Michael Stanfield

The Life of Fibonacci

The Italian mathematician Fibonacci was born as Leonard Pisano in 1170 A.D. His mathematical name, Fibonacci, is short for the Latin expression son of Bonacci. He was born in the city of and later passed away there about 80 years later. Even though Fibonacci is Italian, he received his education in North Africa after his father received a superior profession and moved the both of them there. The North African education he received is what led him to his primary contributions to the modern day mathematical subject.

Guilielmo Bonacci, the father of Fibonacci, was a Holy Roman Empirical diplomat. In Pisa, Guilielmo was the representative of the merchants that resided and transacted there. This was an important profession during the time because this was a time period that the was experiencing increasingly important and persistent sea trade. Although Pisa is not a sea port, a massive amount of traded goods still went through the town as it was close to the coast. When Fibonacci was young, his father became a consul to a North African port of Bugia, in present day , due to his work as a representative. He protected the Roman interest in the port of Bugia.

After Fibonacci and his father moved to North Africa, Fibonacci received his education there. This is when and where he learned from Arab masters of . After learning from the Arab mathematicians, Fibonacci traveled around Northern Africa and Europe to Egypt, Syria, Greece, and other countries to study more about different systems and calculations. Fibonacci enjoyed doing so as he was given the opportunity to learn about how different mathematic systems around the region worked. After Fibonacci learned a subject, he would teach it to other people. It was not until after his travels that his greatest works were released.

After his travels, Fibonacci finally settled in Pisa at about the age of 30. This was the time frame that a majority of his work of accomplished. During his lifetime, Fibonacci wrote four books. His most popular and well known is . Liber abaci, or the book of calculations, was the first book written by Fibonacci. He wrote it in 1202, only a few years after he settled back in Pisa. He also later revised Liber abaci in 1228 after he wrote the rest of his books. Practica geometriae, or Practice of , was written is 1220 when Fibonacci was 50. At the age of 55, he wrote two books. One was called Flos, or the Flower, while the other one was Liber quadratorum, or . The Book of Squares contained many questions involving squares including some work with Pythagorean triples.

Fibonacci was greatly influenced by trading. His entire life had revolved around it even though it was not his profession. He spent his life around merchants. His father was a representative of the merchants after all. A majority of what he introduced greatly helped commerce and trading around the world at that time. He knew it would so that is why he taught it.

Even though Fibonacci is considered one of the best mathematicians during the , we know very little about him. Fortunately, we are still able to received a detailed description due to the autobiographical notes that Fibonacci left in his books. If it were not for those notes, we may not know anything about him. Along with the notes he left, the books are where his mathematical computations come from. The most well known book he wrote was Liber abaci. Liber abaci is what effectively introduced Europe to the Arabic numeral system. His absolute understanding of allowed him to easily and simply explain how they work to the rest of the Europeans. Although Liber abaci was more influential, it is Fibonacci's work in Liber quadratorum that places him as the largest contributor to theory between Diophantus and Fermat. Fibonacci´s Mathematical Works

Fibonacci's greatest contributions to mathematics were not through his computational work, but the notations he provided. Fibonacci standardized multiple notions and the numerical system that we still use today. He greatly influenced the mathematicians of his day and the future. His work is not very well known, but the notations that he left behind, and the Fibonacci sequence, are his greatest legacy.

Fibonacci wrote four mathematical books. The first, and by far the most well known, is Liber abaci. Liber abaci is the book that caused Europe to use the Arabic numeral system. The writing Fibonacci does in this book produces his most well known contributions to mathematics. First and foremost, the introduction of the Arabic numeral system in Europe. In Liber abaci, Fibonacci shows the superiority of the Arabic numeral system. He first shows how to use the system, then explains how it will benefit everyone from mathematicians to merchants, who he had grown up around. Liber abaci shows the flaws of the Roman numeral system including its lack of 0 and a decimal system. Liber abaci also introduced the fraction bar to mathematical notation. Before the fraction bar, people mainly used quotation marks to set off the numerator. The use of the fraction bar allows to a much more comprehendible notation.

Although Liber abaci was not the first time Arabic numerals were used, it is what subsequently standardized the notation in Europe. The excellent portrayal that Fibonacci uses to explain these allow it to have been easily understood by the Europeans. Along with his excellent explanations, the superiority of the system is what eventually led to its widespread use. Soon after the release of Liber abaci, Europeans realized that using LXXXVIII to write 88 was not the best notation, therefore; they switched over to the Arabic numeral system that will still use today.

