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Fibonacci Numbers and
the Golden Ratio
Say What?
What is the Golden Ratio?
Well, before we answer that question let's
examine an interesting sequence (or list)
of numbers.
Actually the series starts with 0, 1 but to
make it easier we’ll just start with:
1, 1
To get the next number we add the previous two
numbers together. So now our sequence becomes
1, 1, 2. The next number will be 3. What do you
think the next number in the sequence will be?
Remember, we add the previous two numbers to
get the next. So the next number should be 2+3, or
5. Here is what our sequence should look like if we
continue on in this fashion for a while:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…
Now, I know what you might be thinking:
"What does this have to do with the Golden
Ratio?
This sequence of numbers was first
“discovered” by a man named Leonardo
Fibonacci, and hence is known as
Fibonacci's sequence.
Math GEEK
Leonardo Fibonacci
The relationship of this sequence to the
Golden Ratio lies not in the actual numbers
of the sequence, but in the ratio of the
consecutive numbers. Let's look at some of
the ratios of these numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…
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….
2/1 = 2.0
3/2 = 1.5
5/3 = 1.67
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618
89/55 = 1.618
2/1 = 2.0
3/2 = 1.5
5/3 = 1.67
8/5 = 1.6(smaller)
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618
89/55 = 1.618
(bigger)
Aha! Notice that as
we continue down
the sequence, the
ratios seem to be
converging upon one
number (from both
sides of the number)!
Fibonacci Number calculator
(smaller)
(bigger)
(bigger)
(smaller)
(bigger)
(smaller)
5/3 = 1.67
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.618
89/55 = 1.618
Notice that I have rounded my ratios to the third decimal
place. If we examine 55/34 and 89/55 more closely, we will
see that their decimal values are actually not the same. But
what do you think will happen if we continue to look at the
ratios as the numbers in the sequence get larger and
larger? That's right: the ratio will eventually become the
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
irrational number: it has an infinite
number of decimal places and it never
repeats itself! Generally, we round
the Golden Ratio to 1.618.
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..
instead of 1·6.. .
Read this as ordinary text, in lines across, so Phi is 1·61803398874...)
Dps: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 50 28621 35448 62270 52604 62818
90244 97072 07204 18939 11374 100 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635
44333 89086 59593 95829 05638 32266 13199 28290 26788 200 06752 08766 89250 17116 96207 03222 10432
16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300 86444 92410
44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217
05751 79788 400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599
14669 79873 17613 56006 70874 80710 500 13179 52368 94275 21948 43530 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 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 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 1000 10767 38937 64556 06060
59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 21221 24154 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 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 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 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 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 83391 83285 99130 39607
20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940 98864 00764 43617 09334
17270 91914 33650 13715 2000
We work with another important
irrational number in Geometry: pi,
which is approximately 3.14. Since
we don't want to make the Golden
Ratio feel left out, we will give it its
own Greek letter: phi.
Φ φ Phi
which is equal to:
One more interesting thing about Phi is its
reciprocal. If you take the ratio of any number in
the Fibonacci sequence to the next number (this is
the reverse of what we did before), the ratio will
approach the approximation 0.618. This is the
reciprocal of Phi: 1 / 1.618 = 0.618. It is highly
unusual for the decimal integers of a number and
its reciprocal to be exactly the same. In fact, I
cannot name another number that has this
property! This only adds to the mystique of the
Golden Ratio and leads us to ask: What makes it
so special?
The Golden Ratio is not just some number that math
teachers think is cool. The interesting thing is that it keeps
popping up in strange places - places that we may not
ordinarily have thought to look for it. It is important to note
that Fibonacci did not "invent" the Golden Ratio; he just
discovered one instance of where it appeared naturally. In
fact civilizations as far back and as far apart as the Ancient
Egyptians, the Mayans, as well as the Greeks discovered
the Golden Ratio and incorporated it into their own art,
architecture, and designs. They discovered that the Golden
Ratio seems to be Nature's perfect number. For some
reason, it just seems to appeal to our natural instincts. The
most basic example is in rectangular objects.
