angle the most likely choice.
They also discovered that the next best choice for packing an angle created by a second very similar sequence called the anomalous series (4, 7, 11, 18, 29…). After inspection of more spirals and more plant this was found to be the 2nd most common choice after the Fibonacci sequence.
Overall, nature has evolved and adapted to use Fibonacci numbers and the golden ratio they approximate, as it gives the most efficient method for survival. Over the years this had been pondered by many people and its frequency in nature has been described as many to be proof of intelligent design and higher power .
Shapes of the Golden ratio
Although undeniably stunning, the sources of the golden ratio and Fibonacci numbers in nature are only half the applications of these phenomena in the real world. As humans, along with the rest of nature, are hotwired to apply the golden proportions, some of the human applications are some of the most remarkable. As a species we are attracted to the shapes they make and therefore adapt it to the structures we built, the way we think and the art we create.
One of the most common shapes is that of the golden rectangle. It is formed from a ratio of length to width of 1.168… : 1 (i.e. the golden ratio). This alone is not that interesting, but remove a square with the same width and height as the width of the golden rectangle (a square ratio 1:1) and you are left with another rectangle. If you take the measurements of this you once again find the ratio 1.168… : 1 the golden rectangle.
Repeat the process and the same happens again and again and again; removing a square ratio 1:1 leaves a smaller golden rectangle. The pattern continues indefinitely and is known in mathematics as a fractal (a geometric pattern that is repeated at every scale). Look at most regular paper sizes, credit cards and company logo s you will find an abundance of golden rectangles. However its man-made applications are not its only uses, it can be applied to create another, much more stunning shape – the logarithmic spiral.
Visually, it can be described as a long, slow spiral and is known as a logarithmic or equiangular spiral. It is known as this as each radii from the centre intercepts the curve at exactly the same angle.
It is created by constructing an arc from the furthest corner of each square in the golden rectangle to the opposing corner of that square. The pattern continues and repeats the further you zoom toward the centre making this yet another Fibonacci fractal. The most stunning example of this is the chambered nautilus (see the image of its shell right). As it grows it must produce more room within its shell, while keeping its original shape. To do this it adds a chamber larger than its previous, with each radii intercepting the curve at the same angle (remaining equiangular), keeping the original shape. There are also numerous other examples including; a rams horn, a galaxy spiral, a sea horse and many more.
Last but not least, the pentagon and pentagram are found to have Fibonacci connections. These shapes have interested humans for many years and have been the insignia of many religious and political groups. The explanation for its popularity however lies with our desire to search for the golden ratio. From the diagram (left), we can see how the ratio 1:F connects the length of the side of the pentagon to the distance between corners of the pentagram. There is however another ratio, the distance between a vertex and the corner of the inscribed pentagon is 1: 1/F. These ratios mean that many pentagons in nature, art and architecture have Fibonacci numbers present in the lengths.
Overall, we can see how many of the regular shapes found both in nature and modern life have been dictated by the Fibonacci sequence. There are thousands of examples of these proportions in the real world and more regular shapes than have been divulged here. As interesting as finding them in the real world is, it doesn t come close the intrigue which lies behind the way we can use them to our own advantage.
Art and Architecture
It is said that renaissance art was inspired by a sense of beauty and proportion . It seems fitting therefore that the dimensions for such art would lie in the ratio s and sequences of the most elaborate and efficient set of numbers known to maths.
The use of the series in art has however been known long before this period with Luca Pacioli stating without mathematics there is no art upon the completion of his work with Leonardo Da Vinci on De Divina Proportione (you may recall this from History of the Sequence). Legend also has it that long before this, Greek mathematician Eudoxus studied human affinity to this proportion by asking a group of his followers to divide sticks into the ratios they found most pleasing. This experiment was later adapted by German psychologist, Gustav Fechner in the 1860 s. He took a series of ten rectangles of different proportion and asked subjects to choose which they found to be the most pleasing, 76% of all participants chose the three rectangles closest to the golden rectangle.
It is clear from this then that we have known for many years that the golden or divine proportion has visually pleasing qualities and unknown to us, it can be found almost everywhere we look as a direct result. One of the earliest and most obvious sightings of this was in the Great Pyramids of 4700BC. Here F is found extensively in its construction but most scholars now believe that this is more coincidence than design, it is however interesting to note that the exact height of the structure is 5813 inches (numbers of the Fibonacci sequence.
1,400 years later and the Tomb of Ramses IV was built, this was later discovered to have several approximations to the golden rectangle as its centre. It had been constructed with a double square (approximation to the golden rectangle, a golden rectangle and a double golden rectangle.
The first people to consciously apply the maths of the golden ratio to their art and architecture were the Greeks. The Parthenon of Greece 440BC is the single finest example of this. The whole structure fits within the golden rectangle proportions as well as each pair of columns and even the sections of sculpture that run above the columns. The designer, Phidias was said to be the greatest and most prolific sculptor of his age. His work was dependent upon extensive use of the golden proportion. Its abundance in his work later meant the ratio was named Phi in his honour.
Many artefacts of the era from urns and vases to Afrodita’s sculpture (right) and temples all extensively used the proportion. It is believed that as Pythagoras linked it to the human body (see next section) it was generally associated with the divine and beautiful, making many associate it with the Gods and good.
One of the most interesting instances of the Fibonacci sequence at work is in the operation of the stock market.

