Fibonacci [Compatibility Mode]

8/7/2019 Fibonacci [Compatibility Mode]

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Fibonacci Numbers andFibonacci Numbers and

Say What?Say What?


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What is theWhat is theGolden RatioGolden Ratio??

'',,

examine an interesting sequence (or list)examine an interesting sequence (or list)..

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

233, 377, 610233, 377, 610


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The series is derived by the following formula:The series is derived by the following formula:

== -- --

oror


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Now, I know what you might be thinking:Now, I know what you might be thinking:

"What does this have to do with the"What does this have to do with the oldenolden

Ratio?Ratio?

s sequence o num ers was rsts sequence o num ers was rstdiscovered by a man named Leonardodiscovered by a man named Leonardo

Fibonacci, and hence is known asFibonacci, and hence is known as

Fibonacci'sFibonacci's sequence.sequence.

Math GEEK

Leonardo FibonacciLeonardo Fibonacci


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The relationship of this sequence to theThe relationship of this sequence to the

GoldenGolden Ratio lies not in the actual numbersRatio lies not in the actual numbers

o t e sequence, ut n t e rat o o t eo t e sequence, ut n t e rat o o t econsecutive numbers. Let's look at some ofconsecutive numbers. Let's look at some of

the ratios of these numbers: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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== ..3/2 = 1.53/2 = 1.5

nce a a o s as ca y a rac on or

a division problem) we will find the

ratios of these numbers by dividing

the larger number by the smaller

= .= .8/5 = 1.68/5 = 1.6

number that fall consecutively in the

series.

So, what is the ratio of the 2nd and

13/8 = 1.62513/8 = 1.625

==

3rd numbers?

Well, 2 is the 3rd number divided by the ..

34/21 = 1.61934/21 = 1.6192nd number which is 1

2 divided by 1 = 2

..89/55 = 1.61889/55 = 1.618

And the ratios continue like this.


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= .= . (bigger)(bigger)

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

we continue downwe continue down 5/3 = 1.675/3 = 1.67(bigger)(bigger)8/5 = 1.68/5 = 1.6(smaller)(smaller)

,,

ratios seem to beratios seem to be

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

==converging upon oneconverging upon one

number (from bothnumber (from both

..

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

sides of the number)!sides of the number)! ..(smaller)(smaller)

89/55 = 1.61889/55 = 1.618

Fibonacci Number calculator


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Depiction of Golden Ratio

2

2.1

1.8

1.9

1.5

1.6

.

1.3

1.4

1.1

1.2

1

0 5 10 15 20 25 30


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irrational number: it has an infiniteirrational number: it has an infinite

number of decimal places and it nevernumber of decimal places and it never

the Golden Ratio to 1.618.the Golden Ratio to 1.618.


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We work with another importantWe work with another important

,,which is approximately 3.14. Sincewhich is approximately 3.14. Since

we don't want to make the Goldenwe don't want to make the Golden

own Greek letter: phi.own Greek letter: phi.

Phi


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which is equal to:


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One more interesting thing about Phi is itsOne more interesting thing about Phi is its

reciprocal. If you take the ratio of any number inreciprocal. If you take the ratio of any number in

the reverse of what we did before), the ratio willthe reverse of what we did before), the ratio will

reciprocal of Phi: 1 / 1.618 = 0.618. It is highlyreciprocal of Phi: 1 / 1.618 = 0.618. It is highly

unusual for the decimal integers of a number andunusual for the decimal integers of a number andits reciprocal to be exactly the same. In fact, Iits reciprocal to be exactly the same. In fact, I

cannot name another number that has thiscannot name another number that has this

proper y s on y a s o e mys que o eproper y s on y a s o e mys que o eGolden Ratio and leads us to ask: What makes itGolden Ratio and leads us to ask: What makes it


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The Golden Ratio is not just some number that mathThe Golden Ratio is not just some number that math

eac ers n s coo . e n eres ng ng s a eepseac ers n s coo . e n eres ng ng s a eepspopping up in strange placespopping up in strange places -- places that we may notplaces that we may not

ordinarily have thought to look for it. It is important to noteordinarily have thought to look for it. It is important to note

that Fibonacci did not "invent" the Golden Ratio; he justthat Fibonacci did not "invent" the Golden Ratio; he just

discovered one instance of where it appeared naturally. Indiscovered one instance of where it appeared naturally. In

Egyptians, the Mayans, as well as the Greeks discoveredEgyptians, the Mayans, as well as the Greeks discovered

the Golden Ratio and incorporated it into their own art,the Golden Ratio and incorporated it into their own art,arc ec ure, an es gns. ey scovere a e o enarc ec ure, an es gns. ey scovere a e o en

Ratio seems to be Nature's perfect number. For someRatio seems to be Nature's perfect number. For some

reason, it just seems to appeal to our natural instincts. Thereason, it just seems to appeal to our natural instincts. The

most basic example is in rectangular objects.most basic example is in rectangular objects.


