the fibonacci numbers and the golden sectionp2 (1) (1)

8/3/2019 The Fibonacci Numbers and the Golden SectionP2 (1) (1)

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The Fibonacci Numbers

andThe Golden Section

By:

Dini Davis 0915020

Geethu K. 0915021

Shophi Philip 0915027


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WhoWas Fibonacci?

~ Born in Pisa, Italy in 1175 AD

~ Full name was Leonardo Pisano~ Grew up with a North African education under the Moors

~ Traveled extensively around the Mediterranean coast

~ Met with many merchants and learned their systems of arithmetic

~ Realized the advantages of the Hindu-Arabic system


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Fibonaccis Mathematical

Contributions~ Introduced the Hindu-Arabic number system into Europe

~ Based on ten digits and a decimal point

~ Europe previously used the Roman number system

~ Consisted of Roman numerals

~ Persuaded mathematicians to use the Hindu-Arabic number system

1 2 3 4 5 6 7 8 9 0 and .

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500

M = 1000


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Fibonaccis Mathematical

Contributions Continued

~Wrote five mathematical works

~ Four books and one preserved letter

~Liber Abbaci (The Book of Calculating) written in 1202

~Practica geometriae (Practical Geometry) written in 1220

~Flos written in 1225

~Liber quadratorum (The Book of Squares) written in 1225

~A letter to Master Theodorus written around 1225


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The number pattern that you have been using is known as theFibonacci sequence.

1 1

}

2

+


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1 1 2

}

3

+


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1 1 2 3

}

5 8 13 21 34 55

These numbers can be seen in many natural situations

+


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~Were introduced in The Book of Calculating

~ Series begins with 0 and 1

~ Next number is found by adding the last two numbers together

~ Number obtained is the next number in the series

~ Pattern is repeated over and over

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

Fn + 2 = Fn + 1 + Fn


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Fibonaccis Rabbits

Problem:

Suppose a newly-born pair of rabbits (one male, one female) are put in

a field. Rabbits are able to mate at the age of one month so that at theend of its second month, a female can produce another pair of rabbits.

Suppose that the rabbits never die and that the female always produces

one new pair (one male, one female) every month from the second

month on. How many pairs will there be in one year?


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Pairs

1 pair

Attheend ofthe first monththereisstill only one pair


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Pairs

1 pair

1 pair

2 pairs

End first month only one pair

Attheend ofthesecond monththe female producesanew pair,

so nowthereare 2 pairs ofrabbits


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Pairs

1 pair

1 pair

2 pairs

3 pairs

Endsecond month 2 pairs ofrabbits

Attheend ofthethird

month,the original

female producesa

second pair, making 3

pairsinallinthe field.



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Pairs

1 pair

1 pair

2 pairs

3 pairsEndthird month3 pairs

5 pairs


Endsecond month 2 pairs ofrabbits

Attheend ofthe fourth month,the first pair producesyetanothernew pair,andthe femaleborntwomonthsago producesher first pair ofrabbitsalso, making 5 pairs.


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Fibonaccis Rabbits Continued

~ End of the first month = 1 pair

~ End of the second month = 2 pair

~ End of the third month = 3 pair~ End of the fourth month = 5 pair

~ 5 pairs of rabbits produced in one year

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


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1

1

2

3

5

8

13

21

34

55


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The Fibonacci Numbers in

Pascals Triangle~ Entry is sum of the two numbers either side of it, but in the row above

~ Diagonal sums in Pascals Triangle are the Fibonacci numbers

~ Fibonacci numbers can also be found using a formula

1

1 11 2 1

1 3 3 1

1 4 6 4 1

Fib(n) =nk

k 1

n

k=1 ( )