Fibonacci Sequence

How Does the Fibonacci Sequence Relate to Nature and Other Math Processes?
Nature is all around us, and because I spend a lot of time outside I have been able to enjoy and observe all that nature has to offer. Due to the fact that I love science and discovering how everything around me functions and relates to everything else, I decided to investigate the relation that Fibonacci has with other math processes—as well as with the environment. I wanted to understand how plants know the best way to form their seeds or outer shell, and why some patterns may repeat in nature in different plants and organic materials. Thus, this exploration looks at two seemingly unrelated topics—Fibonacci and the golden ratio—both of which produce the same number, phi. While this could be mere coincidence, that possibility is negated when the fact that the number produced is irrational is introduced. It was this peculiar discovery, as well as the abundant appearances of Fibonacci in nature, that led me to choose this exploration topic.
To begin, I should start by identifying what initially sparked my curiosity in this subject: a pinecone. As with many other plants, as well as fruits and vegetables, pinecones display the golden ratio. In order to better understand what I am talking about I have included a picture of a pinecone similar to the one that I first inspected.
Labeled below is the noticeable spiral pattern on the pinecone. Counting the number of spirals in that direction produces the number eight, and in the other direction it produces the number thirteen while a third and tighter spiral produces twenty-one. These numbers are situational to the pinecone in the pictures, but the Fibonacci numbers as a whole are far more complex than they first appear to be.
To understand the importance of these numbers it is crucial to understand the fundamentals of the Fibonacci sequence itself. The sequence usually begins with the numbers 1, 1, 2, 3, 5, 8, 13 and follows an easily definable pattern.
1, 1, 2, 3, 5, 8, 13
Start with the number 5, or the nth number in the sequence. We’ll call it n. 5 equals the two numbers before it added together: 2 + 3. Or, in broader terms, a number in the sequence is the sum of the two numbers preceding it.
1, 1, 2, 3, 5, 8, 13n = n-1 + n-2
An interesting idea comes up at the mention of this formula though.
This ratio just so happens to equal a number often notated as, or phi.
> 1/11Phi is greater than one,
> 3/21.5Phi is greater than three halves,
> 8/51.6Phi is greater than eight fifths,
You’ll notice that each fraction listed above is made up of numbers from the original seven number sequence, in other words, each pair of Fibonacci numbers creates a ratio that gets closer and closer to phi as the numbers increase. This is better shown on a graph I created, displayed below.

The ratio created by these sequences as they approach phi is called the golden ratio. The golden ratio, however, is not as important to this study as the lesser known concept of the golden angle. Below is a representation of the golden ratio in relation to the golden angle, the smaller portion of the circle notated using alpha, or α.

α = 137.507764° 137.5°
The reason this conversion is necessary is because the golden angle is present in the next discussion topic: sunflowers. Or, more specifically, their seeds. Sunflowers are another great example of the appearance of Fibonacci in nature, and also led me to an interesting discovery.
In order to plot the distribution of a sunflower’s seeds we need an X and a Y coordinate pair. Using the square roots from an index numbered from one to one thousand and multiplying them by the cosine of the radian of the angle alpha gives us a formula to find x, dependent on the index number used. Y can be calculated with a very similar formula, using sine instead of cosine. The equations are listed in their entirety below.