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Look at the following rectangles:Look at the following rectangles:

ow as yourse , w c o em seems o e e mosow as yourse , w c o em seems o e e mos

naturally attractive rectangle? If you said the first one, thennaturally attractive rectangle? If you said the first one, then

you are probably the type of person who likes everything toyou are probably the type of person who likes everything to

be symmetrical. Most people tend to think that the thirdbe symmetrical. Most people tend to think that the thirdrectangle is the most appealing.rectangle is the most appealing.


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''

and width, and compare the ratio of length to widthand width, and compare the ratio of length to width

for each rectangle you would see the following:for each rectangle you would see the following:Rectangle one: Ratio 1:1Rectangle one: Ratio 1:1

Rectangle two: Ratio 2:1Rectangle two: Ratio 2:1

ec ang e ree: a o . :ec ang e ree: a o . :


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Have you figured out why the third rectangle is theHave you figured out why the third rectangle is the

most appealing? That's rightmost appealing? That's right -- because the ratio ofbecause the ratio ofits length to its width is the Golden Ratio! Forits length to its width is the Golden Ratio! For

centuries, designers of art and architecture havecenturies, designers of art and architecture have

their work.their work.


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Fibonacci made the following assumptionsFibonacci made the following assumptions

based on the breeding habits of rabbitsbased on the breeding habits of rabbits

He imagined a pair of rabbits in a field on theirHe imagined a pair of rabbits in a field on their

own.own.

ne ma e one ema e.ne ma e one ema e.

Rabbits never die.Rabbits never die. Female rabbits always produces one new pairFemale rabbits always produces one new pair

(one male, one female) every month from the(one male, one female) every month from the

.. The gestation period for rabbits is one month.The gestation period for rabbits is one month.

Maths Support Service 2007 Maths Support Service 2007

..


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8

there be after 5 months?



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male and one female.male and one female.

month.month.

n ree ng.n ree ng.

Rabbits die and get killed.Rabbits die and get killed.

EtcEtc



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Practise questionPractise question

How many she calves after 10years?



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In this activity, we study the ancestors of aIn this activity, we study the ancestors of a

oney ee.oney ee.

In a family of honeybees not all of them have twoIn a family of honeybees not all of them have twoarents.arents.

There is a special female called the Queen, who hasThere is a special female called the Queen, who hasbeen fed with a special substance called royal jellybeen fed with a special substance called royal jelly

only the Queen becomes a mother.only the Queen becomes a mother.

Males are produced by the queen's unfertilized eggs,Males are produced by the queen's unfertilized eggs,

, ., . The females are produced when the queen hasThe females are produced when the queen has

mated with a male and so they have two parents.mated with a male and so they have two parents.



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Lets take a look at the iano ke boarddo ou see

Anything familiar?


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oun e num er o eys no es n eac o e rac e soun e num er o eys no es n eac o e rac e sYou will see the numbers 2,3,5,8,13.coincidence?You will see the numbers 2,3,5,8,13.coincidence?

Does it look like the Fibonacci sequenceit shouldDoes it look like the Fibonacci sequenceit should

because it is!because it is!


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How about Architecture?


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Find the Golden Ratio in the Parthenon.Find the Golden Ratio in the Parthenon.

1. Let's start by drawing a rectangle around the Parthenon,1. Let's start by drawing a rectangle around the Parthenon,

rom t e e t most p ar to t e r g t an rom t e ase o t erom t e e t most p ar to t e r g t an rom t e ase o t epillars to the highest point.pillars to the highest point.

2. Measure the len th and the width of this rectan le. Now2. Measure the len th and the width of this rectan le. Now

find the ratio of the length to the width. Is the number fairlyfind the ratio of the length to the width. Is the number fairly

close to the Golden Ratio?close to the Golden Ratio?