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1


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The Fibonacci Numbers and

Pythagorean Triangles

a b a + b a + 2b

1 2 3 5

~ The Fibonacci numbers can be used to generate Pythagorean triangles

~ First side of Pythagorean triangle = 12

~ Second side of Pythagorean triangle = 5

~ Third side of Pythagorean triangle = 13

~ Fibonacci numbers 1, 2, 3, 5 produce Pythagorean triangle 5, 12, 13


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Geometers Sketchpad

Construction:~ On any line segment AB construct any rectangle ABCD

~ Find midpoints P2 of segment CD, M1 of segment AD, and M2 of segment BC

~ Draw segment M1M2

~ Draw segment AP2~ Draw diagonal BD

~ Let S be the point of intersection of segment AP2 and segment BD

~ The foot of the altitude from S to segment CD is P3, where DP = (1/3)DC

~ Let S be the point of reflection of S over segment M1M2

~ Draw segment SD

~ Draw segment AP3

~ Let K be the point of intersection of segment AP3 and segment SD

~ The foot of altitude from K to segment CD is P5, where DP5 = (1/5)DC

~ Let K be the point of reflection of K over segment M1M2

~ Draw segment KD

~ Draw segment AP5

~ LetW be the point of intersection of segment AP5 and segment KD

~ The foot of the altitude from W to segment CD is P8, where DP8 = (1/8)CD


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Geometers Sketchpad Continued~ All points in the pattern are reciprocals of the Fibonacci numbers

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


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The Fibonacci Numbers in Nature

~ Fibonacci spiral found in both snail and sea shells


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Nature Continued

~ Sneezewort (Achillea ptarmica) shows the Fibonacci numbers


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Nature Continued~ Plants show the Fibonacci numbers in the arrangements of their leaves~ Three clockwise rotations, passing five leaves

~ Two counter-clockwise rotations


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Nature Continued~ Pinecones clearly show the Fibonacci spiral


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Nature Continued

Lilies and irises = 3 petals

Black-eyed Susans = 21 petalsCorn marigolds = 13 petals

Buttercups and wild roses = 5 petals


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Nature Continued~ The Fibonacci numbers are found in the arrangement of seeds on flower heads

~ 55 spirals spiraling outwards and 34 spirals spiraling inwards


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Continued~ Fibonacci spirals can be made through the use of visual computer programs

~ Each sequence of layers is a certain linear combination of previous ones


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Continued

~ Fibonacci spiral can be found in cauliflower


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Nature Continued~ The Fibonacci numbers can be found in pineapples and bananas

~ Bananas have 3 or 5 flat sides

~ Pineapple scales have Fibonacci spirals in sets of 8, 13, 21


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Continued~ The Fibonacci numbers can be found in the human hand and fingers

~ Person has 2 hands, which contain 5 fingers

~ Each finger has 3 parts separated by 2 knuckles


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Fibonacci in Music

The intervals between keys on a piano are Fibonacci numbers.

23

5

8 white

13 w & b


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The Golden Section

~ Represented by the Greek letter Phi~ Phi equals s and s 0.6180339887

~ Ratio of Phi is 1 : 1.618 or 0.618 : 1

~ Mathematical definition is Phi2 = Phi + 1

~ Euclid showed how to find the golden section of a line

< - - - - - - - 1 - - - - - - - >

A G B

g 1 -g

GB = AG or 1 g = g

so thatg2

= 1 g

AG AB g 1


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The Golden Section and The

Fibonacci Numbers

~ The Fibonacci numbers arise from the golden section

~ The graph shows a line whose gradient is Phi

~ First point close to the line is (0, 1)

~ Second point close to the line is (1, 2)

~ Third point close to the line is (2, 3)

~ Fourth point close to the line is (3, 5)

~ The coordinates are successive Fibonacci numbers


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The Golden Section and The

Fibonacci Numbers Continued

~ The golden section arises from the Fibonacci numbers

~ Obtained by taking the ratio of successive terms in the Fibonacci series

~ Limit is the positive root of a quadratic equation and is called the golden section


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The relationship of this sequence to the Golden

Ratio lies not in the actual numbers of the

sequence, but in the ratio of the consecutive

numbers. Let's look at some of the ratios of these

numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 6101, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610


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2/1 = 2.0

3/2 = 1.55/3 = 1.67

8/5 = 1.6

13/8 = 1.62521/13 = 1.615

34/21 = 1.619

55/34 = 1.61889/55 = 1.618

Sincea Ratio isbasicallya

fraction (oradivision problem)wewill findtheratios ofthese

numbersbydividingthelarger

numberbythesmallernumber

that fallconsecutivelyinthe

series.So,whatistheratio ofthe 2nd

and 3rd numbers?

Well, 2 isthe 3rd numberdivided

bythe 2

nd

numberwhichis 1

2 dividedby 1 = 2

Andtheratioscontinuelike

this.


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Aha! Notice that as we

continue down the

sequence, the ratiosseem to be converging

upon one number (from

both sides of thenumber)!