This notation has made mathematics substantially simpler to use. It made mathematics an easier to understand concept. The use of 10 characters to quantify any value versus the use of 7 characters whose largest difference in value is 500 made the use of Arabic numerals a much preferred choice. ranged in value, and the largest Roman numeral only stood for 1000, the second largest being 500, third largest being 100, so on. After Fibonacci release Liber abaci, the Arabic numerals were not taken up immediately as you would imagine. Instead, it took a while for Arabic numerals to be adopted. Fibonacci did not only introduce Arab mathematics to Europe, but he also taught it to some Arabs as well. Even though Fibonacci learned mathematics from Arab, it was only the scientists and mathematicians that used those characters. Fibonacci taught this kind of mathematics to Arab businessmen.

Fibonacci did not introduce Arabic numerals to everyone for no reason. He felt they were more effective at displaying information as well as better to use computationally. This is why Fibonacci taught everyone these numerals. He believed these numerals would help commerce and business become easier. This is the case since after Fibonacci introduced Arabic numerals, banking and business became very common. It was due to Fibonacci's introduction of Arabic numerals that led Italy to start making banks. Arabic numerals made things easier. Fibonacci understood this and that is why he introduced them to everyone.

In Liber abaci, the use of the fraction bar presented by Fibonacci is substantial. The fraction bar allows the ability to easily write and comprehend fractions. However in Liber abaci, Fibonacci wrote his fractions a little bit differently. One major difference is the way in which Fibonacci wrote mixed numbers.

3 3 Our mixed numbers:2 4 versus Fibonacci's mixed numbers: 4 2.

2 As shown above, Fibonacci wrote his whole units on the right rather than the left how we do now. Fibonacci was also a major user of unit fractions described as:

1/n

Fibonacci liked to use unit fractions to show other fractions such as:

5/8 = 1/2 + 1/8

17/18 = 1/2 + 1/3 + 1/9

Although this seems messing, this is how Fibonacci wrote his fractions often. He enjoyed using unit fractions so that is what he did.

After Fibonacci wrote Liber abaci, he then wrote a brief book called Practica geometriae, or Practice of Geometry. This book was only eight chapters long. Even though it was short, all eight chapters were filled with theorems designed around Euclid's Elements and On Divisions. He did not release another book until 5 years later.

It was after the release of Practica geometriae that the Holy Roman empire noticed the works that Fibonacci had produced. The Holy Roman emperor wanted to challenge Fibonacci so he sent a member of his court to present Fibonacci with a multitude of mathematical problems. Out of these problems sent, three of them appeared in his next book, Flos. Flos was written in 1225. One of the problems Fibonacci was challenged to solve was as follows:

10x + 2x2 + x3 = 20

Fibonacci successfully answered this problem. Fibonacci accurately solved the problem to nine decimal places. In the same year, Fibonacci wrote another book, this one being his last.

Liber quadratorum was the last book that Fibonacci wrote before his death in 1250. It is this book that makes modern mathematicians consider Fibonacci as a great number theorist. In Liber quadratorum, Fibonacci searches for new and various ways to find Pythagorean triples. Along with working with Pythagorean triples, Fibonacci also worked with Diophantine equations. One of the first occurrences he notes it this book is that squares can be constructed as sums of odd numbers as in the formula below.

n2 + (2n + 1) = (n + 1)2

Along with this, in this book he also discovers that if you take an odd perfect square and then add the odd numbers before it to that square, the sum produces another perfect square. A few examples are shown below:

32 + 7 + 5 + 3 + 1 = 52

72 + 47 + 45 + 43 + 41 + 39 + 37 + 35 + 33... + 1 = 252

3 He also discovered that the sum of two perfect squares equals another perfect square. Along with this he states that:

There exists no natural numbers for x,y such that x2 + y2 and x2 − y2 are both perfect squares.

He also notes a similar occurrence. He shows that:

There exists no natural number x,y, and z such that x4 − y4 = z2.

It is interesting to note that these results are some what similar to Fermat's Last Theorem. Fibonacci is shown to prove a set of exponents are inequivalent if you subtract them almost 500 years before Pierre de Fermat proclaimed this theorem.

