The research inquiry of this drawn-out essay is, “ Is there a relation between the Fibonacci series and the Golden Ratio? If so be the ground, what is it and explicate it. ” The Fibonacci series, which was foremost introduced by Leonardo of Pisa ( Fibonacci ) , was found to hold had a close connexion with the Golden Ratio. The relation found was that the bound of the ratios of the Numberss in the Fibonacci sequence converges to the aureate mean/golden ratio. I decided to transport out a few set of experiments that involved single constructs of both: the Fibonacci series and the Golden Ratio. Using their single applications such as the Golden Rectangle, a computerized computation supported by a sketched graph, I found that I could get at a speculation that linked the two constructs. I besides used the Fibonacci spiral and Golden spiral to happen the bound where the values would be given to run into. After transporting out the experiments, I decided to happen the cogent evidence of the relation utilizing the Binet ‘s expression which is basically the expression for the n-th term of a Fibonacci sequence. However, the Binet ‘s expression was interesting adequate to do me happen its cogent evidence and work out it myself. From at that place, I proceeded on to the cogent evidence of the relation between the Fibonacci series and the Golden Ratio utilizing this expression. The Binet expression is given by ; . Following the cogent evidence, I carried out stairss to verify it by replacing different values to look into its cogency. After turn outing the cogency of the speculation, I arrived at the decision that such a relation does be. I besides 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 farther probe.

## Table of Contentss

## Sl. No.

## Contentss

## Page No.

1.

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Introduction to the Fibonacci Series

4

2.

Introduction to the Golden Ratio

5

3.

The Relationship between them

6

4.

Forming the speculation

6

5.

Testing the speculation

7

6.

The cogent evidence

15

7.

Confirmation of the cogent evidence

20

8.

Decision

22

9.

Further Probe

22

10.

Bibliography

23

## Introduction

## The Fibonacci Series

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

0, 1, 1, 2, 3, 5, 8, 13, 21, 34 aˆ¦

Where the first two footings are 0 and 1 and the term following it is the amount of the two footings predating it, which in this instance are 0 and 1. Hence, 0 + 1 = 1 ( 3rd term )

Similarly,

Fourth term = 3rd term + 2nd term

Fourth term = 1 + 1 = 2

And so the sequence follows.

The series was foremost invented by an Italian by the name of Leonardo Pisano Bigollo ( 1180 – 1250 ) in 1202. He is better known as ‘Fibonacci ‘ which basically means the ‘son of Bonacci. ‘ In his book, Liber Arci, there was a mystifier refering the genteelness of coneies and the solution to this mystifier resulted in the find of the Fibonacci series. The job was based on the entire figure of coneies that would be born get downing with a brace of coneies foremost followed by the genteelness of new coneies which would besides get down giving birth one month after they were born themselves.[ 1 ]

The job was broken down into parts and the reply that was obtained gave rise to the Fibonacci series. The Fibonacci series gained a world-wide credence shortly every bit after its find and was used in many Fieldss. It had its utilizations and applications in nature ( such as the petals of a helianthus and the nautilus shell ) . Shown below is the application of the series on the commotion of a pine cone.[ 2 ]

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

## The Golden Mean / Golden Ratio

The aureate mean, besides known as the aureate ratio, as the name suggests is a ratio of distances in simple geometric figures[ 3 ]. This is merely one of the many definitions found for the term. It is non entirely restricted to geometric figures but the proportion is used for art, nature and architecture every bit good. From pine cones to the pictures of Leonardo Da Vinci, the aureate proportion is found about everyplace. Another definition of the aureate ratio is – “ a precise manner of spliting a line ”[ 4 ]

There has ne’er been one concrete definition for the aureate ratio which makes it susceptible to different definitions utilizing the same construct. First claimed to be known by Pythagoreans around 500 B.C. , the aureate proportion was established in print in one of Euclid ‘s major plants viz. , Elementss, one time and for all in 300 B.C. Euclid, the celebrated Grecian mathematician was the first to set up what the aureate subdivision truly was with regard to a line. Harmonizing to him, the division of a line in a “ mean and extreme ratio ”[ 5 ]such a manner that the point where this division takes topographic point, the ratio of the parts of the line would be the Golden proportion. He determined that the Golden Ratio was such that –

The aureate ratio is denoted by the Greek alphabet which has a value of 1.6180339aˆ¦

