Fibonacci Series And The Golden Ratio Engineering Essay

Fibonacci Series And The lucky balance Engineering EssayThe enquiry question of this extended essay is, Is there a carnal knowledge mingled with the Fibonacci serial and the fortunate Ratio? If so be the reason, what is it and explain it. The Fibonacci series, which was first introduced by da Vinci of Pisa (Fibonacci), was found to seduce had a close connection with the halcyon Ratio. The relation found was that the limit of the balances of the numbers in the Fibonacci sequence converges to the florid immoral/golden ratio. I decided to carry unwrap a some set of experiments that involved oneness-on-one concepts of both the Fibonacci series and the well-to-do Ratio. exploitation their individual applications such as the Golden Rectangle, a computerized t aloney supported by a sketched graph, I found that I could arrive at a supposition that linked the two concepts. I as well as used the Fibonacci spiral and Golden spiral to find the limit where the harbors would t end to meet. After carrying out the experiments, I decided to find the inference of the relation using the Binets expression which is essentially the variety showula for the ordinal margin of a Fibonacci sequence. However, the Binets formula was interesting enough to shuffle me find its proof and solve it myself. From there, I proceeded on to the proof of the relation between the Fibonacci series and the Golden Ratio using this formula. The Binet formula is given by . Following the proof, I carried out steps to verify it by substituting different look ons to crack its validity. After proving the validity of the conjecture, I arrived at the end point that such a relation does exist. I also learned that this relation had applications in nature, art and architecture. Apart from these, there is a possibility that there be other applications which can be subjected to further probe.Table of ContentsSl. No.ContentsPage No.1.Introduction to the Fibonacci Series42.Introduction t o the Golden Ratio53.The Relationship between them64.Forming the conjecture65.Testing the conjecture76.The proof157.Verification of the proof208.Conclusion229.Further Investigation2210.Bibliography23IntroductionThe Fibonacci SeriesThe Fibonacci series is that sequence where every term is the sum of the two toll that precedes it (in the Hindu-Arabic system) where the first two hurt of the sequence are 0 and 1. The Fibonacci series is shown below 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 Where the first two scathe are 0 and 1 and the term following it is the sum of the two foothold preceding it, which in this case are 0 and 1. Hence, 0 + 1 = 1 ( terzetto term)Similarly,Fourth term = third term + second termFourth term = 1 + 1 = 2And so the sequence follows.The series was first invented by an Italian by the name of Leonardo Pisano Bigollo (1180 1250) in 1202. He is better know as Fibonacci which essentially remembers the son of Bonacci. In his book, Liber Arci, there was a crush concernin g the breeding of rabbits and the solution to this puzzle answered in the discovery of the Fibonacci series. The problem was based on the total number of rabbits that would be born starting with a couple up of rabbits first followed by the breeding of new rabbits which would also start giving birth one month after they were born themselves.1The problem was broken downhearted into parts and the answer that was obtained gave rise to the Fibonacci series. The Fibonacci series gained a worldwide acceptance soon as after its discovery and was used in some fields. It had its uses and applications in nature (such as the petals of a sunflower and the nautilus shell). Shown below is the application of the series on the whirls of a pine cone.2http//www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib2.jpghttp//www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib3.jpgThe Golden average / Golden RatioThe golden mean, also known as the golden ratio, as the name suggests is a rat io of distances in simple geometric figures3. This is but one of the many definitions found for the term. It is not solely confine to geometric figures but the proportion is used for art, nature and architecture as well. From pine cones to the paintings of Leonardo Da Vinci, the golden proportion is found almost everywhere. Another definition of the golden ratio is a precise way of dividing a line4 in that location has never been one concrete definition for the golden ratio which makes it susceptible to different definitions using the same concept. First claimed to be known by Pythagoreans round 500 B.C., the golden proportion was established in print in one of Euclids major works namely, Elements, once and for all in 300 B.C. Euclid, the renowned Greek mathematician was the first to establish what the golden section really was with respect to a line. According to him, the division of a line in a mean and extreme ratio5such a way that the point where this division takes place, the ratio of the parts of the line would be the Golden proportion. He determined that the Golden Ratio was such that The golden ratio is denoted by the Greek alphabet which has a order of 1.6180339Since then, the golden ratio has been used in confused fields. In art, Leonardo Da Vinci coined the ratio as the Divine Proportion and used it to define the fundamental proportions