. .. .

rectangles on the face of the Parthenon. Find the ratio ofrectangles on the face of the Parthenon. Find the ratio of

the length to the width of one of these rectangles. Noticethe length to the width of one of these rectangles. Noticeany ngany ng

There are many other places where the Golden RatioThere are many other places where the Golden Ratio

appears in the Parthenon, all of which we cannot seeappears in the Parthenon, all of which we cannot see

because we only have a frontal view of the structure. Thebecause we only have a frontal view of the structure. Thebuilding is built on a rectangular plot of land which happensbuilding is built on a rectangular plot of land which happens

--......


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Once its ruined trian ular

pediment is restored, ...


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the ancient tem le fits almost

precisely into a golden

rectangle.


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Further classic subdivisions of the rectan le ali n

perfectly with major architectural features of the

structure.

Further classic subdivisions of the rectangle align perfectly with major architectural features of the structure.


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The Golden Ratio in ArtThe Golden Ratio in Art

discover the Golden Ratio in art.discover the Golden Ratio in art.

We will concentrate on the works ofWe will concentrate on the works of

,,only a great artist but also a geniusonly a great artist but also a genius

when it came to mathematics andwhen it came to mathematics andinvention.invention.


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The AnnunciationThe Annunciation --Using the left side of the painting as aUsing the left side of the painting as a

side, create a square on the left of the painting by insertingside, create a square on the left of the painting by inserting

..

rectangle. The rectangle turns out to be a Goldenrectangle. The rectangle turns out to be a GoldenRectangle, of course. Also, draw in a horizontal line that isRectangle, of course. Also, draw in a horizontal line that is

. o t e way own t e pa nt ng .. o t e way own t e pa nt ng . -- t e nverse ot e nverse o

the Golden Ratio). Draw another line that is 61.8% of thethe Golden Ratio). Draw another line that is 61.8% of the

wa u the aintin . Tr a ain with vertical lines that arewa u the aintin . Tr a ain with vertical lines that are61.8% of the way across both from left to right and from61.8% of the way across both from left to right and from

right to left. You should now have four lines drawn acrossright to left. You should now have four lines drawn across

..

parts of the painting, such as the angel, the woman, etc.parts of the painting, such as the angel, the woman, etc.

Coincidence? I think not!Coincidence? I think not!


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e ona sae ona sa -- easure t e engt aneasure t e engt an

the width of the painting itself. The ratio is, ofthe width of the painting itself. The ratio is, of

course, olden. Draw a rectangle aroundcourse, olden. Draw a rectangle around

Mona's face (from the top of the forehead toMona's face (from the top of the forehead tot e ase o t e c n, an rom e t c ee tot e ase o t e c n, an rom e t c ee to

right cheek) and notice that this, too, is aright cheek) and notice that this, too, is a

Golden rectangle.Golden rectangle.


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Leonardo da Vinci's talent as an artist may wellLeonardo da Vinci's talent as an artist may well

have been outweighed by his talents as ahave been outweighed by his talents as a

mathematician. He incorporated geometry intomathematician. He incorporated geometry into

many of his paintings, with the Golden Ratio beingmany of his paintings, with the Golden Ratio being

us one o s many ma ema ca oo s. y ous one o s many ma ema ca oo s. y o

you think he used it so much? Experts agree thatyou think he used it so much? Experts agree that

made his paintings more attractive. Maybe he wasmade his paintings more attractive. Maybe he was

ust a little too obsessed with erfection. Howeverust a little too obsessed with erfection. However

he was not the only one to use Golden propertieshe was not the only one to use Golden propertiesin his work.in his work.


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Isn't it strange that the Golden Ratio came up in suchIsn't it strange that the Golden Ratio came up in such

unexpected places? Well let's see if we can find out why.unexpected places? Well let's see if we can find out why.

The Greeks were the first to call phi the Golden Ratio. TheyThe Greeks were the first to call phi the Golden Ratio. They..

of human nature or instinct for us to find things that containof human nature or instinct for us to find things that contain

the Golden Ratio naturally attractivethe Golden Ratio naturally attractive -- such as the "perfect"such as the "perfect"

. ,. ,

incorporate the Golden Ratio into their designs so as toincorporate the Golden Ratio into their designs so as to

make them more pleasing to the eye. Doors, notebookmake them more pleasing to the eye. Doors, notebook

paper, textbooks, etc. all seem more attractive if their sidespaper, textbooks, etc. all seem more attractive if their sideshave a ratio close to phi. Now, let's see if we can constructhave a ratio close to phi. Now, let's see if we can construct

our own " erfect" rectan le.our own " erfect" rectan le.


