Fibonacci Numbers and Fibonacci Numbers and the the GoldenGolden Ratio Ratio

Say What?Say What?

What is the What is the Golden RatioGolden Ratio??

Well, before we answer that question let's Well, before we answer that question let's examine an interesting sequence (or list) examine an interesting sequence (or list)

of numbers. of numbers.

Actually the series starts with 0, 1 but to Actually the series starts with 0, 1 but to make it easier we’ll just start with:make it easier we’ll just start with:

1, 1 1, 1

To get the next number we add the previous two To get the next number we add the previous two numbers together. So now our sequence becomes numbers together. So now our sequence becomes

1, 1, 2. The next number will be 3. What do you 1, 1, 2. The next number will be 3. What do you think the next number in the sequence will be? think the next number in the sequence will be?

Remember, we add the previous two numbers to Remember, we add the previous two numbers to get the next. So the next number should be 2+3, or get the next. So the next number should be 2+3, or 5. Here is what our sequence should look like if we 5. Here is what our sequence should look like if we

continue on in this fashion for a while:continue on in this fashion for a while:

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

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 GoldenGolden

Ratio?Ratio? This sequence of numbers was first This sequence of numbers was first

“discovered” by a man named Leonardo “discovered” by a man named Leonardo Fibonacci, and hence is known as Fibonacci, and hence is known as

Fibonacci's sequence. Fibonacci's sequence.

Math GEEKReally FamousReally Smart

Leonardo FibonacciLeonardo Fibonacci

The relationship of this sequence to the The relationship of this sequence to the GoldenGolden Ratio lies not in the actual numbers Ratio lies not in the actual numbers

of the sequence, but in the ratio of the of the sequence, but in the ratio of the consecutive numbers. Let's look at some of consecutive 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, 610…1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…

2/1 = 2.02/1 = 2.03/2 = 1.53/2 = 1.5

5/3 = 1.675/3 = 1.678/5 = 1.68/5 = 1.6

13/8 = 1.62513/8 = 1.62521/13 = 1.61521/13 = 1.61534/21 = 1.61934/21 = 1.61955/34 = 1.61855/34 = 1.61889/55 = 1.61889/55 = 1.618

Since a Ratio is basically a fraction (or a division problem) we will find the ratios of these numbers by dividing the larger number by the smaller number that fall consecutively in the series.

So, what is the ratio of the 2nd and 3rd numbers?

Well, 2 is the 3rd number divided by the 2nd number which is 1

2 divided by 1 = 2

And the ratios continue like this….

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

converging upon one converging upon one number (from both number (from both

sides of the number)!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

Fibonacci Number calculator

5/3 = 1.675/3 = 1.678/5 = 1.68/5 = 1.6

13/8 = 1.62513/8 = 1.62521/13 = 1.61521/13 = 1.61534/21 = 1.61934/21 = 1.61955/34 = 1.61855/34 = 1.618

89/55 = 1.61889/55 = 1.618 Notice that I have rounded my ratios to the third decimal Notice that I have rounded my ratios to the third decimal

place. If we examine 55/34 and 89/55 more closely, we will place. If we examine 55/34 and 89/55 more closely, we will see that their decimal values are actually not the same. But see that their decimal values are actually not the same. But what do you think will happen if we continue to look at the what do you think will happen if we continue to look at the

ratios as the numbers in the sequence get larger and ratios as the numbers in the sequence get larger and larger? That's right: the ratio will eventually become the larger? That's right: the ratio will eventually become the

same number, and that number is the Golden Ratio! same number, and that number is the Golden 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

The Golden Ratio is what we call an The Golden Ratio is what we call an irrational number: it has an infinite irrational number: it has an infinite

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

repeats itself! repeats itself! Generally, we round Generally, we round the Golden Ratio to 1.618.the Golden Ratio to 1.618.

Here is the decimal value of Phi to 2000 places grouped in blocks of 5 Here is the decimal value of Phi to 2000 places grouped in blocks of 5 decimal digits. The value of phi is the same but begins with 0·6.. decimal digits. The value of phi is the same but begins with 0·6..

instead of 1·6.. . instead of 1·6.. . Read this as ordinary text, in lines across, so Phi is 1·61803398874...)Read this as ordinary text, in lines across, so Phi is 1·61803398874...)