Since so, the aureate ratio has been used in assorted Fieldss. In art, Leonardo Da Vinci coined the ratio as the “ Divine Proportion ” and used it to specify the cardinal proportions of his celebrated picture of “ The Last Supper ” every bit good as “ Mona Lisa ” . hypertext transfer protocol: //goldennumber.net/images/davinciman.gif

Finally, it was in the 1900 ‘s that the term ‘Phi ‘ was coined and used for the first clip by an American mathematician – Mark Barr who used the Grecian missive ‘phi ‘ to call this ratio.[ 6 ]Hence, the term obtained a concatenation of different names such as the aureate mean, aureate subdivision and aureate ratio every bit good as the Divine proportion.A

## The Relation between the Fibonacci series and the Golden Ratio

After the find of the Fibonacci series and the aureate ratio, a relation between the two was established. Whether this relation was a happenstance or non, no 1 was able to reply this inquiry. However, today, the relation between the two is a really close one and it is seeable in assorted Fieldss. The relation is said to be –

“ The bound of the ratios of the Numberss in the Fibonacci sequence converges to the aureate ratio ” .

This means that as we move to the n-th term in the Fibonacci sequence, the ratios of the back-to-back footings of the Fibonacci series arrive closer to the value of the aureate mean ( ) .[ 7 ]

## Forming the Speculation

The Fibonacci series and the aureate ratio have been linked together in many ways. Hence, I shall now bring forth the same statement as a speculation as I am about to turn out the relation through a set of experiments and finally turn outing the speculation ( right or incorrect ) .

The speculation is stated below –

## The bound of the ratios of the footings of the Fibonacci series converge to the aureate mean as n a†’ , where ‘n ‘ is the n-th term of the Fibonacci sequence.

In order to turn out this speculation, I have carried out a few experiments below that shall impute to the consequence of the above speculation.

## Testing the Speculation

Experiment No. 1:

The first set of experiments trade with the Golden Rectangle. The aureate rectangle is that rectangle whose dimensions are in the ratio ( where ‘y ‘ is the length of the rectangle and ‘x ‘ is the comprehensiveness of the rectangle ) , and when a square of dimensions is removed from the original rectangle, another aureate rectangle is left buttocks. Besides, the ratio of the dimensions ( is equal to the aureate mean ( ) . I have used the construct of the Golden Rectangle to prove whether the ratios of the dimensions of the two aureate rectangles, when equated to each other, give the value of the aureate ratio or non which is besides said to be the expression for the n-th term of the Fibonacci series. The latter portion of the statement is in conformity with Binet ‘s expression.

The undermentioned experiment shows how this works.

Let us see a rectangle with dimensions. The flecked line is the line that has divided the rectangle in such a manner 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 aureate rectangle formed and ( y-x ) is the comprehensiveness.

## Golden Rectangle 1:

Y

ten

y-x

The ground why this rectangle is called a ‘Golden Rectangle ‘ is because the ratio of its dimensions gives the value of I† . Hence, the information we can garner from the above figure is that

( 1 )

The new aureate rectangle formed from the above 1 is shown below with dimensions

## Golden Rectangle 2:

y – ten

ten

The above new aureate rectangle shown must therefore besides have the same belongings as that of any other aureate rectangle. Therefore,

From the above experiments we can set up the undermentioned relation –

( 2 )

For convenience interest, I have decided to take so as to do ‘y ‘ the topic of the equation. Hence, the above equation can now be re-written as –

On cross-multiplying the footings above we get –

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

Using the quadratic expression, , we get

Hence, the two roots obtained are

However, the 2nd root is rejected as a value as ‘y ‘ is a dimension of the rectangle and hence can non be a negative value. Hence we have,

Measuring this value we have –

But, from equation 1, we know that –

However, the value of ‘x ‘ was restricted to ‘1 ‘ in the above trial. So as to extinguish the variable in order to maintain merely ‘y ‘ as the topic, I carried out the computations below that aid in making so –

Rewriting the equation –

Cross-multiplying the variables –

Dividing the equation by, we get –

But we know that. Therefore, utilizing this permutation in the above equation we have –

This is the same quadratic that we obtained earlier and therefore the uncertainty for the presence of ‘x ‘ clears out.

Experiment No. 2:

For my 2nd experiment, I have decided to utilize the construct of the Fibonacci spiral and that of the Golden Spiral. The stairss on how to pull these spirals are given below –

A Fibonacci spiral is formed by pulling squares with dimensions equal to the footings of he Fibonacci series.

