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Fibonacci Fibonacci Sequence Sequence & & Golden Ratio Golden Ratio Monika Bała

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The Fibonacci Sequence In the XIII century an Italian mathematician Leonardo Fibonacci discovered a series of numbers that have very intresting properties. This is definitely one of the most interesting mathematical sequences. The Fibonacci Sequence is closely related with the Golden Ratio. Both of them are equally common in nature.

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Page 1: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Fibonacci SequenceFibonacci Sequence&&

Golden Ratio Golden RatioMonika Bała

Page 2: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

PLAN OF THE PLAN OF THE PRESENTATION:PRESENTATION:

Definition of the Fibonacci Sequence Definition of the Fibonacci Sequence and its propertiesand its properties

Pascal’s trianglePascal’s triangle Golden ratio and the number Golden ratio and the number φ Properties of the number Properties of the number φ Examples of occurrences of the Examples of occurrences of the

number number φ in real life in real life

Page 3: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

The Fibonacci SequenceThe Fibonacci Sequence

In the XIII century an Italian mathematician Leonardo Fibonacci discovered a series of numbers that have very intresting properties. This is definitely one of the most interesting mathematical sequences.

The Fibonacci Sequence is closely related with the Golden Ratio. Both of them are equally common in nature.

Page 4: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Definition and properties of Definition and properties of the Fibonacci Sequencethe Fibonacci Sequence

This is the kind of recurrent sequence This is the kind of recurrent sequence in which every following term is in which every following term is equal to the sum of the two previous equal to the sum of the two previous terms (except for the first two).terms (except for the first two).

Page 5: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Sometimes this sequence is called the Sometimes this sequence is called the

golden seqence because if we divide golden seqence because if we divide successive terms by the previous we get successive terms by the previous we get the value the value oscillating around the numberoscillating around the number φ..

The greater terms of the series we divide The greater terms of the series we divide by each other, the better approximation by each other, the better approximation we have. However this value will never be we have. However this value will never be equal to the golden number because equal to the golden number because φ

is irrational.is irrational.

2 / 1 = 2.0000000000000000003 / 2 = 1.5000000000000000005 / 3 = 1.6666666666666666708 / 5 = 1.60000000000000000013 / 8 = 1.62500000000000000021 / 13 = 1.61538461538461538034 / 21 = 1.61904761904761905055 / 34 = 1.61764705882352941089 / 55 = 1.618181818181818180144 / 89 = 1.617977528089887640

233 / 144 = 1.618055555555555560377 / 233 = 1.618025751072961370610 / 377 = 1.618037135278514590987 / 610 = 1.6180327868852459001597 / 987 = 1.6180344478216818602584 / 1597 = 1.6180338134001252304181 / 2584 = 1.6180340557275541806765 / 4181 = 1.61803396316670653010946 / 6765 = 1.618033998521803400

Page 6: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Pascal’s Triangle The Pascal’s triangle triangle is a triangular array of is a triangular array of

the binomial coefficientsthe binomial coefficients. . Each number in the triangle Each number in the triangle is the sum of two numbers that are directly above itis the sum of two numbers that are directly above it. . The numbers in the rows correspond to the The numbers in the rows correspond to the coefficients in the expansion of coefficients in the expansion of (a+b)n.

What the Fibonacci sequence and Pascal’s triangle have in common?

So, calculating the sum of the elements in the diagonal columns we get the following numbers of the sequence.

Page 7: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Fibonacci Spiral This spiral is based on the squares

of the lengths of the sides equal the following terms of the Fibonacci sequence (except for te first two except for te first two squares).squares).

Page 8: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

The Golden Ratio of a line The Golden Ratio of a line segmentsegment

This is a division of a line segment into two parts in such a way that the ratio of the length of the longer part by the shorter is the same as the ratio of the length of the whole line segment by the longer part of the line segment.

This ratio we denote by the Greek letter φ.

This is exactly the golden number which is equal 2

51

Page 9: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

The Golden number is an irrational number.Its approximation to 32 decimal places is the

number:

1,61803398874989484820458683436564 

Golden numberGolden number

Page 10: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

45

21

045

21

01

11

2

2

x

x

xxxxx

25

21

x25

21

x

The point E divide in the golden ratio the line segment AB if:

251

x

251

or

Because is less than zero, so the only solution is:

Page 11: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Properties of the number φ: Properties of the number φ:

To get its square root we only need to add one:

12

To get the inverse number we only need to subtract one:

11

11

11

Page 12: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

...111

11

11

11

11

x

The following fraction is equal to the golden number:

Page 13: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Golden triangleGolden triangle The Golden triangle The Golden triangle is an isosceles triangle, is an isosceles triangle,

which has an acute angle of measure 36° at which has an acute angle of measure 36° at the vertex and two acute angles of measure the vertex and two acute angles of measure 72° at the base. 72° at the base.

The ratio of the length of a side of a triangle by the length of the base of this triangle is the golden number.

Page 14: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

PentagramPentagram To draw a perfect pentagram we need to draw a regular

pentagon and lead diagonals or extend his sides until the intersection. The interior angle of the pentagram is 36°.

The whole pentagram has hidden in itself the golden ratio.

1. The ratio of the yellow segment by the blue segment = φ2. The ratio of the blue segment by the green segment = φ3. The ratio of the green segment by the red segment = φ

Page 15: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Golden proportionsGolden proportions Leonardo da Vinci noted that if Leonardo da Vinci noted that if

the human body is built the human body is built proportionally then it is proportionally then it is inscribed in a square and a inscribed in a square and a circle. Such a square and a circle. Such a square and a circle define a rectangle ABCD, circle define a rectangle ABCD, which for a man of the correct which for a man of the correct proportions is goldproportions is gold,, which is which is the height of a man to the the height of a man to the length of the lower part of the length of the lower part of the body (from the navel down) is body (from the navel down) is the golden numbethe golden number. r. (the ratio of (the ratio of the lower body to the upper is the lower body to the upper is the golden numberthe golden number))

Page 16: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci
Page 17: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

PARTHENON-PARTHENON- Athena’s temple Athena’s temple on the Acropolis in on the Acropolis in Athens, built in the Athens, built in the years 448-432 years 448-432 before Christ. before Christ. Fronton of the Fronton of the temple was located temple was located in a rectangle in in a rectangle in which the ratio of which the ratio of the sides is equal the sides is equal to the golden to the golden number.number.

Page 18: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

PYRAMIDS PYRAMIDS IN GIZA IN GIZA

If we take a cross-If we take a cross-section the Great section the Great Pyramid we get a right-Pyramid we get a right-angled triangle which is angled triangle which is called the Triangle of called the Triangle of Egypt. The ratio of the Egypt. The ratio of the hypotenuse to the base is hypotenuse to the base is 1,61804 and differs from 1,61804 and differs from the golden number of the golden number of only about 1 in fifth only about 1 in fifth place after the decimal place after the decimal point.point.

Page 19: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

BIBLIOGRAPHY:BIBLIOGRAPHY: http://www.naucz31.republika.pl/

http://www.jakubas.pl/matematyka/1/zlota-liczba.htm

http://www.zobaczycmatematyke.krk.pl/003-Golonka-Kalwaria/index.html

Page 20: Fibonacci Sequence & Golden Ratio Monika Bała. PLAN OF THE PRESENTATION: Definition of the Fibonacci Sequence and its properties Definition of the Fibonacci

Thank You for your Thank You for your attention! attention!