of his famous painting of The Last Supper as well as Mona Lisa. http//goldennumber.net/images/davinciman.gifFinally, it was in the 1900s that the term Phi was coined and used for the first clock time by an American mathematician Mark Barr who used the Greek letter phi to name this ratio.6Hence, the term obtained a grasp of different names such as the golden mean, golden section and golden ratio as well as the Divine proportion.The Relation between the Fibonacci series and the Golden RatioAfter the discovery of the Fibonacci series and the golden ratio, a relation between the two was established. Whether thi s relation was a coincidence or not, no one was able to answer this question. However, today, the relation between the two is a very close one and it is visible in various fields. The relation is said to be The limit of the ratios of the numbers in the Fibonacci sequence converges to the golden ratio.This means that as we move to the nth term in the Fibonacci sequence, the ratios of the consecutive terms of the Fibonacci series arrive closer to the think of of the golden mean ().7Forming the ConjectureThe Fibonacci series and the golden ratio have been linked together in many ways. Hence, I shall now produce the same statement as a conjecture as I am about to prove the relation by means of a set of experiments and eventually proving the conjecture (right or wrong).The conjecture is stated below The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n , where n is the nth term of the Fibonacci sequence.In order to prove this conjecture, I have carried out a few experiments below that shall attribute to the result of the preceding(prenominal) conjecture.Testing the ConjectureExperiment No. 1The first set of experiments deal with the Golden Rectangle. The golden rectangle is that rectangle whose holdings are in the ratio (where y is the length of the rectangle and x is the breadth of the rectangle), and when a upstanding of dimensions is removed from the original rectangle, another golden rectangle is left field behind. Also, the ratio of the dimensions ( is equal to the golden mean (). I have used the concept of the Golden Rectangle to test whether the ratios of the dimensions of the two golden rectangles, when equated to each other, give the value of the golden ratio or not which is also said to be the formula for the nth term of the Fibonacci series. The latter part of the statement is in accordance with Binets formula.The following experiment shows how this works.Let us turn a rectangle with dimensions . The dotted line is the line that has divided the rectangle in such a way that the square on the left has dimensions of . Now, the rectangle on the right has the dimensions of where x is now the length of the new golden rectangle form and (y-x) is the breadth.Golden Rectangle 1yxy-xThe reason why this rectangle is called a Golden Rectangle is because the ratio of its dimensions gives the value of . Hence, the information we can gather from the to a higher place figure is that(1)The new golden rectangle formed from the to a higher place one is shown below with dimensionsGolden Rectangle 2y xxThe in a higher place new golden rectangle shown must thus also have the same property as that of any other golden rectangle. Therefore,From the higher up experiments we can establish the following relation (2)For convenience sake, I have decided to take so as to make y the subject of the equation. Hence, the above equation can now be re-written as On cross-procreateing the terms above we get Writing th e above equation in the form of a quadratic equation, we get Using the quadratic formula, , we getHence, the two roots obtained areHowever, the second root is rejected as a value as y is a dimension of the rectangle and hence cannot be a negative value. Hence we have,Evaluating this value we have But, from equation 1, we know that However, the value of x was restricted to 1 in the above test. So as to eliminate the variable in order to keep only y as the subject, I carried out the calculations below that help in doing so Rewriting the equation Cross-multiplying the variables Dividing the equation by , we get But we know that . Thus, using this substitution in the above equation we have This is the same quadratic that we obtained earlier and hence the doubt for the presence of x clears out.Experiment No. 2For my second experiment, I have decided to use the concept of the Fibonacci spiral and that of the Golden Spiral. The steps on how to draw these spirals are given below A Fibonacci spiral is formed by drawing squares with dimensions equal to the terms of he Fibonacci series.We start by first drawing a 1 x 1 square1 x 1Next, another 1 x 1 square is drawn on the left of the first square. (every new square is bordered in red)Now, a 2 x 2 square is drawn below the two 1 x 1 squares.Next, a 3 x 3 square is drawn to the right of the above figure.Now, a 5 x 5 square is adjoined to the top of the figure.Next, a 8 x 8 square is adjoined to the left of the figure.And so the figure continues in the same manner. The squares are adjoined to the original shape in a left to right spiral (from down to up) and each time the square gets bigger but with dimensions equal to the numbers in the Fibonacci series. Starting from the inner square, a quarter of an arc of a circle is drawn within the square. This step is repeated