Even though the Liber quadratorum is considered his main work of , what Fibonacci is known most for is the sequence of numbers he shows in Liber abaci. The Fibonacci sequence is a sequence of numbers that gives almost exact numerical value to the . The Fibonacci sequence didn't receive its name until the 1870s, when a French mathematician, Edouard Lucas, coined the term. The Fibonacci sequence has a surprising amount of appearances in nature. Any place that has natural spirals can be quantified by the Fibonacci sequence. It is quite remarkable how these spirals do quantify to the exact ratio set by the Fibonacci sequence. The Fibonacci sequence is a major part of the beauty in math because it helps prove that math is in nature. The Fibonacci sequence may not be the most difficult result to appear in math, but it is still extraordinary. The mathematical formula for the Fibonacci sequence be be written as:

xn = xn−1 + xn−2

One major use that the Fibonacci is used for is forex trading. Forex, or foreign exchange, trading is the center of trading of currencies. The Fibonacci sequence makes a showing in the market that trades world currencies. Although it is not perfect, it is believed that the Fibonacci sequence can be used to predict the oscillation of the forex charts. The Fibonacci sequence is extremely crucial to know and therefore an elementary concept that needs to be understood before someone begins to forex trade.

Possibly the most stunning result of the Fibonacci sequence is its relevance to the world around us. Anywhere that there are spirals, the sequence can be shown. Pineapples, conch shells, and even the way leaves are arranged on a branch are consistent with the Fibonacci sequence. The Fibonacci sequence played in scale to music is audibly appealing as well. The reason that the Fibonacci sequence is so commonplace in nature may have to do with its relation with the Golden Ratio. The Golden ratio is 1.618..., and it is often used in architecture and paintings as well. Artists believe the Golden ratio makes their paintings the most visually appealing. What is probably the most fascinating thing about the Fibonacci sequence and the Golden ratio is their relation to each other. The Fibonacci sequence gives an approximation to the Golden ratio, as the terms grow larger, the ratio grows more accurate.

Golden ratio=1.61803398875...

or the formula

If a/b = (a + b)/a then these numbers produce the Golden ratio which can also be denoted as Φ

1/1 = 1

4 2/1 = 2

3/2 = 1.5

610/377 = 1.618037...

Also just as interesting is the fact that it doesn't have to be the exact Fibonacci sequence for this to happen. If you start with any two numbers, the result will still be the same. As the terms of that sequence grow larger, the ratio will begin to show.

It is believed that this ratio is used in plants because it allows for the most adequate amount of sunlight to hit the leaves of the plant. This precise leave placement also allows for the most rainfall to be collected by the leaves. The more rainwater the leaves collect, the more water that the roots receive. The reason that seeds may form in spirals of Fibonacci numbers is believed to be for optimal packing. These perfect spirals allow for more seeds to be packed in on one flower head than any other shape. Despite this, there are still a few plants that do not use Fibonacci sequence. Instead, some of these plants use what are called the Lucas numbers. They sequence is the same except that it starts with 2 and 1, not 1 and 1.

Many arithmetical ideas stemmed from the Fibonacci sequence. The Fibonacci sequences produces multiple results that are interesting to say the least. One such idea is called the Pisano period. When dividing the Fibonacci sequence by any number, a sequence results by taking the remainder of these numbers. This sequence will inevitably repeat. The amount of digits that the sequence takes to repeat is called a Pisano period, after Fibonacci's real name.

Take this portion of the Fibonacci sequence:

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

Divide this portion of the Fibonacci sequence by 2 and write down the remainder

(1, 1, 0), (1, 1, 0), (1, 1, 0)

The parentheses mark the Pisano period of 3 digits for diving by 2.

Using a bigger portion, we can show the Pisano period of dividing by 3:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

Dividing this portion of the sequence by 3, the remainders are:

(1, 1, 2, 0, 2, 2, 1, 0), (1, 1, 2, 0, 2, 2, 1, 0)

The parentheses mark the Pisano period of 7 digits for dividing by 3.

5 To discover something else interesting about the Fibonacci sequence, look at the last digit of each .

Take this portion of the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

The last digits of these numbers being,

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 1, 5, 9, 4, 3, 7, 0, 7

While it is only a small portion, it is noticeable that makes a bit of a showing in this sequence. 3, 1, 4, 1, 5, 9, show up in order as they are at the beginning of pi. Although is may be purely coincidental, it is still interesting to see the sequence and pi linked.

