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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 Fibonacci Numbers
The number pattern that you have been using is known as theFibonacci sequence.
1 1
}
2
+
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The Fibonacci Numbers
The number pattern that you have been using is known as theFibonacci sequence.
1 1 2
}
3
+
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The Fibonacci Numbers
The number pattern that you have been using is known as theFibonacci sequence.
1 1 2 3
}
5 8 13 21 34 55
These numbers can be seen in many natural situations
+
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The Fibonacci Numbers
~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.
End first month only one pair
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Pairs
1 pair
1 pair
2 pairs
3 pairsEndthird month3 pairs
5 pairs
End first month only one pair
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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The Fibonacci Numbers and
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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The Fibonacci Numbers and
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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The Fibonacci Numbers in
Nature Continued
~ Sneezewort (Achillea ptarmica) shows the Fibonacci numbers
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The Fibonacci Numbers in
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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The Fibonacci Numbers in
Nature Continued~ Pinecones clearly show the Fibonacci spiral
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The Fibonacci Numbers in
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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The Fibonacci Numbers in
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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The Fibonacci Numbers in Nature
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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The Fibonacci Numbers in Nature
Continued
~ Fibonacci spiral can be found in cauliflower
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The Fibonacci Numbers in
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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The Fibonacci Numbers in Nature
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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The Golden Section in
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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The Golden Section in
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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The Golden Section in Art
Continued
Madonna with Child and Saints
The Last Supper
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The Golden Section in Art
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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The Golden Section in Music
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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The Golden Section in Music
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
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