We start by first pulling a 1 ten 1 square

1 x 1

Following, another 1 ten 1 square is drawn on the left of the first square. ( every new square is bordered in ruddy )

Now, a 2 ten 2 square is drawn below the two 1 tens 1 squares.

Next, a 3 ten 3 square is drawn to the right of the above figure.

Now, a 5 ten 5 square is adjoined to the top of the figure.

Next, a 8 ten 8 square is adjoined to the left of the figure.

And so the figure continues in the same mode. The squares are adjoined to the original form in a left to right spiral ( from down to up ) and each clip the square gets bigger but with dimensions equal to the Numberss in the Fibonacci series. Get downing from the inner square, a one-fourth of an discharge of a circle is drawn within the square. This measure is repeated as we move outward, towards the bigger square. The coiling finally looks like this –

hypertext transfer protocol: //library.thinkquest.org/27890/media/fibonacciSpiralBoxes.gif

The form shown below is the Fibonacci spiral without the squares

hypertext transfer protocol: //library.thinkquest.org/27890/media/fibonacciSpiral2.gif

A similar procedure is followed for organizing the aureate spiral. However, the lone difference is that we draw the outer squares foremost and so pull the discharge get downing from the larger squares. Hence, the coiling bends inwards all the manner to the interior squares.

Golden Spiral

The Golden spiral finally looks like this –

Golden Spiral

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

hypertext transfer protocol: //library.thinkquest.org/27890/media/spirals.gif

From the above experiment, it can be seen that there is a connexion between the Fibonacci series and the Golden Mean as their single spirals overlap each other as the ‘n ‘ ( which is the n-th term in the series ) tends to eternity.

Experiment No. 3:

My 3rd experiment involves engineering. In this experiment, I decided to utilize a plan of Microsoft Office, viz. , Microsoft Excel in order to enter the values obtained on ciphering the ratio of the back-to-back footings of the Fibonacci series. In the tabular array below, I have recorded the footings of the Fibonacci series in the first column, the value of the ratio of the back-to-back footings in the Fibonacci sequence in the 2nd column, the value of[ 8 ]in the 3rd column and the fluctuation of the value of the ration from the value of I† in the last column.

Term of Fibonacci Series

Value of ratio of back-to-back footings

value of

fluctuation of value calculated from value of

0

## –

## –

## –

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 purpose of the tabular array is to happen out whether the value of the ratio reaches the value of I† or non, as the figure of footings additions boundlessly.

Observation:

From the above tabular array, it can be seen that as we reach the n-th term of the Fibonacci series, the fluctuation in the value of the ratios from the value of I† , decreases. This observation is in understanding with the speculation – “ The bound of the ratios of the footings of the Fibonacci series converge to the aureate mean as n a†’ , where ‘n ‘ is the n-th term of the Fibonacci sequence. ”

Inference:

From the above 3 experiments, I have found that the speculation holds true for them all. Hence, I would wish to province that the trials for the speculations have been significantly successful.

## The Proof

In order to happen the relation between the Fibonacci series and the Golden Ratio, I followed the cogent evidence below that uses calculus to set up the needed relation.

The Fibonacci series is given by,

Assuming that 0, 1, and 1 are the first three footings of the sequence:

( 3 )

This finally goes on to organize the well – known sequence: 0, 1, 1, 2, 3, 5, 8, 13aˆ¦

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 replacing the bound of the ratios of the footings ( as n a†’ ) of the Fibonacci series with ‘A ‘ , the bound is taken on both sides such that n a†’

The above is true as the ratio

Hence, the below quadratic equation is formed

We can happen the roots of A by utilizing the quadratic expression, .

or

From this we find that

This value of is easy come-at-able utilizing the Binet expression. The Binet expression is that expression which gives the value of by replacing the variable ‘x ‘ with one of the n footings of the Fibonacci series. Using the construct of the aureate rectangle, the quadratic that was obtained earlier –

Give the value of. The cogent evidence of the Binet expression shows another possibility to get at the relation between the Fibonacci series and the Golden Ratio. The beauty of this cogent evidence is that the quadratic first arose from the Fibonacci series computation and the root that was obtained gave the value of phi. This is from the cogent evidence that was written above. Under the header ‘Testing the Conjecture ‘ that was done earlier, the quadratic arose from the dimensions of the Golden Rectangle and the equation therefore obtained gave the value of phi. Using this construct, I have followed the cogent evidence below which was solved by older mathematicians.