as we move outward, towards the bigger square. The spiral eventually looks like this http//library.thinkquest.org/27890/media/fibonacciSpiralBoxes.gifThe shap e shown below is the Fibonacci spiral without the squareshttp//library.thinkquest.org/27890/media/fibonacciSpiral2.gifA similar process is followed for forming the golden spiral. However, the only difference is that we draw the outer squares first and then draw the arcs starting from the larger squares. Hence, the spiral turns inwards all the way to the inner squares.Golden SpiralThe Golden spiral eventually looks like this Golden SpiralOn comparing the two spirals, it can be seen that they overlap as the arcs guide the squares with dimensions of the latter terms of the Fibonacci series. An image of how the two spirals look is shown below http//library.thinkquest.org/27890/media/spirals.gifFrom the above experiment, it can be seen that there is a connection between the Fibonacci series and the Golden Mean as their individual spirals overlap each other as the n (which is the nth term in the series) tends to infinity.Experiment No. 3My third experiment involves technology. In this ex periment, I decided to use a program of Microsoft Office, namely, Microsoft Excel in order to record the values obtained on calculating the ratio of the consecutive terms of the Fibonacci series. In the table below, I have recorded the terms of the Fibonacci series in the first column, the value of the ratio of the consecutive terms in the Fibonacci sequence in the second column, the value of8in the third column and the variation of the value of the ration from the value of in the last column. consideration of Fibonacci SeriesValue of ratio of consecutive termsvalue ofvariation of value calculated from value of0111.000000000000001.618033988749890.6180339887498922.000000000000001.61803398874989-0.3819660112501131.500000000000001.618033988749890.1180339887498951.666666666666671.61803398874989-0.0486326779167881.600000000000001.618033988749890.01803398874989131.625000000000001.61803398874989-0.00696601125011211.615384615384621.618033988749890.00264937336527341.619047619047621.61803398 874989-0.00101363029773551.617647058823531.618033988749890.00038692992636891.618181818181821.61803398874989-0.000147829431931441.617977528089891.618033988749890.000056460660002331.618055555555561.61803398874989-0.000021566805673771.618025751072961.618033988749890.000008237676936101.618037135278511.61803398874989-0.000003146528629871.618032786885251.618033988749890.0000012018646415971.618034447821681.61803398874989-0.0000004590717925841.618033813400131.618033988749890.0000001753497641811.618034055727551.61803398874989-0.0000000669776667651.618033963166711.618033988749890.00000002558318109461.618033998521801.61803398874989-0.00000000977191177111.618033985017361.618033988749890.00000000373253286571.618033990175601.61803398874989-0.00000000142571463681.618033988205321.618033988749890.00000000054457750251.618033988957901.61803398874989-0.000000000208011213931.618033988670441.618033988749890.000000000079451964181.618033988780241.61803398874989-0.000000000030353178111.618033988738301.61803 3988749890.000000000011595142291.618033988754321.61803398874989-0.00000000000443The aim of the table is to find out whether the value of the ratio reaches the value of or not, as the number of terms increases infinitely.ObservationFrom the above table, it can be seen that as we reach the nth term of the Fibonacci series, the variation in the value of the ratios from the value of , decreases. This observation is in agreement with the conjecture The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n , where n is the nth term of the Fibonacci sequence.InferenceFrom the above 3 experiments, I have found that the conjecture holds true for them all. Hence, I would like to state that the tests for the conjectures have been significantly successful.The proofreadIn order to find the relation between the Fibonacci series and the Golden Ratio, I followed the proof below that uses calculus to establish the required relation.The Fibonacci series is give n by,Assuming that 0, 1, and 1 are the first three terms of the sequence(3)This eventually goes on to form the well known sequence 0, 1, 1, 2, 3, 5, 8, 13Dividing the left(a) Hand Side (or LHS) and the Right Hand Side (or RHS) of equation 3 by F(n), gives(By taking the numerator as the denominator of F(n))By substituting the limit of the ratios of the terms (as n ) of the Fibonacci series with A, the limit is taken on both sides such that n The above is true as the ratioHence, the below quadratic equation is formedWe can find the roots of A by using the quadratic formula, .orFrom this we find thatThis value of is easily attainable using the Binet formula. The Binet formula is that formula which gives the value of by substituting the variable x with one of the n terms of the Fibonacci series. Using the concept of the golden rectangle, the quadratic that was obtained earlier Gave the value of . The proof of the Binet formula shows another possibility to arrive at the relation betwe en the Fibonacci series and the Golden Ratio. The beauty of this proof is that the quadratic first arose from the Fibonacci series calculation and the root that was obtained gave the value of phi. This is from the proof that was written above. Under the heading Testing the Conjecture that