Fibonacci And The Golden Ratio Mathematics Essay

Fibonacci And The Golden Ratio Mathematics EssaySome aspects of mathematics finish be dull and tedious from start to end, much of it further 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 meter. If you stool ever looked at a winding-sheet of paper and wondered Why do we uptake those dimensions? or looked at the leaf or an attractive plant and wondered Why can I never control a four leaved clover? then this may be of some interest. M either of these things atomic number 18 quite interconnected in a counseling you would not realise, and most of them ar connected by the Fibonacci term.If I re unit of ammunition to one of my original questions Why can I never reign a four leaved clover? it seems mindable, that if you can date 3 leaved clover and 5 leaved clover, you would be able to mother the more radial 4 leaved clover. Why then is it so rare to find one?If we look closely at other examples of personality, we can perhaps find the state. If you were to search through your average garden, you would find the bulk of flowers select 5 petals, many a(prenominal) bugger off 3 or 8 or more just now if you look closely, you will always find more of certain numbers pool, compared to others. These numbers just so happen to be part of the Fibonacci sequence1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89Although, why does nature choose these numbers over others? In addition, the connection between the real world and this sequence does not just end in that respect it can be embed almost everywhere we look spirals on a snail shell, the core of an apple, geometry, art, architecture, the stock market and even the gentle body. So what makes it so useful? Why is it so special?My project intends to answer these questions and a enormous the way discover natural applications and more examples. I will be delving into the numerical 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 cereb score to this.The Fibonacci sequence is institute by adding the previous term to the term before that. For example0, 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 twain. Also, as the sequence progresses the ratio between each ensuant term seems to converge upon a single number.1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 892/1=2 3/2=1.5 5/3=1.667 8/5=1.6 13/8=1.625 21/13=1.615F=1.618034Eventually, 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 crouch the index finger as full as possible, m easuring the dimensions of the rect bung 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 53 or simply 1.6671 (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 adjudge 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 game 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 aloofness 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 espouse it?History of the Sequ ence and RatioFrom 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 rough 6 metres long and 1/3 of a metre wide, it is one of the counterbalance mathematical handbooks. It was discovered by Scottish Egyptologist Henry Rhind in 1858 and is bankd 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 sixth and 3rd centuries, Greek philosophers, mathematicians and artists used and anal ysed 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 excogitate 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 compute he creates puzzles based on the ratio in pentagons, equi lateral 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 da Vinci of Pisa (full name, Leonardo Pis ano 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 musical arrangement of counting, this was quite complex and meant most calculations required an abacus. While growing up in Algeria he versed 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 renowned immortal pikas problem (see next section of more detail). This permited him to explain addition, subtraction and division using the Hindu- Arabic system and in turn allow ed 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 joint 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 sublunar 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 peal 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 sc holars investigated the sequence, etymologizing 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 place of any Fibonacci number without the need for the previous two.nth number= 1/(v5) ((1+v5)/2)n- 1/(v5) ((1-v5)/2)nIn 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 principles 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 conne cted with the sequence.The Immortal Rabbits ProblemTo explain his mathematical theorems, Fibonacci care 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 describe 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 ifEach rabbit is immortalThey stay within their pairsThey breed once per month and produce a pair each timeEach new pair takes 1 month to mature, and then breeds to form a new pair the next monthJanuary, we start with 2 rabbits, these then take one month to breed..February, there is now one big(a) pair and a new born pair of immature rabbits.March, the new born pair have now matured, and the adult pair have reproducedApril, the new b orn pair from March have now developed, the first pair reproduce again and the second pair reproduce for the first time..The human body continues untilMonth Pairs of mature rabbits Pairs of immature rabbits Overall Number of PairsJanuary 1 0 1February 1 1 2March 2 1 3April 3 2 5May 5 3 8June 8 5 13July 13 8 21August 21 13 34September 34 21 55October 55 34 89November 89 55 144 declination 144 89 233After a while, we begin to notice a pattern, the make out number of rabbits in any prevailn month is a Fibonacci number. This is because the total is formed from the number of immature rabbits (the alike 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 preeminence 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. blossom forth patterns and primordaAs 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 state 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. bearing carefully at the picture of the pinea pple 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 kindred 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 whimsical spiral display. Starting from the affection 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) fo cussing 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 bigger 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 moldiness 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 core they have the maximum amount of pose and light to grow, ultimately making the plant stronger and more promising 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.. 582.5i.e. 582.5 -360 = 222.5 away clockwise (or 137.5 anticlockwise).As they grow at their angles the leaves have enough light and stead 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 seeming 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 als o has connections to the kindred 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 tha t using the golden angle allowed the seed head to pack together with merely 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 serial (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 ratioAlthough 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. atomic number 53 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 11) and you a re 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 11 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 unreal 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 patte rn 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 spiritual 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 1F connects the length of the side of the pentagon to the distance between corner s 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 youthful 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 computer architectureIt 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 b efore 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). story 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 G reat Pyramids of 4700BC. Here F is found drawn-outly 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 w as 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 Afroditas 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.