When these formulas are used and input into Microsoft Excel they produce a graph similar to the following.

Wow! That graph bears a striking resemblance to the original Fibonacci spirals that appeared in the pinecones, and as mentioned earlier it is not mere coincidence.
While the use of the golden ratio is apparent, there is another aspect of it that I wish to address, the golden spiral. Its formulae are given by the following equations, and are readily apparent in nature as well (nautilus shells for example).

In these equations is the undetermined scaling factor and is the growth factor of the spiral. In the instance of the golden spiral, is equal to the operation below.

At first, these formulae appeared to be a strange smattering of numbers, and one I didn’t understand at all. However, upon noticing the appearance of a natural log in the formula for I made a connection to the letter , better known as Euler’s number, that is present in both the X and Y formulae. After thorough searches of many sources I discovered another math process that bares resemblance to the above formulae.

This is Euler’s formula. It becomes increasingly apparent that its resemblance is not coincidental when the formula is transformed into the final formula shown below.

While the visual similarities may be obvious when the formula is displayed as it is above, the importance of each variable can be clarified with simple explanations. is the arbitrary scaling factor, responsible for determining the scale of the spiral. dictates the rotation of the spiral, and remains constant. The in dictates the growth of the spiral, and the dictates the speed—together representing the speed of the growth of the spiral. More simply put, any given ordered pair can be found by multiplying the growth of the spiral by its rotation (as shown in the originally given formulae for finding said coordinates.)
What is produced, however, after inputting over two thousand pieces of data, derived from the coordinates calculated using the formulae above, into Microsoft Excel, is shown in the graph below.

After putting in the Fibonacci squares (using the original golden ratio) into the spiral its appearance and relation to Fibonacci become even clearer.

Very simply put, my investigation yielded the result that the Fibonacci sequence, the golden spiral, and Euler’s number are all related to one another in nature. The results are eye opening for me, as I am beginning to realize just how much of the world is made up of math—rather than my previous belief that everything natural occurred randomly. My exploration only stemmed into plants, and while that may only have practical use in fields such as botany, all three have great value in many fields. To begin with, Fibonacci appears in bee populations, proportions of the human body, formation of cells, and possibly more practically in code and the stock market. Any of these fields could present an interesting extension to my exploration, and because they all stem from Fibonacci they all have roots in combinatorics and number theory. The implications of this are staggering! Simply the thought that all of these vastly different fields are related to one another by one sequence of numbers discovered by Leonardo of Pisa, better known as Fibonacci himself, is baffling considering that he discovered them while looking at the breeding patterns of rabbits. There are so many other areas in nature that Fibonacci appears in, and I’m so excited that I have the opportunity to discover and study them now that I know more about them.
Works Cited
Azad, Kalid. “Intuitive Understanding Of Euler’s Formula.” Better Explained. N.p., n.d. Web. 23 Feb. 2015.>.
“Nature by Numbers.” Eterea. N.p., n.d. Web. 3 Feb. 2015.>.
Wolverson, Tim. “Plot a Fibonacci Spiral in Excel.” Reviews and Guides. WordPress, 08 Feb. 2014. Web. Jan.-Feb. 2015.>.
McDonald 1

An Explanation of the Fibonacci Sequence

In this paper, I will examine the properties of Fibonacci Sequence and some identities related to Fibonacci sequence. I will also touch the applications of Fibonacci numbers, recurrence and closed forms of Fibonacci sequence.
1 Introduction
2 Fibonacci Numbers in Pascal’s Triangle
3 Cassini’s Identity
4 Difference of Square of Fibonacci Numbers
5 Sum of Odd Fibonacci Numbers
6 Golden Ratio
7 Golden Spiral
8 Fibonacci Recurrence Relation
9 Fibonacci Closed Form
10 Conclusion

Fibonacci numbers were discovered by Leonardo Pisano. His nickname is Fibonacci. The Fibonacci Sequence is the series of numbers:
Each number in the sequence is sum of the previous two terms.
u1 = 1 u2 = 1 un = un−1+un−2,n > 2
Fibonacci numbers first used in ‘rabbit problem’. At the beginning, Fibonacci had one male and one female rabbits. He supposed that, every month. Next month more babies (male and female) were born. The next month these babies were grown and the first pair had two more babies (again male and female) . The next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature. Fibonacci asked how many rabbits a single can produce after a year with this breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month). Fibonacci asked how many would be formed in a year. Following the Fibonacci sequence perfectly the rabbits reproduction was determined 144 rabbits (Sinha,2017).