was done earlier, the quadratic arose from the dimensions of the Golden Rectangle and the equation thus obtained gave the value of phi. Using this concept, I have followed the proof below which was solved by older mathematicians.The Binet formula is given by Now, from the above tests, we got However, there were 2 values that were obtained on calculating the value of y. The value of y that was negative was rejected then as it was incorrect to consider it a valid answer for a dimension of a geometric figure. Calling this negative root as , we can rewrite the Binet formula as Going back to the quadratic equation, we can transmute in place of y and so the quadratic equation is (4)This quadratic was o btained from the Golden Rectangle. In order to arrive at the Fibonacci sequence, a series of algebraic manipulations will help us reach that step. To start off with, we have the value of in terms of . Now, to get the value of in terms of , we multiply equation (4) into .Using equation (4), we substitute for and we get Using the same method to find the value for raised to higher powers, we have Similarly,Writing the various values for raised to higher powers (5)Now if we look at the coefficients closely, we see that they are the consecutive terms of the Fibonacci series. This can be written as (6)However, the above trend is not enough proof for oecumenicalizing the above statement. Hence, I decided to prove it by using the principle of mathematical induction. measurement 1Step 2To prove that P(1) is true.Hence, P(1) is true (from equation 5)Step 3Hence, P(k) is true whereStep 4To prove that P(k+1) is true.Starting from the RHS,(from equation 3)(from equation 4)(from P(k))= RHSHence , P(k+1) is true.Therefore, P(n) is true for allNow that we have proved that P(n) is trueis true in its generalized form.Also, we know that is the other root of the quadratic equation and so the above general equation can be written in the above form as well (7)In order to obtain the Binet formula in the form of We can subtract equation (7) from equation (6) to get Substituting the original values of and in denominator of the above equation, we get Substituting the value of and in the above equation, we get This is the Binet formula which we started to prove. Hence, the formula is valid.Verifying the ProofIn order to validate a proof, it must be tested in order to check whether the conjecture is valid and can be generalized. For this reason, I have decided to use the Binet formula (that was proved above) to check the validity of the relation between the Fibonacci series and the Golden Ratio by substituting values for x in the equation UsingCase 1,Which is the first term of the Fibon acci series.Case 2,Which is the second term of the Fibonacci series.Case 3,Which is the third term of the Fibonacci series.Case 4,Which is the fourth term of the Fibonacci series.From these substitutions it is clear that the formula is a valid one which gives the desired result.Also, the above calculations have proved to be substantial examples for proving the validity of the proofs shown above. However, an important note to remember in the Binet formula is that the value of x starts from 0 and increases. So it can be said that (x belongs to the set of whole numbers). This is to account for the fact that the Fibonacci series starts from 0 and then continues.Hence, the conjecture is true and can be generalized. Hence the conjecture below can be considered true.The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n , where n is the nth term of the Fibonacci sequence.ConclusionFrom the above tests and verifications, it is clear that a relation be tween the Fibonacci series and the Golden Ratio does truly exist. The relation being The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n , where n is the nth term of the Fibonacci sequence.The Fibonacci series as well as the Golden Ratio have their individual applications as well as combined applications in various fields of nature, art, etc. As mentioned earlier, the Fibonacci series was used to find a solution to the rabbit problem. The relation between the two concepts was an integral part of the central idea in the novel The Da Vinci Code.Along with these well known ideas, other applications of the two concepts are present in the whirls of a pine cone, the paintings of Leonardo Da Vinci, the spiral of the nautilus shell, the petals of the sunflower. These are only very few examples regarding the applications of the two concepts.However, this relation has proved to be useful to environmentalists, artists and many other researches. For ex ample, artists were able to use the study of the concept in the paintings of Leonardo Da Vinci and draw old symbols. It also has given them the chance to create art of their own that by using this concept in their procedure of creating.Further InvestigationWith the prominent number of applications that were found regarding the Fibonacci series and the Golden Ratio, there is a possibility that there are other applications of the concept as well. The convergence of the ratios of the values to the value of phi may prove to be of great significance if applied to another theory that has boggled minds of mathematicians for years. Possibilities such as these give rise to the question of further investigation in this aspect of the relationship between the two concepts.