Seena.VAssistant ProfessorDepartment of Mathematics

Fibonacci Numbers And Golden Ratio

Who Was Fibonacci?

• A Famous Mathematician

• Fibonacci (1170-1250) is a short for theLatin "filius Bonacci" which means "the son of Bonacci“ but his full name was Leonardo Pisano

• He introduced the Hindu-Arabic number system into Europe

About the

Origin of

Fibonacci Sequence

Fibonacci Sequence was discovered after an investigation on the reproduction of rabbits.

Fibonacci’s 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 the end 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?

Pairs

1 pair

At the end of the first month there is still only one pair

Pairs

1 pair

1 pair

2 pairs

End first month… only one pair

At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits

Pairs

1 pair

1 pair

2 pairs

3 pairs

End second month… 2 pairs of rabbits

At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

End first month… only one pair

Pairs

1 pair

1 pair

2 pairs

3 pairsEnd third month… 3 pairs

5 pairs

End first month… only one pair

End second month… 2 pairs of rabbits

At the end of the fourth month, the first pair produces yet another new pair, and the female born two months ago produces her first pair of rabbits also, making 5 pairs.

Thus We get the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34 ,55,89,144....

This sequence, in which each number is a sum of two

previous is called Fibonacci sequence

so there is the

simple rule: add the last two to get the next!

 

So 144 Pairs will be there

at the end of One Year….

Fibonacci sequence in

Nature

Spirals seen in the

arrangement of seeds in the head of

this sunflower number

Spirals seen in the arrangement of seeds in the head of this sunflower

number 34 in a counterclockwise direction

and 55 in a clockwise direction

Note that 34 and 55 are the ninth and tenth Fibonacci numbers respectively.

Also note that The flower itself has 34 petals.

Pinecones clearly show the Fibonacci

Spiral

Note that 8 and 13 are Consecutive Fibonacci

numbers

The number of petals on a flower are The number of petals on a flower are often often

Fibonacci numbers.Fibonacci numbers.

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

The

Golden Ratio

The The golden ratiogolden ratio is an irrational is an irrational mathematical constant, mathematical constant,

approximately equals toapproximately equals to

1.61803398871.6180339887

The The golden ratiogolden ratio is is often denoted by the often denoted by the

GreekGreek letter letter φ φ (Phi)(Phi)

So So φ = φ = 1.61803398871.6180339887

Also known as: • Golden Ratio,• Golden Section,• Golden cut,• Divine proportion, • Divine section, • Mean of Phidias• Extreme and mean ratio,• Medial section,

Two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and

the smaller

.

a b

a+b

a+b

a=

a

b= φ

φ =1+√5

2 = 1.618

One interesting thing about Phi is its reciprocal

1/φ = 1/1.618 = 0.618.

It is highly unusual for the decimal integers of a number and

its reciprocal to be exactly the same.

A golden rectangle is a rectangle where the ratio of its

length to width is the golden ratio. That is whose sides are

in the ratio 1:1.618

The golden rectangle has the property that it can be further subdivided in to two portions a square and a golden rectangle

This smaller rectangle can similarly be subdivided in to another set of smaller golden rectangle and smaller square.

And this process can be done repeatedly to produce smaller versions of squares

and golden rectangles

Golden Rectangle

Golden Spiral

Start with the smallest one on the right connect the lower right corner to the upper right corner with an arc that is one fourth of a circle. Then continue your line in to the second square on the with an arc that is one fourth of a circle , we will continue this process until each square has an arc inside of it, with all of them connected as a continues line. The line should look like a spiral when we are done .

Golden Triangle

The Golden triangle is a special isosceles triangle. The top angle is 360 while the bottom two angles are 720 each

Relation between Fibonacci

Sequence and Golden ratio

Aha! Notice that as we continue down the sequence, the ratios seem to be

converging upon one number (from both

sides of the number)!

2/1 = 2.0 2/1 = 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 = 1.625 13/8 = 1.625 (bigger)(bigger)

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

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

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

89/55 = 1.61889/55 = 1.618

The Fibonacci sequence is 1,1,2,3,5,8,13,21,34,55,….

If we continue to look at the If we continue to look at the ratios as the numbers in the ratios as the numbers in the sequence get larger and larger sequence get larger and larger the ratio will eventually become the ratio will eventually become the same number, and that the same number, and that number is the number is the Golden RatioGolden Ratio!!

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

Golden ratio

in

Nature

Nautilus Shell

Golden ratio in Art

Many artists who lived after Phidias have used this proportion. Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings

Mona Lisa's face is a perfect golden rectangle, according to the ratio of the width of her forehead compared to the length from the top of her head to her chin.

Golden Ratio

in the

Human Body

Golden Ratio in Fingers

Golden Ratio in Hands

Golden ratio in the Face

• The blue line defines a perfect square of the pupils The blue line defines a perfect square of the pupils and outside corners of the mouth. The golden and outside corners of the mouth. The golden section of these four blue lines defines the nose, the section of these four blue lines defines the nose, the tip of the nose, the inside of the nostrils, the two tip of the nose, the inside of the nostrils, the two rises of the upper lip and the inner points of the ear. rises of the upper lip and the inner points of the ear. The blue line also defines the distance from the The blue line also defines the distance from the upper lip to the bottom of the chin.upper lip to the bottom of the chin.

• The yellow line, a golden section of the blue line, The yellow line, a golden section of the blue line, defines the width of the nose, the distance between defines the width of the nose, the distance between the eyes and eye brows and the distance from the the eyes and eye brows and the distance from the pupils to the tip of the nose.pupils to the tip of the nose.

• The green line, a golden section of the yellow line The green line, a golden section of the yellow line defines the width of the eye, the distance at the defines the width of the eye, the distance at the pupil from the eye lash to the eye brow and the pupil from the eye lash to the eye brow and the distance between the nostrils.distance between the nostrils.

• The magenta line, a golden section of the green line, The magenta line, a golden section of the green line, defines the distance from the upper lip to the defines the distance from the upper lip to the bottom of the nose and several dimensions of the bottom of the nose and several dimensions of the eye.eye.

• The front two incisor teeth form a golden rectangle, with a phi ratio in the heighth to the width.The ratio of the width of the first tooth to the second tooth from the center is also phi.

• The ratio of the width of the smile to the third tooth from the center is phi as well.

Golden Ratio in Human body

• The white line is the body's height.

• The blue line, a golden section of the white line, defines the distance from the head to the finger tips

• The yellow line, a golden section of the blue line, defines the distance from the head to the navel and the elbows.

• The green line, a golden section of the yellow line, defines the distance from the head to the pectorals and inside top of the arms, the width of the shoulders, the length of the forearm and the shin bone.

• The magenta line, a golden section of the green line, defines the distance from the head to the base of the skull and the width of the abdomen. The sectioned portions of the magenta line determine the position of the nose and the hairline.

• Although not shown, the golden section of the magenta line (also the short section of the green line) defines the width of the head and half the width of the chest and the hips.

Golden Mean Gauge

More Examples of Golden Sections