Look at the following rectangles:
Now ask yourself, which of them seems to be the most
naturally attractive rectangle? If you said the first one, then
you are probably the type of person who likes everything to
be symmetrical. Most people tend to think that the third
rectangle is the most appealing.
If you were to measure each rectangle's length
and width, and compare the ratio of length to width
for each rectangle you would see the following:
Rectangle one: Ratio 1:1
Rectangle two: Ratio 2:1
Rectangle Three: Ratio 1.618:1
Have you figured out why the third rectangle is the
most appealing? That's right - because the ratio of
its length to its width is the Golden Ratio! For
centuries, designers of art and architecture have
recognized the significance of the Golden Ratio in
their work.
Let's see if we can discover where
the Golden Ratio appears in
everyday objects. Use your
measuring tool to compare the
length and the width of rectangular
objects in the classroom or in your
house (depending on where you
are right now). Try to choose
objects that are meant to be
visually appealing.
Object
Length
Width
Ratio
index card
photograph
picture frame
textbook
door frame
computer
screen
TV screen
Were you surprised to find the Golden Ratio in so many
places? It's hard to believe that we have taken it for granted
for so long, isn't it?
How about in music?
Lets take a look at the piano keyboard…do you see
Anything familiar?
Count the number of keys (notes) in each of the brackets…
You will see the numbers 2,3,5,8,13….coincidence?
Does it look like the Fibonacci sequence…it should
because it is!
How about Architecture?
Find the Golden Ratio in 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
pillars to the highest point.
2. Measure the length and the width of this rectangle. Now
find the ratio of the length to the width. Is the number fairly
close to the Golden Ratio?
3. Now look above the pillars. You should notice some
rectangles on the face of the Parthenon. Find the ratio of
the length to the width of one of these rectangles. Notice
anything?
There are many other places where the Golden Ratio
appears in the Parthenon, all of which we cannot see
because we only have a frontal view of the structure. The
building is built on a rectangular plot of land which happens
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 Art
Now let's go back and try to
discover the Golden Ratio in art.
We will concentrate on the works of
Leonardo da Vinci, as he was not
only a great artist but also a genius
when it came to mathematics and
invention.
The Annunciation - Using the left side of the painting as a
side, create a square on the left of the painting by inserting
a vertical line. Notice that you have created a square and a
rectangle. The rectangle turns out to be a Golden
Rectangle, of course. Also, draw in a horizontal line that is
61.8% of the way down the painting (.618 - the inverse of
the Golden Ratio). Draw another line that is 61.8% of the
way up the painting. Try again with vertical lines that are
61.8% of the way across both from left to right and from
right to left. You should now have four lines drawn across
the painting. Notice that these lines intersect important
parts of the painting, such as the angel, the woman, etc.
Coincidence? I think not!
The Mona Lisa - Measure the length and
the width of the painting itself. The ratio is, of
course, Golden. Draw a rectangle around
Mona's face (from the top of the forehead to
the base of the chin, and from left cheek to
right cheek) and notice that this, too, is a
Golden rectangle.
Leonardo da Vinci's talent as an artist may well
have been outweighed by his talents as a
mathematician. He incorporated geometry into
many of his paintings, with the Golden Ratio being
just one of his many mathematical tools. Why do
you think he used it so much? Experts agree that
he probably thought that Golden measurements
made his paintings more attractive. Maybe he was
just a little too obsessed with perfection. However,
he was not the only one to use Golden properties
in his work.
Constructing A Golden Rectangle
Isn't it strange that the Golden Ratio came up in such
unexpected places? Well let's see if we can find out why.