Dps:Dps: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 5050 28621 35448 62270 52604 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 62818 90244 97072 07204 18939 11374 100100 84754 08807 53868 91752 12663 38622 23536 93179 31800 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 200200 06752 08766 89250 17116 06752 08766 89250 17116

96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300300 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 98871 24007 65217 05751 79788 400400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 500500 13179 52368 94275 21948 43530 13179 52368 94275 21948 43530

56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 10001000

10767 38937 64556 06060 59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 10767 38937 64556 06060 59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 2122121221 24154 24154 44006 47034 05657 34797 66397 23949 49946 58457 88730 39623 09037 50339 93856 21024 23690 25138 44006 47034 05657 34797 66397 23949 49946 58457 88730 39623 09037 50339 93856 21024 23690 25138 68041 45779 95698 12244 57471 78034 17312 64532 20416 39723 21340 44449 48730 23154 17676 89375 68041 45779 95698 12244 57471 78034 17312 64532 20416 39723 21340 44449 48730 23154 17676 89375 21030 68737 88034 41700 93954 40962 79558 98678 72320 95124 26893 55730 97045 09595 68440 17555 21030 68737 88034 41700 93954 40962 79558 98678 72320 95124 26893 55730 97045 09595 68440 17555 19881 92180 20640 52905 51893 49475 92600 73485 22821 01088 19464 45442 22318 89131 92946 89622 19881 92180 20640 52905 51893 49475 92600 73485 22821 01088 19464 45442 22318 89131 92946 89622 00230 14437 70269 92300 78030 85261 18075 45192 88770 50210 96842 49362 71359 25187 60777 88466 00230 14437 70269 92300 78030 85261 18075 45192 88770 50210 96842 49362 71359 25187 60777 88466 58361 50238 91349 33331 22310 53392 32136 24319 26372 89106 70503 39928 22652 63556 20902 97986 58361 50238 91349 33331 22310 53392 32136 24319 26372 89106 70503 39928 22652 63556 20902 97986 42472 75977 25655 08615 48754 35748 26471 81414 51270 00602 38901 62077 73224 49943 53088 99909 42472 75977 25655 08615 48754 35748 26471 81414 51270 00602 38901 62077 73224 49943 53088 99909 50168 03281 12194 32048 19643 87675 86331 47985 71911 39781 53978 07476 15077 22117 50826 94586 50168 03281 12194 32048 19643 87675 86331 47985 71911 39781 53978 07476 15077 22117 50826 94586 39320 45652 09896 98555 67814 10696 83728 84058 74610 33781 05444 39094 36835 83581 38113 11689 39320 45652 09896 98555 67814 10696 83728 84058 74610 33781 05444 39094 36835 83581 38113 11689 93855 57697 54841 49144 53415 09129 54070 05019 47754 86163 07542 26417 29394 68036 73198 05861 93855 57697 54841 49144 53415 09129 54070 05019 47754 86163 07542 26417 29394 68036 73198 05861 83391 83285 99130 39607 20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940 83391 83285 99130 39607 20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940

98864 00764 43617 09334 17270 91914 33650 13715 98864 00764 43617 09334 17270 91914 33650 13715 20002000

We work with another important We work with another important irrational number in Geometry: pi, irrational number in Geometry: pi, which is approximately 3.14. Since which is approximately 3.14. Since we don't want to make the Golden we don't want to make the Golden Ratio feel left out, we will give it its Ratio feel left out, we will give it its

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

Φ φ Phi

which is equal to:                                       

One more interesting thing about Phi is its One more interesting thing about Phi is its reciprocal. If you take the ratio of any number in reciprocal. If you take the ratio of any number in

the Fibonacci sequence to the next number (this is the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will the reverse of what we did before), the ratio will approach the approximation 0.618. This is the approach the approximation 0.618. This is the reciprocal of Phi: 1 / 1.618 = 0.618. It is highly reciprocal of Phi: 1 / 1.618 = 0.618. It is highly

unusual for the decimal integers of a number and unusual for the decimal integers of a number and its reciprocal to be exactly the same. In fact, I its reciprocal to be exactly the same. In fact, I

cannot name another number that has this cannot name another number that has this property! This only adds to the mystique of the property! This only adds to the mystique of the

Golden Ratio and leads us to ask: What makes it Golden Ratio and leads us to ask: What makes it so special?so special?