The Binet expression is given by –

Now, from the above trials, we got –

However, there were 2 values that were obtained on ciphering the value of ‘y ‘ . The value of ‘y ‘ that was negative was rejected so as it was wrong to see it a valid reply for a dimension of a geometric figure. Naming this negative root as ” , we can rewrite the Binet expression as –

Traveling back to the quadratic equation, we can replace ” in topographic point of ‘y ‘ and so the quadratic equation is –

( 4 )

This quadratic was obtained from the Golden Rectangle. In order to get at the Fibonacci sequence, a series of algebraic uses will assist us make that measure. To get down off with, we have the value of in footings of. Now, to acquire the value of in footings of, we multiply equation ( 4 ) into.

Using equation ( 4 ) , we substitute for and we get –

Using the same method to happen the value for raised to higher powers, we have –

Similarly,

Writing the assorted values for raised to higher powers –

( 5 )

## aˆ¦

Now if we look at the coefficients closely, we see that they are the back-to-back footings of the Fibonacci series. This can be written as –

( 6 )

However, the above tendency is non adequate cogent evidence for generalising the above statement. Hence, I decided to turn out it by utilizing the rule of mathematical initiation.

Measure 1:

Measure 2:

To turn out that P ( 1 ) is true.

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

Measure 3:

Hence, P ( K ) is true where

Measure 4:

To turn out that P ( k+1 ) is true.

Get downing 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 generalised signifier.

Besides, we know that is the other root of the quadratic equation and so the above general equation can be written in the above signifier every bit good –

( 7 )

In order to obtain the Binet expression in the signifier of –

We can deduct equation ( 7 ) from equation ( 6 ) to acquire –

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 expression which we started to turn out. Hence, the expression is valid.

## Verifying the Proof

In order to formalize a cogent evidence, it must be tested in order to look into whether the speculation is valid and can be generalized. For this ground, I have decided to utilize the Binet expression ( that was proved supra ) to look into the cogency of the relation between the Fibonacci series and the Golden Ratio by replacing values for ‘x ‘ in the equation –

Using

Case 1:

## ,

Which is the first term of the Fibonacci series.

Case 2:

## ,

Which is the 2nd term of the Fibonacci series.

Case 3:

## ,

Which is the 3rd term of the Fibonacci series.

Case 4:

## ,

Which is the 4th term of the Fibonacci series.

From these permutations it is clear that the expression is a valid one which gives the coveted consequence.

Besides, the above computations have proved to be significant illustrations for turn outing the cogency of the cogent evidence shown above. However, an of import note to retrieve in the Binet expression is that the value of ‘x ‘ starts from 0 and additions. So it can be said that ( ten belongs to the set of whole Numberss ) . This is to account for the fact that the Fibonacci series starts from 0 and so continues.

Hence, the speculation is true and can be generalized. Hence the speculation below can be considered true.

## “ The bound of the ratios of the footings of the Fibonacci series converge to the aureate mean as n a†’ , where ‘n ‘ is the n-th term of the Fibonacci sequence. ”

## Decision

From the above trials and confirmations, it is clear that a relation between the Fibonacci series and the Golden Ratio does genuinely be. The relation being –

## “ The bound of the ratios of the footings of the Fibonacci series converge to the aureate mean as n a†’ , where ‘n ‘ is the n-th term of the Fibonacci sequence. ”

The Fibonacci series every bit good as the Golden Ratio have their single applications every bit good as combined applications in assorted Fieldss of nature, art, etc. As mentioned earlier, the Fibonacci series was used to happen a solution to the “ coney job ” . The relation between the two constructs was an built-in portion of the cardinal thought in the novel ‘The Da Vinci Code ‘ .

Along with these good known thoughts, other applications of the two constructs are present in the commotion of a pine cone, the pictures of Leonardo Da Vinci, the spiral of the nautilus shell, the petals of the helianthus. These are merely really few illustrations sing the applications of the two constructs.

However, this relation has proved to be utile to conservationists, creative persons and many other researches. For illustration, creative persons were able to utilize the survey of the construct in the pictures of Leonardo Da Vinci and decipher old symbols. It besides has given them the opportunity to make art of their ain that by utilizing this construct in their process of making.

## Further Probe

With the great figure of applications that were found sing the Fibonacci series and the Golden Ratio, there is a possibility that there are other applications of the construct every bit good. The convergence of the ratios of the values to the value of phi may turn out to be of great significance if applied to another theory that has boggled heads of mathematicians for old ages. Possibilities such as these give rise to the inquiry of farther probe in this facet of the relationship between the two constructs.