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Method OneMethod One1. We'll start by making a square, any square (just remember that all sides have to1. We'll start by making a square, any square (just remember that all sides have to

have the same length, and all angles have to measure 90 degrees!):have the same length, and all angles have to measure 90 degrees!):

2.Now, let's divide the square in half (bisect it). Be sure to use your protractor to2.Now, let's divide the square in half (bisect it). Be sure to use your protractor todivide the base and to form another 90 degree angle:divide the base and to form another 90 degree angle:


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Now draw in one of the dia onals of one of the rectan les

Measure the length of the diagonal and make a noteMeasure the length of the diagonal and make a note

o t.o t.


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Now extend the base of the square from theNow extend the base of the square from the

midpoint of the base by a distance equal to themidpoint of the base by a distance equal to the

length of the diagonallength of the diagonal


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Construct a new line perpendicular to the base at the end ofConstruct a new line perpendicular to the base at the end of

our new line, and then connect to form a rectangle:our new line, and then connect to form a rectangle:

Measure the length and the width of your rectangle.Measure the length and the width of your rectangle.

Now, find the ratio of the length to the width.

Are you surprised by the result? The rectangle you have made is called a

Golden Rectangle because it is "perfectly" proportional.

Constructing a Golden RectangleConstructing a Golden Rectangle -- Method TwoMethod Two


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g gg g

Now, let's try a different method that will relate theNow, let's try a different method that will relate the

'' ..with a square. The size does not matter, as long as allwith a square. The size does not matter, as long as all

sides are congruent. We'll use a small square to conservesides are congruent. We'll use a small square to conserve

space, because we are going to build our golden rectanglespace, because we are going to build our golden rectangle

around this square. Please note that the golden area isaround this square. Please note that the golden area is..


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Now, let's build another, congruent square right next to theNow, let's build another, congruent square right next to the

Now we have a rectangle with a width 1 and length 2 units.Now we have a rectangle with a width 1 and length 2 units.

Let's build a square on top of this rectangle, so that the newLet's build a square on top of this rectangle, so that the new

square will have a side of 2 units:square will have a side of 2 units:

Notice that we have a new rectangle with width 2 andNotice that we have a new rectangle with width 2 and

length 3.length 3.


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Let's continue the process, building another square on theLet's continue the process, building another square on the

right of our rectangle. This square will have a side of 3:right of our rectangle. This square will have a side of 3:

Now we have a rectangle of width 3 and length 5.Now we have a rectangle of width 3 and length 5.


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A ain, let's build u on thisA ain, let's build u on this

rectangle and construct a squarerectangle and construct a square

,,


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The new rectan le has a width of 5The new rectan le has a width of 5

and a length of 8. Let's continue toand a length of 8. Let's continue to

Have you noticed the pattern yet?Have you noticed the pattern yet?


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y p yy p y

and a length of 13. Let's continueand a length of 13. Let's continue

with one final square on top, with awith one final square on top, with a

, , , , , , , ..., , , , , , , ...

is golden! Each successive rectangle that we constructedis golden! Each successive rectangle that we constructed

h d idth d l th th t ti t i thh d idth d l th th t ti t i th


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had a width and length that were consecutive terms in thehad a width and length that were consecutive terms in the

onacc sequence. o we v e e eng y eonacc sequence. o we v e e eng y ewidth, we will arrive at the Golden Ratio! Of course, ourwidth, we will arrive at the Golden Ratio! Of course, our

rectangle is not "perfectly" golden. We could keep therectangle is not "perfectly" golden. We could keep the

process going until the sides approximated the ratio better,process going until the sides approximated the ratio better,

but for our purposes a length of 21 and a width of 13 arebut for our purposes a length of 21 and a width of 13 are..

34


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34

21

Do the Math!! 34 divided by 21 =1.61904761904

go the closer the ratio gets to being perfect!


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well proportioned to you, i.e. itwell proportioned to you, i.e. itshould be pleasing to the eye. If itshould be pleasing to the eye. If it

'' ,,checked!checked!