2/1 = 2.02/1 = 2.0 (bigger)(bigger)

3/2 = 1.53/2 = 1.5 (smaller)(smaller)

5/3 = 1.675/3 = 1.67(bigger)(bigger)

8/5 = 1.68/5 = 1.6(smaller)(smaller)

13/8 = 1.62513/8 = 1.625 (bigger)(bigger)

21/13 = 1.61521/13 = 1.615 (smaller)(smaller)

34/21 = 1.61934/21 = 1.619 (bigger)(bigger)

55/34 = 1.61855/34 = 1.618(smaller)(smaller)

89/55 = 1.61889/55 = 1.618

Fibonacci Number calculator


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5/3 = 1.67

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615

34/21 = 1.619

55/34 = 1.618

89/55 = 1.618Notice that I have rounded my ratios to the third decimal place. If

we examine 55/34 and 89/55 more closely, we will see that their

decimal values are actually not the same. But what do you think

will happen if we continue to look at the ratios as the numbers in

the sequence get larger and larger? That's right: the ratio willeventually become the same number, and that number is the

Golden Ratio!

1


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1

1

2

3 1.5000000000000000

5 1.6666666666666700

8 1.6000000000000000

13 1.6250000000000000

21 1.6153846153846200

34 1.6190476190476200

55 1.6176470588235300

89 1.6181818181818200

144 1.6179775280898900

233 1.6180555555555600

377 1.6180257510729600

610 1.6180371352785100

987 1.6180327868852500

1,597 1.6180344478216800

2,584 1.6180338134001300

4,181 1.6180340557275500

6,765 1.6180339631667100

10,946 1.6180339985218000

17,711 1.6180339850173600

28,657 1.6180339901756000

46,368 1.6180339882053200

75,025 1.6180339889579000


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The Golden Ratio is what we call an

irrational number: it has an infinite number

of decimal places and it never repeatsitself! Generally, we round the

Golden Ratio to 1.618.


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whichisequalto:


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The Golden Section and

Geometry

~ Is the ratio of the side of a regular pentagon to its diagonal

~ The diagonals cut each other with the golden ratio

~ Pentagram describes a star which forms parts of many flags

European Union

United States


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The Golden Section in Nature

~ Arrangements of leaves are the same as for seeds and petals

~ All are placed at 0.618 per turn

~ Is 0.618 of 360o which is 222.5o

~ One sees the smaller angle of 137.5o

~ Plants seem to produce their leaves, petals, and seeds based upon the golden section


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The Golden Section in

Architecture

~ Golden section appears in many of the proportions of the Parthenon in Greece~ Front elevation is built on the golden section (0.618 times as wide as it is tall)


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The ancient temple fits almost precisely into a golden

rectangle.

Further classic subdivisions of the rectangle alignperfectly with major architectural features of the

structure.


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Architecture Continued

~ Golden section can be found in the Great pyramid in Egypt

~ Perimeter of the pyramid, divided by twice its vertical height is the value of Phi


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Architecture Continued

~ Golden section can be found in the design of Notre Dame in Paris

~ Golden section continues to be used today in modern architecture

United Nations Headquarters Secretariat building


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The Golden Section in Art

~ Golden section can be found in Leonardo da Vincis artwork

The Annunciation


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Continued

Madonna with Child and Saints

The Last Supper


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Continued~ Golden section can be seen in Albrecht Durers paintings

TrentoNurnberg


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The Golden Section in Music

~ Stradivari used the golden section to place the f-holes in his famous violins

~ Baginsky used the golden section to construct the contour and arch of violins


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Continued~ Mozart used the golden section when composing music

~ Divided sonatas according to the golden section

~ Exposition consisted of 38 measures

~ Development and recapitulation consisted of 62 measures~ Is a perfect division according to the golden section


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Continued

~ Beethoven used the golden section in his famousFifth Symphony

~ Opening of the piece appears at the golden section point (0.618)

~ Also appears at the recapitulation, which is Phi of the way through the piece


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Bibliography

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

http://evolutionoftruth.com/goldensection/goldsect.htmhttp://pass.maths.org.uk/issue3/fiibonacci/

http://www.sigmaxi.org/amsci/issues/Sciobs96/Sciobs96-03MM.html

http://www.violin.odessa.ua/method.html

the fibonacci numbers and the golden sectionp2 (1) (1)

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