The Golden Ratio is not just some number that math The Golden Ratio is not just some number that math teachers think is cool. The interesting thing is that it keeps teachers think is cool. The interesting thing is that it keeps

popping up in strange places - places that we may not popping up in strange places - places that we may not ordinarily have thought to look for it. It is important to note ordinarily have thought to look for it. It is important to note that Fibonacci did not "invent" the Golden Ratio; he just that Fibonacci did not "invent" the Golden Ratio; he just

discovered one instance of where it appeared naturally. In discovered one instance of where it appeared naturally. In fact civilizations as far back and as far apart as the Ancient fact civilizations as far back and as far apart as the Ancient Egyptians, the Mayans, as well as the Greeks discovered Egyptians, 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, architecture, and designs. They discovered that the Golden architecture, and designs. They discovered that the Golden

Ratio seems to be Nature's perfect number. For some Ratio seems to be Nature's perfect number. For some reason, it just seems to appeal to our natural instincts. The reason, it just seems to appeal to our natural instincts. The

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

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

    

Now ask yourself, which of them seems to be the most Now ask yourself, which of them seems to be the most naturally attractive rectangle? If you said the first one, then naturally attractive rectangle? If you said the first one, then you are probably the type of person who likes everything to you are probably the type of person who likes everything to

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

 

If you were to measure each rectangle's length If you were to measure each rectangle's length and width, and compare the ratio of length to width and 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 1:1Rectangle two: Ratio 2:1Rectangle two: Ratio 2:1

Rectangle Three: Ratio 1.618:1Rectangle Three: Ratio 1.618:1

 

Have you figured out why the third rectangle is the Have you figured out why the third rectangle is the most appealing? That's right - because the ratio of most appealing? That's right - because the ratio of

its length to its width is the Golden Ratio! For its length to its width is the Golden Ratio! For centuries, designers of art and architecture have centuries, designers of art and architecture have

recognized the significance of the Golden Ratio in recognized the significance of the Golden Ratio in

their work.their work.

 

Let's see if we can discover where Let's see if we can discover where the Golden Ratio appears in the Golden Ratio appears in everyday objects. Use your everyday objects. Use your

measuring tool to compare the measuring tool to compare the length and the width of rectangular length and the width of rectangular objects in the classroom or in your objects in the classroom or in your house (depending on where you house (depending on where you

are right now). Try to choose are right now). Try to choose objects that are meant to be objects that are meant to be

visually appealing. visually appealing.

Were you surprised to find the Golden Ratio in so many Were you surprised to find the Golden Ratio in so many places? It's hard to believe that we have taken it for granted places? It's hard to believe that we have taken it for granted

for so long, isn't it?for so long, isn't it?

ObjectObject LengthLength WidthWidth RatioRatio

index cardindex card

photographphotograph

picture framepicture frame

textbooktextbook

door framedoor frame

computer computer screenscreen

TV screenTV screen

How about in music?

Lets take a look at the piano keyboard…do you seeAnything familiar?

Count the number of keys (notes) in each of the brackets…Count the number of keys (notes) in each of the brackets…You 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 sequence…it should Does it look like the Fibonacci sequence…it should because it is!because it is!

How about Architecture?