Fig.1 rabbit problem diagram

The Fibonacci Numbers are also applied in Pascal’s Triangle. Entry is sum of the two numbers either side of it, but in the row above. Diagonal sums in Pascal’s Triangle are the Fibonacci numbers.
An interesting property of Pascal’s Triangle is that its diagonals sum to the Fibonacci sequence, as shown in the picture below:

Fig.2 diagonal sums in Pascal’s triangle
The sum of the entries in the nth diagonal of Pascal’s triangle is equal to the nth Fibonacci numbers for all positive integers n. Suppose ∑dn sum of the numbers at nth diagonal and fn = nth Fibonacci number, n ≥ 0.
Principle of Mathematical Induction will be used for proof.
For n = 0, ∑d0 = 0, f0 = 0 (The result is true for n = 0) Assume that ∑dk = fk fk+1 = fk+ fk−1, Property of Fibonacci Sequence f(k) = ∑dk and f(k−1) = ∑dk−1, inductive hypothesis
Therefore, ∑dk+1 = ∑dk+∑dk−1
F12+F22 = 12

Cassini(1625-1712) is the astronomer who found Cassini Identity. Cassini’s identity is a mathematical identity for Fibonacci sequence.
The history of science is silent as to why Cassini took such a great interest in Fibonacci numbers. Most likely it was simply a hobby of the Great astronomer. At that time many serious scientist took a great interest in Fibonacci numbers and the golden mean. These mathematical objects were also a hobby of Cassini’s contemporary, Kepler(Stahbov,2012).
Proof by induction
Let’s check that if the identity works for n=1.
For n = 1,2×1−12 = 1 = 11+1
Assume that the identity works for some n=k,
We want to show that also
To prove this we will establish
Here is how:
= Fk+1Fk−Fk−FkFk−1−Fk+1Fk−1
= Fk+1Fk−2FkFk

The difference of the square of two consecutive Fibonacci numbers is equal to the product of two Fibonacci numbers.
F FnFn−1
Use the following identities.
F FnFn−1
Fn+1 = Fn+Fn−1
an = F2n−1 bn = 2FnFn−1
a2n = b2n+c2n
Consider separately
= (Fn+Fn−1)4−4(Fn+Fn−1)FnFn−1
After simplifying this by using identities
We can show that FFnFn−1)2 = (FnFn−1)2
Hence F(FnFn−1)2 So, F FnFn−1.

The sum of odd Fibonacci numbers is also a Fibonacci number.
F1+F3+F5+…+F2n−1 = F2n,F0 = 0,F1 = 1
Proof by induction. base case: for n=1 F1 = 1 = F2 This proves that the base case is true.
inductive step: Lets assume that it is true for some n ≥ 1.
F1+F3+F5+…+F2n−1 = F2n
We need to prove that it is also true for n+1.
F1+F3+F5+…+F2(n+1)−1 = (F1+F3+F5+…+F2n−1)+F2(n+1)−1
= F2n+F2(n+1)−1
= F2n+F2n+1
= F2n+2
= F2(n+1)
This proves that it is true for n+1. Hence it is true for all n ≥ 1.

Any Fibonacci number divided by the previous Fibonacci number has a quotient of approximately 1.618034…
FFn−n1 ≈ 1.618034(φ)

Fig.3 ratio of consecutive Fibonacci numbers
The Golden Ratio is also found all throughout the physical universe. Weather patterns, Whirlpool have almost the same form like the golden spiral. Even the Sea Wave sometimes shows almost the same spiral pattern. The three rings of Saturn are designed naturally based on the Golden Ratio. The Galaxy, Milky Way, also has the spiral pattern almost like golden spiral. Relative planetary distances of the Solar System also have the golden ratio properties. (2011)

