Fibonacci Series And The Golden Ratio Engineering Essay

The research question of this extended essay is, “Is there a relation between the Fibonacci series and the Golden Ratio? If so be the reason, what is it and explain it.” The Fibonacci series, which was first introduced by Leonardo of Pisa (Fibonacci), was found to have had a close connection with the Golden Ratio. The relation found was that the limit of the ratios of the numbers in the Fibonacci sequence converges to the golden mean/golden ratio. I decided to carry out a few set of experiments that involved individual concepts of both: the Fibonacci series and the Golden Ratio. Using their individual applications such as the Golden Rectangle, a computerized calculation supported by a sketched graph, I found that I could arrive at a conjecture that linked the two concepts. I also used the Fibonacci spiral and Golden spiral to find the limit where the values would tend to meet. After carrying out the experiments, I decided to find the proof of the relation using the Binet’s formula which is essentially the formula for the nth term of a Fibonacci sequence. However, the Binet’s formula was interesting enough to make 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 values to check its validity. After proving the validity of the conjecture, I arrived at the conclusion 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 are other applications which can be subjected to further investigation.

Table of Contents

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Contents

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Introduction to the Fibonacci Series

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Introduction to the Golden Ratio

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The Relationship between them

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Forming the conjecture

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Testing the conjecture

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The proof

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Verification of the proof

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Conclusion

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Further Investigation

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Bibliography

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Introduction

The Fibonacci Series

The Fibonacci series is that sequence where every term is the sum of the two terms that precedes it (in the Hindu-Arabic system) where the first two terms of the sequence are 0 and 1. The Fibonacci series is shown below –

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

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Where the first two terms are 0 and 1 and the term following it is the sum of the two terms preceding it, which in this case are 0 and 1. Hence, 0 + 1 = 1 (third term)

Similarly,

Fourth term = third term + second term

Fourth term = 1 + 1 = 2

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And 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 known as ‘Fibonacci’ which essentially means the ‘son of Bonacci.’ In his book, Liber Arci, there was a puzzle concerning the breeding of rabbits and the solution to this puzzle resulted in the discovery of the Fibonacci series. The problem was based on the total number of rabbits that would be born starting with a pair of rabbits first followed by the breeding of new rabbits which would also start giving birth one month after they were born themselves. [1] 

The problem was broken down 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 many 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. [2] 

http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib2.jpghttp://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib3.jpg

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The Golden Mean / Golden Ratio

The golden mean, also known as the golden ratio, as the name suggests is a ratio of distances in simple geometric figures [3] . This is only one of the many definitions found for the term. It is not solely restricted 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 line” [4] 

There 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 around 500 B.C., the golden proportion was established in print in one of Euclid’s major works namely, Elements, once and for all in 300 B.C. Euclid, the famous 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 ratio” [5] such 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 value of 1.6180339…

Since then, the golden ratio has been used in various 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.gif

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Finally, it was in the 1900’s that the term ‘Phi’ was coined and used for the first time by an American mathematician – Mark Barr who used the Greek letter ‘phi’ to name this ratio. [6] Hence, the term obtained a chain 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 Ratio

After the discovery of the Fibonacci series and the golden ratio, a relation between the two was established. Whether this 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 value of the golden mean (). [7] 

Forming the Conjecture

The 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 through a set of experiments and eventually proving the conjecture (right or wrong).

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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 above conjecture.

Testing the Conjecture

Experiment No. 1:

The first set of experiments deal with the Golden Rectangle. The golden rectangle is that rectangle whose dimensions are in the ratio (where ‘y’ is the length of the rectangle and ‘x’ is the breadth of the rectangle), and when a square of dimensions is removed from the original rectangle, another golden rectangle is left 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 Binet’s formula.

The following experiment shows how this works.

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Let us consider 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 formed and (y-x) is the breadth.