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, from the left most pillar to the right and from the base of the from the left most pillar to the right and from the base of the

pillars to the highest point.pillars to the highest point.2. Measure the length and the width of this rectangle. Now 2. Measure the length and the width of this rectangle. Now find the ratio of the length to the width. Is the number fairly find the ratio of the length to the width. Is the number fairly

close to the Golden Ratio?close to the Golden Ratio?3. Now look above the pillars. You should notice some 3. Now look above the pillars. You should notice some

rectangles on the face of the Parthenon. Find the ratio of rectangles on the face of the Parthenon. Find the ratio of the length to the width of one of these rectangles. Notice the length to the width of one of these rectangles. Notice

anything?anything?There are many other places where the Golden Ratio There are many other places where the Golden Ratio appears in the Parthenon, all of which we cannot see appears in the Parthenon, all of which we cannot see

because we only have a frontal view of the structure. The because we only have a frontal view of the structure. The building is built on a rectangular plot of land which happens building is built on a rectangular plot of land which happens

to be ... you guessed it - a Golden Rectangle!to be ... you guessed it - a Golden Rectangle!

Once its ruined triangular pediment is restored, ...

the ancient temple fits almost precisely into a golden rectangle.

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

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

The Golden Ratio in ArtThe Golden Ratio in ArtNow let's go back and try to Now let's go back and try to

discover the Golden Ratio in art. discover the Golden Ratio in art. We will concentrate on the works of We will concentrate on the works of Leonardo da Vinci, as he was not Leonardo da Vinci, as he was not

only a great artist but also a genius only a great artist but also a genius when it came to mathematics and when it came to mathematics and

invention. invention.

The Annunciation - The Annunciation - Using the left side of the painting as a Using the left side of the painting as a side, create a square on the left of the painting by inserting side, create a square on the left of the painting by inserting a vertical line. Notice that you have created a square and a a vertical line. Notice that you have created a square and a

rectangle. The rectangle turns out to be a Golden rectangle. The rectangle turns out to be a Golden Rectangle, of course. Also, draw in a horizontal line that is Rectangle, of course. Also, draw in a horizontal line that is 61.8% of the way down the painting (.618 - the inverse of 61.8% of the way down the painting (.618 - the inverse of the Golden Ratio). Draw another line that is 61.8% of the the Golden Ratio). Draw another line that is 61.8% of the way up the painting. Try again with vertical lines that are way up the painting. Try again with vertical lines that are 61.8% of the way across both from left to right and from 61.8% of the way across both from left to right and from

right to left. You should now have four lines drawn across right to left. You should now have four lines drawn across the painting. Notice that these lines intersect important the painting. Notice that these lines intersect important

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!

The Mona Lisa - The Mona Lisa - Measure the length and Measure the length and the width of the painting itself. The ratio is, of the width of the painting itself. The ratio is, of

course, Golden. Draw a rectangle around course, Golden. Draw a rectangle around Mona's face (from the top of the forehead to Mona's face (from the top of the forehead to the base of the chin, and from left cheek to the base of the chin, and from left cheek to right cheek) and notice that this, too, is a right cheek) and notice that this, too, is a

Golden rectangle.Golden rectangle.

Leonardo da Vinci's talent as an artist may well Leonardo da Vinci's talent as an artist may well have been outweighed by his talents as a have been outweighed by his talents as a

mathematician. He incorporated geometry into mathematician. He incorporated geometry into many of his paintings, with the Golden Ratio being many of his paintings, with the Golden Ratio being just one of his many mathematical tools. Why do just one of his many mathematical tools. Why do you think he used it so much? Experts agree that you think he used it so much? Experts agree that he probably thought that Golden measurements he probably thought that Golden measurements

made his paintings more attractive. Maybe he was made his paintings more attractive. Maybe he was just a little too obsessed with perfection. However, just a little too obsessed with perfection. However, he was not the only one to use Golden properties he was not the only one to use Golden properties

in his work.in his work.

Constructing A Golden Rectangle Constructing A Golden Rectangle

Isn't it strange that the Golden Ratio came up in such Isn'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. They The Greeks were the first to call phi the Golden Ratio. They associated the number with perfection. It seems to be part associated the number with perfection. It seems to be part of human nature or instinct for us to find things that contain of human nature or instinct for us to find things that contain the Golden Ratio naturally attractive - such as the "perfect" the Golden Ratio naturally attractive - such as the "perfect"

rectangle. Realizing this, designers have tried to rectangle. Realizing this, designers have tried to incorporate the Golden Ratio into their designs so as to incorporate the Golden Ratio into their designs so as to make them more pleasing to the eye. Doors, notebook make them more pleasing to the eye. Doors, notebook

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

our own "perfect" rectangle. our own "perfect" rectangle.