There is a good change of variable analysis in textbook ‘An Introduction to the Analysis of Algorithms’ (2013).
an = 1/(1+an−1) (2013)
Let’s write some terms.
a0 = 1
a1 = =
a2 = 1+1+1 = 1+2 =
When we continue, we will see the pattern to continued fractions continues.
a3 = 1+1+1 1 = 1+3 =
a4 = 1+1 = 1+5 =
1+ 1
The form an = bbn−n1 is certainly suggested:
substituting this equation into the recurrences gives
bbn−1 = 1/(1+ bbnn−−21) for n > 1 with b0 = b1 = 1.
Dividing both sides by bn−1gives
bn−1+1bn−2 for n > 1 with b0 = b1 = 1,
which implies that bn = Fn+1, the Fibonacci sequence. This argument generalizes to give a way to express general classes of “continues fraction” representations as solutions to recurrences (2013).

Golden spiral is derived from golden ratio.The growth factor of golden spiral is φ , the golden ratio.

Fig.4 golden spiral
To draw Golden Spiral, draw squares by using Fibonacci sequence. This works by drawing squares which have the side lengths of Fibonacci numbers and combining them.
Golden spiral is used in real life in many areas like art, beauty, clothing, architecture,logos etc.
The Golden ratio is prevalent in Da Vinci’s The Annunciation, Madonna with Child and Saints, The Mona Lisa and St. Jerome. He was famous for using the Golden ratio in his works. The Mona Lisa, a well known portrait of a woman with a coy smile, is embedded with Golden rectangles (2018).

Fig.5 The Mona Lisa

We can find recurrence formula of Fibonacci sequence by using its definition.
Fn = Fn−1+Fn−2 xn = xn−1+xn−2
We will divide each term by xn−2
x2 = x+1 x2−x−1 = 0
We will solve the quadratic equation by using quadratic formula
x x
Fn n
F0 = 0,F1 = 1
c c
Fibonacci Recurrence Relation

Let G(z) be closed form of Fibonacci sequence.
G(z) = F0+F1z+F2z2+F3z3+F4z4+…
Now let’s find zG(z) and z2G(z) by multiplying G(z) by z and z2.
zG(z) = F0z+F1z2+F2z3+F3z4+F4z5+… z2G(z) = F0z2+F1z3+F2z4+F3z5+F4z6+…
Now we will find the difference of G(z)−zG(z)−z2G(z)
(1−z−z2)G(z) = F0+(F1−F0)z+(F2−F1−F0)z2+(F3−F2−F1)z3+…
We know that F1 = F2 = 1
F2−F1 = 0.
Also since for n > 2 we have that
Fn = Fn−1+Fn−2 then
F3−F2−F1 = 0,F4−F3−F2 = 0,…,
Fn+1−Fn−Fn−1 = 0.
(1−z−z2)G(z) = z
This will give us the closed form of Fibonacci sequence as:
G(z) = 1−zz−z2

In conclusion, studying Fibonacci numbers was great experience. It was interesting and joyful to see the harmony of numbers and the beauty they made. I searched the history of Fibonacci numbers, some identities related to Fibonacci numbers and the applications of Fibonacci numbers. Seeing the applications of Fibonacci numbers was eye opening.
1. Sinha, Sudipta, 2017, The Fibonacci Numbers and Its Amazing Applications, International Journal of Engineering Science Invention, Volume 6, Issue6
2. Stakhov, Alexey, 2012, A generalization of the Cassini Formula, The International Clubof the Golden Section
3. Akhtaruzzaman, Md , Shafie, Amir A. ,2011, Geometrical Substantiation of Phi, theGolden Ratio and the Baroque of Nature, Architecture, Design and Engineering International Journal of Arts
4. Thapa ,Gyan Bahadur , Thapa , Rena, 2018, Journal of the Institute of Engineering, TheRelation of Golden Ratio, Mathematics and Aesthetics
5. Sedgewick, R., & Flajolet, P. (2013). An introduction to the analysis of Algorithms, (Second Edition). Addision-Wesley Professional.
6. Wiki page Golden Ratio.
7. Fig.1 Knott, Ron, 2016, Fibonacci Numbers and Nature.
8. Fig.2
9. Fig.3 Ancient Architecture(ratio of consecutive Fibonacci numbers).
10. Fig.4. Wikimedia commons.
11. Fig.5. The Mona Lisa