Collaboration with Other Scholars

Although none of Fibonacci's books were collaborative, there are some aspects of his work that were. Such as the use of Arabic numerals. Fibonacci didn't come up with these numerals himself, he simply set out to teach the use of this notation to the rest of Europe. When Fibonacci released Liber abaci, it contained a problem in which the Fibonacci sequence was used.

This problem included reproducing rabbits. Although the sequence was named after Fibonacci because of this problem, he wasn't the first to introduce the sequence or the problem. It is believed that Indian mathematicians had first proposed the problem and its sequential solution. The reason it is believed that Indian mathematicians were the first to propose the problem is that historians believe it was proposed to the mentoring student, Fibonacci. So ultimately, the problem and sequence that bears Fibonacci's name, was neither proposed nor solved by Fibonacci himself.

A majority of Fibonacci's collaborations were in a learning setting. Whether he is the student or the master, when Fibonacci talked about mathematics with someone, someone was learning. This can be said because for the first thirty years of his life, Fibonacci traveled about Northern Africa and Europe. He taught people the Arabic numerals that he met before writing Liber abaci. Even though he was European, he taught Arab businessmen Arabic numerals. Fibonacci introduced the Arabic numerals to a lot of people, but he made an effort to teach it to merchants. He felt the Arabic numeral system was substantially better for commerce than other numerical systems such as Roman numerals.

Historical Events that Marked Fibonacci´s Life

Fibonacci's father's move to Northern Africa helped Fibonacci develop his arithmetical skills. This move is what caused Fibonacci to learn mathematics the way he did; therefore, leading him become well known and popularize the Arabic numeral system in Europe. The reason these events were set in motion was that during the Middle Ages, Italy's trading increased significantly. Trading among the seas made Pisa a major port town. As a Holy Roman Empire secretary in Pisa, Fibonacci's father was appointed as the consul of a North African trade port as Italy's trading increased.

6 After learning for many years, Fibonacci continued on to travel around learning mathematics from various places. He learned from Greeks, Egyptians, Syrians, and others. This is how some people believe he received the nickname, Bigollo, or well traveled. Historians believe that Fibonacci finally settled in Pisa around 1200. From then on, he continued to work on his math and write.

Mentioned early, it was not until about 1225 that Fibonacci became known by the Holy Roman empire. At this point, the Holy Roman empire challenged Fibonacci to do a multitude of problems. This event marks the recognition of Fibonacci by the world around him. This challenge shows that Fibonacci was a great mathematician, not only of his time, but throughout history. The results that Fibonacci produces in Liber quadratorum pushes this statement to be even more true. One book of calculations is what places his name in history, but one book on squares placed his name amongst the greatest number theorists of all time.

Significant Historical Events Around the World During Fibonacci´s Life

When Fibonacci was born, Henry II , the king of England, was in power. Henry was in power from 1154 until him death in 1189. Although it is described that Henry didn't have the look of a king, his personality commanded respect and attention. While he could be a good friend, he had a very nasty temper leading to the death of Archbishop Beckett, who was a friend of Henry. Along with the quarrels with the Archbishop, he also had feuds with his wife and children, both of which tried to overthrow him but failed. Although Henry was harsh, historians do not classify him as a tyrant. But they do say that there was really nothing special about him, he wasn't exceptionally intelligent, he wasn't sympathetic for England, and he didn't take into consideration the country's interest in his actions, only his own. Henry II is nevertheless important because of the impact he had on England.

At the beginning of the next century, the Ormulum was written. Ormulum was written by the Augustinian monk Orm. Ormulum receives its name from the author, Orm. Linguists are able to date the book to around the 1200s due to the language in it. Ormulum was created to help preachers enunciate better. So from this aspect, the interest it provides to readers is virtually nonexistent. However, it does provide greater understanding for the speech and writing at that time.

Famine also struck in one of the countries that Fibonacci had previously visited. Famine struck Egypt, and they didn’t recover for about two years. Less water than what usually flooded the Nile led to less water for the Egyptians therefore meaning less food. An account of this famine was written by Abdel-Latif Al-Baghdadi. He was a scholar from Baghdad and wrote over various topics. He wrote over diabetes, archeology, and Egyptology. He wrote over the famine and Egyptian monuments. In his writing, he says that the famine was so extreme that the Egyptian people resorted to cannibalism. Graves were stolen from as well. This famine wasn't the last or even the worst that Egypt would go through.