Method OneMethod One1. We'll start by making a square, any square (just remember 1. We'll start by making a square, any square (just remember

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

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

degree angle:degree angle:

Measure the length of the diagonal and Measure the length of the diagonal and make a note of it. make a note of it.

Now, draw in one of the diagonals of one of the rectangles

Now extend the base of the square from Now extend the base of the square from the midpoint of the base by a distance the midpoint of the base by a distance

equal to the length of the diagonalequal to the length of the diagonal

Construct a new line perpendicular to the base Construct a new line perpendicular to the base at the end of our new line, and then connect to at the end of our new line, and then connect to

form a rectangle:form a rectangle:Measure the length and the width of your Measure the length and the width of your

rectangle.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 Rectangle - Method TwoConstructing a Golden Rectangle - Method TwoNow, let's try a different method that will relate the Now, let's try a different method that will relate the

rectangle to the Fibonacci series we looked at. We'll start rectangle to the Fibonacci series we looked at. We'll start with a square. The size does not matter, as long as all with a square. The size does not matter, as long as all

sides are congruent. We'll use a small square to conserve sides are congruent. We'll use a small square to conserve space, because we are going to build our golden rectangle space, because we are going to build our golden rectangle

around this square. Please note that the golden area is around this square. Please note that the golden area is what your rectangle will eventually look like.what your rectangle will eventually look like.

Now, let's build another, congruent square right next to the Now, let's build another, congruent square right next to the first one:first one:

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 new Let'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 and Notice that we have a new rectangle with width 2 and

length 3.length 3.

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

Again, let's build upon this Again, let's build upon this rectangle and construct a square rectangle and construct a square

underneath, with a side of 5:underneath, with a side of 5:

The new rectangle has a width of 5 The new rectangle has a width of 5 and a length of 8. Let's continue to and a length of 8. Let's continue to the left with a square with side 8:the left with a square with side 8:

Have you noticed the pattern yet? Have you noticed the pattern yet? The new rectangle has a width of 8 The new rectangle has a width of 8 and a length of 13. Let's continue and a length of 13. Let's continue

with one final square on top, with a with one final square on top, with a side of 13:side of 13:

Our final rectangle has a width of 13 and a length of 21. Our final rectangle has a width of 13 and a length of 21. Notice that we have constructed our golden rectangle using Notice that we have constructed our golden rectangle using square that had successive side lengths from the Fibonacci square that had successive side lengths from the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...)! No wonder our rectangle sequence (1, 1, 2, 3, 5, 8, 13, ...)! No wonder our rectangle is golden! Each successive rectangle that we constructed is golden! Each successive rectangle that we constructed had a width and length that were consecutive terms in the had a width and length that were consecutive terms in the

Fibonacci sequence. So if we divide the length by the Fibonacci sequence. So if we divide the length by the width, we will arrive at the Golden Ratio! Of course, our width, we will arrive at the Golden Ratio! Of course, our rectangle is not "perfectly" golden. We could keep the rectangle 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 are but for our purposes a length of 21 and a width of 13 are

sufficient. sufficient.

3421

Do the Math!! 34 divided by 21 =1.61904761904Remember that the farther into the sequence we go the closer the ratio gets to being perfect!

This rectangle should seem very This rectangle should seem very well proportioned to you, i.e. it well proportioned to you, i.e. it

should be pleasing to the eye. If it should be pleasing to the eye. If it isn't, maybe you need your eyes isn't, maybe you need your eyes

checked! checked!