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Constructing a Golden SpiralConstructing a Golden Spiral


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Notice how we built our rectangle inNotice how we built our rectangle in

a counterclockwise direction. Thisa counterclockwise direction. This

characteristic of the Golden Ratio.characteristic of the Golden Ratio.

e s oo a e rec ang e w a oe s oo a e rec ang e w a oour construction lines drawn in:our construction lines drawn in:


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We are going to concentrate on the squares that we drew,We are going to concentrate on the squares that we drew,


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We are going to concentrate on the squares that we drew,We are going to concentrate on the squares that we drew,

''..on the right. Connect the upper right corner to the lower lefton the right. Connect the upper right corner to the lower left


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we drew, starting with the two smallest ones. Let'swe drew, starting with the two smallest ones. Let's

start with the one on the right. Connect the upperstart with the one on the right. Connect the upper

right corner to the lower left corner with an arc thatright corner to the lower left corner with an arc that

is one fourth of a circle:is one fourth of a circle:

en con nue your ne n o e secon square onen con nue your ne n o e secon square on

the left, again with an arc that is one fourth of athe left, again with an arc that is one fourth of a

We will continue this process until each squareWe will continue this process until each square

has an arc inside of it with all of them connectedhas an arc inside of it with all of them connected

as a continuous line. The line should look like aas a continuous line. The line should look like aspiral when we are done. Here is an example ofspiral when we are done. Here is an example of

what your spiral should look like:what your spiral should look like:


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N w wh w h in f h ?N w wh w h in f h ?


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N w wh w h in f h ?N w wh w h in f h ?

The point is that this "golden spiral"The point is that this "golden spiral"

..

look closely enough, you might findlook closely enough, you might finda golden spiral in the head of aa golden spiral in the head of a

, , ,, , ,or in a nautilus shell that you mightor in a nautilus shell that you might

find on a beach or even in your ear!find on a beach or even in your ear!Here are some exam les:Here are some exam les:


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So, why do shapes that exhibit theSo, why do shapes that exhibit the

Golden Ratio seem more appealingGolden Ratio seem more appealing

knows for sure. But we do haveknows for sure. But we do have

ev ence a e o en a oev ence a e o en a oseems to be Nature's perfectseems to be Nature's perfect

number.number.

Somebody with a lot of time on their hands discovered thatSomebody with a lot of time on their hands discovered that

the individual florets of the daisy (and of a sunflower asthe individual florets of the daisy (and of a sunflower as


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y (y (

we grow n wo sp ra s ex en ng ou rom e cen er. ewe grow n wo sp ra s ex en ng ou rom e cen er. efirst spiral has 21 arms, while the other has 34. Do thesefirst spiral has 21 arms, while the other has 34. Do these


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--

numbers! And their ratio, of course,numbers! And their ratio, of course,

s t e o en at o.s t e o en at o.

We can say the same thing about the spiralsWe can say the same thing about the spirals


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,,have 5 and 8 arms, respectively (or of 8 andhave 5 and 8 arms, respectively (or of 8 and

--,, ,,

Fibonacci numbers:Fibonacci numbers:

A pineapple has three arms of 5, 8, and 13A pineapple has three arms of 5, 8, and 13 --


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coincidence. Now is Nature playing somecoincidence. Now is Nature playing some

for sure, but scientists speculate that plantsfor sure, but scientists speculate that plants

Fibonacci numbers because thisFibonacci numbers because this

for growth. So for some reason, thesefor growth. So for some reason, these

for maximum growth potential and survivalfor maximum growth potential and survival..


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Do these faces seem attractive to you?Do these faces seem attractive to you?

Man eo le seem to think so. But wh ? IsMan eo le seem to think so. But wh ? Is


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Man eo le seem to think so. But wh ? IsMan eo le seem to think so. But wh ? Is

there something specific in each of theirthere something specific in each of theirfaces that attracts us to them, or is ourfaces that attracts us to them, or is our

attraction governed by one of Nature'sattraction governed by one of Nature's

rules? Does this have anything to do withrules? Does this have anything to do with

the Golden Ratio? I think you already knowthe Golden Ratio? I think you already knowthe answer to that question. Let's try tothe answer to that question. Let's try to

ana yze t ese aces to see t e o enana yze t ese aces to see t e o en

Ratio is present or not. Here's how we areRatio is present or not. Here's how we are

going to conduct our search for the Goldengoing to conduct our search for the Golden

each person's face. Then we will compareeach person's face. Then we will compare

their ratios. Let's begin. We will need thetheir ratios. Let's begin. We will need the