In Asia, during this timeframe, Japan entered a new period, something that only happens about every 200- 300 years. This period was called the Kamakura period. The Kamakura period lasted until 1333 when the shogunate was destroyed, and imperial rule was reestablished. During this period, Japan focused more upon land-based economies and military technology.

Also in Asia at this time, Genghis Khan is taking over. The Mongols have invaded China and Russia, captured the Jin capital of Zhongdu, the Persian empire, Bokhora, and the Iranian empire of Khwarizm under the rule of Genghis Khan. In 1227 though, Genghis Khan dies in battle. After his death, his sons split the empire amongst themselves and return to conquering. The Mongols continue on to defeat a Christian army in Poland then invade Hungary. However, Ogadai Khan, the son of Genghis Khan who became the leader of the Mongols, dies in 1241. This leaves the Mongols without a leader so they decide to leave Europe.

7 Throughout his life, four crusades take place, the third, fourth, fifth, and sixth. The crusades were military campaigns that were sanctioned by the Catholic church in order to spread Christianity around the world. They began in 1095 and lasted throughout the Middle Ages until 1290.

The Third Crusade lasted from 1189 until 1192. Fibonacci would have been 22 at the time of the end of the Third Crusade. The Third Crusade was a response to the numerous takeovers of the Crusaders of Jerusalem. The Syrian section of the Turkish empire began to won multiple battles between Jewish and Christian armies. The only battle that actually happened during the Third Crusade was a battle between King Richard, son of King Henry II, and these Syrian forces. In 1192, Richard and Saladin, the leader of the Syrian forces, signed a treaty reestablishing the Kingdom of Jerusalem.

The Fourth Crusade began 6 years later after the Pope called for it after power struggles between Europe and the Byzantium empire. The Crusaders toppled the leader of the Byzantine emperor and replaced him with his nephew. The rest of the Byzantium empire disliked this as the nephew tried to submit to church. He was assassinated later on. This led to the conquering of Constantinople, which ended the Fourth Crusade.

The Fifth Crusade was enacted by Pope Innocent III before he died. Egypt was attacked from land and sea during this crusaders. In this crusade, the Muslim defenders successfully did so driving out the invaders. The Sixth Crusade was the name given to the peaceful transfer of Jerusalem to Crusader control. After a decade, the treaty expired, and the Muslims regained control.

Also in England in 1215, the Magna Carta became the law of the land. The Magna Carta was introduced due to the mistreatment of the people of England that King John had placed upon the people. The Magna Carta enumerated certain freedoms of the people of England, the freedom of church, protection from excessive taxes, and due process were included in the Magna Carta as well as other human rights ideas.

King John is considered one of the worst kings of England. He imprisoned his own wife, starved imprisoned opponents, and collected outrageous taxes from the community to pay for foreign conflicts. Along with that, if they refused to pay, he took their property. Insurgent barons demanded that King John follow the law. Seeing himself as greater than the common people, he disagreed. Soon after these barons gained control over London and forced King John to sign the Magna Carta that guaranteed the English people different rights, most of which where made because of his rule.

After signing the Magna Carta, King John quickly asked for the Pope to annul the document. This in turn led the barons to refuse to surrender London back to the Crown until the Magna Carta was implemented. The Pope agreed with the King calling it illegal, harmful to royal rights, and even shameful to the English people. The Pope declared it null and void forever. A civil war between the barons and the King broke out after this. King John later died of dysentery thus appointing King Henry III. Henry was just 9 when he became King. In order to regain the support of the barons, revised and issued the Magna Carta. When was 18, it was reissued and revised again.

There were also inventions and places that we still use today created during this time period. The Chinese invented the rocket, and thankfully also the parachute, in 1180. The French also invented the water powered saw mill in 1204. In 1209, Cambridge University was founded. This means that the University of Cambridge is the fourth oldest university still in use in the world. The second and third oldest, University of Oxford and University of Salamanca respectively, were founded before Fibonacci was born, but they were chartered during his lifetime. The oldest university is in the home country of Fibonacci, Italy. It is the University of Bologna. This university is about 100 miles away from Pisa. While these had virtually no effect on Fibonacci, they had an effect on the world around him and still affect us today.