Constructing a Golden SpiralConstructing a Golden Spiral   

Notice how we built our rectangle in Notice how we built our rectangle in a counterclockwise direction. This a counterclockwise direction. This leads us into another interesting leads us into another interesting

characteristic of the Golden Ratio. characteristic of the Golden Ratio. Let's look at the rectangle with all of Let's look at the rectangle with all of

our construction lines drawn in:our construction lines drawn in:

We are going to concentrate on the squares that we drew, We are going to concentrate on the squares that we drew, starting with the two smallest ones. Let's start with the one starting with the two smallest ones. Let's start with the one on the right. Connect the upper right corner to the lower left on the right. Connect the upper right corner to the lower left

corner with an arc that is one fourth of a circlecorner with an arc that is one fourth of a circle

We are going to concentrate on the squares that We are going to concentrate on the squares that we drew, starting with the two smallest ones. Let's we drew, starting with the two smallest ones. Let's start with the one on the right. Connect the upper start with the one on the right. Connect the upper

right corner to the lower left corner with an arc that right corner to the lower left corner with an arc that is one fourth of a circle:is one fourth of a circle:

Then continue your line into the second square on Then continue your line into the second square on the left, again with an arc that is one fourth of a the left, again with an arc that is one fourth of a

circle:circle:We will continue this process until each square We will continue this process until each square

has an arc inside of it, with all of them connected has an arc inside of it, with all of them connected as a continuous line. The line should look like a as a continuous line. The line should look like a spiral when we are done. Here is an example of spiral when we are done. Here is an example of

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

Golden Spiral manipulative

Now what was the point of that? Now what was the point of that? The point is that this "golden spiral" The point is that this "golden spiral" occurs frequently in nature. If you occurs frequently in nature. If you

look closely enough, you might find look closely enough, you might find a golden spiral in the head of a a golden spiral in the head of a

daisy, in a pinecone, in sunflowers, daisy, in a pinecone, in sunflowers, or in a nautilus shell that you might or 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 examples:Here are some examples:

So, why do shapes that exhibit the So, why do shapes that exhibit the Golden Ratio seem more appealing Golden Ratio seem more appealing

to the human eye? No one really to the human eye? No one really knows for sure. But we do have knows for sure. But we do have evidence that the Golden Ratio evidence that the Golden Ratio seems to be Nature's perfect seems to be Nature's perfect

number. number.

Somebody with a lot of time on their hands discovered that Somebody with a lot of time on their hands discovered that the individual florets of the daisy (and of a sunflower as the individual florets of the daisy (and of a sunflower as

well) grow in two spirals extending out from the center. The well) grow in two spirals extending out from the center. The first spiral has 21 arms, while the other has 34. Do these first spiral has 21 arms, while the other has 34. Do these

numbers sound familiar?numbers sound familiar?

They should - they are Fibonacci They should - they are Fibonacci numbers! And their ratio, of course, numbers! And their ratio, of course,

is the Golden Ratio. is the Golden Ratio.

We can say the same thing about the spirals We can say the same thing about the spirals of a pinecone, where spirals from the center of a pinecone, where spirals from the center have 5 and 8 arms, respectively (or of 8 and have 5 and 8 arms, respectively (or of 8 and

13, depending on the size)- again, two 13, depending on the size)- again, two Fibonacci numbers:Fibonacci numbers:

A pineapple has three arms of 5, 8, and 13 - A pineapple has three arms of 5, 8, and 13 - even more evidence that this is not a even more evidence that this is not a

coincidence. Now is Nature playing some coincidence. Now is Nature playing some kind of cruel game with us? No one knows kind of cruel game with us? No one knows for sure, but scientists speculate that plants for sure, but scientists speculate that plants

that grow in spiral formation do so in that grow in spiral formation do so in Fibonacci numbers because this Fibonacci numbers because this

arrangement makes for the perfect spacing arrangement makes for the perfect spacing for growth. So for some reason, these for growth. So for some reason, these

numbers provide the perfect arrangement numbers provide the perfect arrangement for maximum growth potential and survival for maximum growth potential and survival

of the plant.of the plant.