Golden Rectangle 1:

y

x

y-x

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The reason why this rectangle is called a ‘Golden Rectangle’ is because the ratio of its dimensions gives the value of φ. Hence, the information we can gather from the above figure is that

(1)

The new golden rectangle formed from the above one is shown below with dimensions

Golden Rectangle 2:

y – x

x

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The above new golden rectangle shown must thus also have the same property as that of any other golden rectangle. Therefore,

From the above 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-multiplying the terms above we get –

Writing the above equation in the form of a quadratic equation, we get –

Using the quadratic formula, , we get

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Hence, the two roots obtained are

However, 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 –

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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. 2:

For 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 square

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1 x 1

Next, 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.gif

The shape shown below is the Fibonacci spiral without the squares

http://library.thinkquest.org/27890/media/fibonacciSpiral2.gif

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A 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 Spiral

The Golden spiral eventually looks like this –

Golden Spiral

On comparing the two spirals, it can be seen that they overlap as the arcs occupy the squares with dimensions of the latter terms of the Fibonacci series. An image of how the two spirals look is shown below –

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http://library.thinkquest.org/27890/media/spirals.gif

From 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. 3:

My third experiment involves technology. In this experiment, 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 of [8] in the third column and the variation of the value of the ration from the value of φ in the last column.

Term of Fibonacci Series

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Value of ratio of consecutive terms

value of

variation of value calculated from value of

0

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–

–

–

1

–

–

–

1

1.00000000000000

1.61803398874989

0.61803398874989

2

2.00000000000000

1.61803398874989

-0.38196601125011

3

1.50000000000000

1.61803398874989

0.11803398874989

5

1.66666666666667

1.61803398874989

-0.04863267791678

8

1.60000000000000

1.61803398874989

0.01803398874989

13

1.62500000000000

1.61803398874989

-0.00696601125011

21

1.61538461538462

1.61803398874989

0.00264937336527

34

1.61904761904762

1.61803398874989

-0.00101363029773

55

1.61764705882353

1.61803398874989

0.00038692992636

89

1.61818181818182

1.61803398874989

-0.00014782943193

144

1.61797752808989

1.61803398874989

0.00005646066000

233

1.61805555555556

1.61803398874989

-0.00002156680567

377

1.61802575107296

1.61803398874989

0.00000823767693

610

1.61803713527851

1.61803398874989

-0.00000314652862

987

1.61803278688525

1.61803398874989

0.00000120186464

1597

1.61803444782168

1.61803398874989

-0.00000045907179

2584

1.61803381340013

1.61803398874989

0.00000017534976

4181

1.61803405572755

1.61803398874989

-0.00000006697766

6765

1.61803396316671

1.61803398874989

0.00000002558318

10946

1.61803399852180

1.61803398874989

-0.00000000977191

17711

1.61803398501736

1.61803398874989

0.00000000373253

28657

1.61803399017560

1.61803398874989

-0.00000000142571

46368

1.61803398820532

1.61803398874989

0.00000000054457

75025

1.61803398895790

1.61803398874989

-0.00000000020801

121393

1.61803398867044

1.61803398874989

0.00000000007945

196418

1.61803398878024

1.61803398874989

-0.00000000003035

317811

1.61803398873830

1.61803398874989

0.00000000001159

514229

1.61803398875432

1.61803398874989

-0.00000000000443

The aim of the table is to find out whether the value of the ratio reaches the value of φ or not, as the number of terms increases infinitely.

Observation:

From 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 φ, 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.”

Inference:

From 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 Proof

In 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 given 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, 13…

Dividing the Left 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 ratio

Hence, the below quadratic equation is formed

We can find the roots of A by using the quadratic formula, .

or

From this we find that

This 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 between 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 substitute ” in place of ‘y’ and so the quadratic equation is –

(4)

This quadratic was obtained 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 generalizing the above statement. Hence, I decided to prove it by using the principle of mathematical induction.

Step 1:

Step 2:

To prove that P(1) is true.

Hence, P(1) is true (from equation 5)

Step 3:

Hence, P(k) is true where

Step 4:

To prove that P(k+1) is true.

Starting from the RHS,

(from equation 3)

(from equation 4)

(from P(k))

= RHS

Hence, P(k+1) is true.

Therefore, P(n) is true for all

Now that we have proved that P(n) is true

is 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 Proof

In 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 –

Using

Case 1:

,

Which is the first term of the Fibonacci 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.”

Conclusion

From the above tests and verifications, it is clear that a relation between 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 example, artists were able to use the study of the concept in the paintings of Leonardo Da Vinci and decipher 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 Investigation

With the great 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.

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