8 Significant Mathematical Progress During the Fibonacci´s lifetime

There were a few mathematicians during his lifetime that discovered important properties. An Indian mathe- matician by the name Bhaskara II established that dividing by zero equated to infinity. He also found solutions to quadratic, cubic, quartic equations. His answers even included negative and irrational solutions. Bhaskara lived from 1114-1185 to an astrologer. Bhaskara became the leader of an astronomical observatory. There are ways that Bhaskara can be represented as the peak of mathematician of the 12th century. His understanding of the number systems was far more advanced than anyone in Europe. Bhaskara wrote over general mathe- matics, , mathematical astronomy, spheres, etc. He completely understood the concept of zero as well as negative numbers. He also studied Diophantine problems. Along with that, he also had developed different concepts of calculus.

During the life of Fibonacci, a Persian mathematician, Nasir al-Din al-Tusi, developed spherical trigonometry as well as the formula for law of sines for triangles. al-Tusi learned primarily learned from his uncle. He was only 13 in 1214, when Genghis Khan began taking his conquests to the West. It was around 1220 when the Mongols finally invaded his home area of Tus. 36 years later, al-Tusi was in Alamut working possibly against his will. Then the Mongols, under the control of the grandson of Genghis Khan, Hulegu invaded and conquered Alamut. al-Tusi quickly joined the Mongols and was appointed as their scientific advisor. Hulegu was interested in science and treated al-Tusi with great respect because of it. During his time with the Mongols, it is believed he discovered trigonometry as an arithmetical discipline.

Connections between History and the Development of Mathematics

While some wars increase the rate that which science and math increases, the Crusades did not produce the same result. Fibonacci was essentially the center of the mathematical universe at that time. He was one of few people who made major mathematical contributions at the time. Although he was one of few, the amount of work and standards he produced is astounding.

The Crusades prevented the use of Arabic numerals sooner. Even though Fibonacci showed how superior of a system it was, the Crusades made Europeans uneasy to use anything Arabic. In 1299, the city of Florence even banned the use of Arabic numerals on the basis that they were easier to falsify than Roman numerals. Well after the death of Fibonacci, the suspicion passed, thus allowing the Arabic numerical system to grow the use that it has today.

Although the invention of the rocket had math to do with it, it is unknown. The rocket was mostly used as fireworks and weapons. These used solid fuel to deliver explosives that contained shrapnel. When the Mongols fought the Chinese, the Chinese used this weaponry against them. This allowed them to understand how to use it and take it to the West to attack Europe.

Remarks

Fibonacci was raised in a time where mathematical evolution was very slow. He was the leading mathematical mind for hundreds of years. He single-handedly introduced the continent of Europe to the Arabic numerals that we still use today. While his computational work is not extremely well known, it is that work that causes us to believe he was a great mathematical mind. He introduced the division notation that we still use today as well as the square root notation. His contribution to the mathematical world transcends formulas. His contributions to the notation of math has made it more universal amongst everyone. The notations he created made mathematics easier to learn and understand. He is the most influential mathematician that some people have not heard of.

9 Fibonacci lived in a time with a lot of conflict going on. While he was never in the middle of it, it was going on around him. Four separate Crusades took place during his life. The Mongols did a lot of their conquering during his time as well. The world was switching back and forth between different kingdoms, religions, and people. England had two tyrannical rulers during his lifetime, one so much so that it led to the Magna Carta. The time throughout the life of Fibonacci was very influential for the future. Out of his time came Arabic numerals, massive conquest, and human rights.

The Fibonacci sequence has allowed us to see the beauty of mathematics. Mathematics is not just numbers and figures, and the Fibonacci sequence helps prove that. Mathematics is what helps us understand the world around us. We quantified different aspects to helps us understand them, and now we are so knowledgable, that we know things that cannot be quantified or seen in nature. The Fibonacci sequence assists us in quantifying the beauty of nature. The Golden ratio is naturally appealing to us as humans so knowing what numbers to use to get this ratio can help artists paint more visually appealing works or to help architects create works of art and mathematics. The Golden ratio, and therefore the Fibonacci sequence, is everywhere we look.

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