Do these faces seem attractive to you? Do these faces seem attractive to you? Many people seem to think so. But why? Is Many people seem to think so. But why? Is

there something specific in each of their there something specific in each of their faces that attracts us to them, or is our faces that attracts us to them, or is our attraction governed by one of Nature's attraction governed by one of Nature's

rules? Does this have anything to do with rules? Does this have anything to do with the Golden Ratio? I think you already know the Golden Ratio? I think you already know

the answer to that question. Let's try to the answer to that question. Let's try to analyze these faces to see if the Golden analyze these faces to see if the Golden

Ratio is present or not. Here's how we are Ratio is present or not. Here's how we are going to conduct our search for the Golden going to conduct our search for the Golden Ratio: we will measure certain aspects of Ratio: we will measure certain aspects of each person's face. Then we will compare each person's face. Then we will compare their ratios. Let's begin. We will need the their ratios. Let's begin. We will need the following measurements, to the nearest following measurements, to the nearest

tenth of a centimeter:tenth of a centimeter:

a = Top-of-head to chin = cma = Top-of-head to chin = cmb = Top-of-head to pupil = cmb = Top-of-head to pupil = cm

c = Pupil to nosetip = cmc = Pupil to nosetip = cmd = Pupil to lip = cmd = Pupil to lip = cm

e = Width of nose = cme = Width of nose = cmf = Outside distance between eyes = cmf = Outside distance between eyes = cm

g = Width of head = cmg = Width of head = cmh = Hairline to pupil = cmh = Hairline to pupil = cmi = Nosetip to chin = cmi = Nosetip to chin = cm

j = Lips to chin = cmj = Lips to chin = cmk = Length of lips = cmk = Length of lips = cml = Nosetip to lips = cml = Nosetip to lips = cm

  

a/g = cma/g = cmb/d = cmb/d = cmi/j = cmi/j = cmi/c = cmi/c = cme/l = cme/l = cmf/h = cmf/h = cmk/e = cmk/e = cm

Now find the following ratios:

face applet

The blue line defines a perfect square of the pupils and outside corners of

the mouth. The golden section of these four blue lines defines the nose,

the tip of the nose, the inside of the nostrils, the two rises of the upper lip

and the inner points of the ear. The blue line also defines the distance

from the upper lip to the bottom of the chin.

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

the pupils to the tip of the nose.The green line, a golden section of the

yellow line defines the width of the eye, the distance at the pupil from the

eye lash to the eye brow and the distance between the nostrils.

The magenta line, a golden section of the green line, defines the distance

from the upper lip to the bottom of the nose and several dimensions

Even when viewed Even when viewed from the side, the from the side, the human head human head illustrates the illustrates the Divine Proportion. Divine Proportion. The first golden section The first golden section ((blueblue) from the front of the ) from the front of the head defines the position of head defines the position of the ear opening. The the ear opening. The successive golden sections successive golden sections define the neck (define the neck (yellowyellow), the ), the back of the eye (back of the eye (greengreen) and ) and the front of the eye and back the front of the eye and back of the nose and mouth of the nose and mouth ((magentamagenta).  The dimensions ).  The dimensions of the face from top to of the face from top to bottom also exhibit the bottom also exhibit the Divine Proportion, in the Divine Proportion, in the positions of the eye brow positions of the eye brow ((blueblue), nose (), nose (yellowyellow) and ) and

mouth (mouth (greengreen and and magentamagenta).). The ear reflects the shape of The ear reflects the shape of a Fibonacci spiral.a Fibonacci spiral.

The front two incisor teeth form a The front two incisor teeth form a golden rectangle, with a phi ratio in the golden rectangle, with a phi ratio in the heighth to the width. heighth to the width. The ratio of the width of the first tooth The ratio of the width of the first tooth to the second tooth from the center is to the second tooth from the center is also phi.also phi.The ratio of the width of the smile to The ratio of the width of the smile to the third tooth from the center is phi as the third tooth from the center is phi as well.well.Visit the site of Dr. Eddy Levin for more Visit the site of Dr. Eddy Levin for more on the on the Golden Section and DentistryGolden Section and Dentistry..

Your hand shows Phi and the Fibonacci Series

your index finger.

 

The ratio of your forearm to hand is Phi

The Human Body

The human body is based on Phi and 5The 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 white line, defines the distance from the head to the finger tipsThe 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.