,,

tenth of a centimeter:tenth of a centimeter:

a = Toa = To --ofof--head to chin = cmhead to chin = cm


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b = Topb = Top--ofof--head to pupil = cmhead to pupil = cmc = Pupil to nosetip = cmc = Pupil to nosetip = cm

d = Pupil to lip = cmd = Pupil to lip = cm

e = Width of nose = cme = Width of nose = cm

g = Width of head = cmg = Width of head = cm

= == =i = Nosetip to chin = cmi = Nosetip to chin = cm

j = Lips to chin = cmj = Lips to chin = cm

k = Length of lips = cmk = Length of lips = cml = Nosetip to lips = cml = Nosetip to lips = cm

Now find the following ratios:


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a/g = cma/g = cm==

i/j = cmi/j = cm==

e/l = cme/l = cmf/h = cmf/h = cm

face applet

The blue line defines a perfect square

of the pupils and outside corners of


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.

these four blue lines defines the nose,the tip of the nose, the inside of the

nostrils, the two rises of the upper lip

.

blue line also defines the distance

from the upper lip to the bottom of thechin.

,

the blue line, defines the width of the

nose, the distance between the eyes

and eye brows and the distance from

.The green line, a golden section of

the yellow line defines the width of

the eye, the distance at the pupil from

distance between the nostrils.

The magenta line, a golden section of

the green line, defines the distance

nose and several dimensions


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ven w en v eweven w en v ewefrom the side, thefrom the side, thehuman headhuman headillustrates theillustrates theDivine Proportion.Divine Proportion.The first golden sectionThe first golden section

((blueblue) from the front of the) from the front of thehead defines the position ofhead defines the position of

..successive golden sectionssuccessive golden sectionsdefine the neck (define the neck (yellowyellow), the), theback of the eye (back of the eye (greengreen) and) andthe front of the eye and backthe front of the eye and back

of the nose and mouthof the nose and mouth((magentamagenta).). The dimensionsThe dimensionsof the face from top to bottomof the face from top to bottomalso exhibit the Divinealso exhibit the DivineProportion, in the positions ofProportion, in the positions ofthe eye brow (the eye brow (blueblue), nose), noseye owye ow an mouan mou greengreen

andand magentamagenta).).The ear reflects the shape ofThe ear reflects the shape ofa Fibonacci spiral.a Fibonacci spiral.


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golden rectangle, w ith a phi ratio in thegolden rectangle, w ith a phi ratio in theheighth to the w idth.heighth to the w idth.

The ratio of the w idth of the first toothThe ratio of the w idth of the first toothto the second tooth from the center isto the second tooth from the center is

also phi.also phi.The ratio of the w idth of the smile to theThe ratio of the w idth of the smile to the

well.well.

Visit the site of Dr. Eddy Levin for moreVisit the site of Dr. Eddy Levin for more

on theon the Golden Section and DentistryGolden Section and Dentistry..


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Your hand shows Phi and the Fibonacci Series

your index finger.

The ratio of your forearm to hand is Phi


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The Human Body

The human body is based on Phi and 5


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The human body illustrates the Golden Section. We'll use the same building blocks again:

The Proportions in the BodyThe white line is the body's height.

The blue line, a golden section of the w hite line, defines the

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

The green line, a golden section of the yellow line, defines thedistance from the head to the pectorals and inside top of thearms, the width of the shoulders, the length of the forearmand the shin bone.

The magenta line, a golden section of the green line, definesthe distance from the head to the base of the skull and thew idth of the abdomen. The sectioned portions of the magentaline determine the position of the nose and the hairline.

A t oug not s own, t e go en section o t e magenta ine(also the short section of the green line) defines the w idth ofthe head and half the width of the chest and the hips.

A LIFE WITHOUT FIBONACCI ANDA LIFE WITHOUT FIBONACCI AND

GOLDEN SECTION WOULD BEGOLDEN SECTION WOULD BE


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REALLY UNIMPRESSIVE .BUT ITSREALLY UNIMPRESSIVE .BUT ITSREALLY AMAZING TO SEE THATREALLY AMAZING TO SEE THAT

NATURE WITHOUT ANATURE WITHOUT A

MATHMETICIAN HAS DEVELOPEDMATHMETICIAN HAS DEVELOPEDTHIS GOLDEN SECTION,DIVINETHIS GOLDEN SECTION,DIVINE

PROPORTION.PROPORTION.

AMAZING ISNT.IT?????????AMAZING ISNT.IT?????????


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Fibonacci [Compatibility Mode]