The Greeks were the first to call phi the Golden Ratio. They
associated the number with perfection. It seems to be part
of human nature or instinct for us to find things that contain
the Golden Ratio naturally attractive - such as the "perfect"
rectangle. Realizing this, designers have tried to
incorporate the Golden Ratio into their designs so as to
make them more pleasing to the eye. Doors, notebook
paper, textbooks, etc. all seem more attractive if their sides
have a ratio close to phi. Now, let's see if we can construct
our own "perfect" rectangle.
Method One
1. 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!):
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 degree angle:
Now, draw in one of the diagonals of one of the rectangles
Measure the length of the diagonal and make a note
of it.
Now extend the base of the square from the
midpoint of the base by a distance equal to the
length of the diagonal
Construct a new line perpendicular to the base at the end of
our new line, and then connect to form a 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 Rectangle - Method Two
Now, let's try a different method that will relate the
rectangle to the Fibonacci series we looked at. We'll start
with a square. The size does not matter, as long as all
sides are congruent. We'll use a small square to conserve
space, because we are going to build our golden rectangle
around this square. Please note that the golden area is
what your rectangle will eventually look like.
Now, let's build another, congruent square right next to the
first one:
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
square will have a side of 2 units:
Notice that we have a new rectangle with width 2 and
length 3.
Let's continue the process, building another square on the
right of our rectangle. This square will have a side of 3:
Now we have a rectangle of width 3 and length 5.
Again, let's build upon this
rectangle and construct a square
underneath, with a side of 5:
The new rectangle has a width of 5
and a length of 8. Let's continue to
the left with a square with side 8:
Have you noticed the pattern yet?
The new rectangle has a width of 8
and a length of 13. Let's continue
with one final square on top, with a
side of 13:
sequence (1, 1, 2, 3, 5, 8, 13, ...)! No wonder our rectangle
is golden! Each successive rectangle that we constructed
had a width and length that were consecutive terms in the
Fibonacci sequence. So if we divide the length by the
width, we will arrive at the Golden Ratio! Of course, our
rectangle is not "perfectly" golden. We could keep the
process going until the sides approximated the ratio better,
but for our purposes a length of 21 and a width of 13 are
sufficient.
34
21
Do the Math!! 34 divided by 21 =1.61904761904
Remember that the farther into the sequence we
go the closer the ratio gets to being perfect!
This rectangle should seem very
well proportioned to you, i.e. it
should be pleasing to the eye. If it
isn't, maybe you need your eyes
checked!
Constructing a Golden Spiral
Notice how we built our rectangle in
a counterclockwise direction. This
leads us into another interesting
characteristic of the Golden Ratio.
Let's look at the rectangle with all of
our construction lines drawn in:
We are going to concentrate on the squares that we drew,
starting with the two smallest ones. Let's start with the one
on the right. Connect the upper right corner to the lower left
corner with an arc that is one fourth of a circle
We are going to concentrate on the squares that
we drew, starting with the two smallest ones. Let's
start with the one on the right. Connect the upper
right corner to the lower left corner with an arc that
is one fourth of a circle:
Then continue your line into the second square on
the left, again 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 continuous line. The line should look like a
spiral when we are done. Here is an example of
what your spiral should look like:
Golden Spiral manipulative
Now what was the point of that?
The point is that this "golden spiral"
occurs frequently in nature. If you
look closely enough, you might find
a golden spiral in the head of a
daisy, in a pinecone, in sunflowers,
or in a nautilus shell that you might
find on a beach or even in your ear!
Here are some examples:
So, why do shapes that exhibit the
Golden Ratio seem more appealing
to the human eye? No one really
knows for sure. But we do have
evidence that the Golden Ratio
seems to be Nature's perfect
number.
Somebody with a lot of time on their hands discovered that
the individual florets of the daisy (and of a sunflower as
well) grow in two spirals extending out from the center. The
first spiral has 21 arms, while the other has 34. Do these
numbers sound familiar?
They should - they are Fibonacci
numbers! And their ratio, of course,
is the Golden Ratio.
We can say the same thing about the spirals
of a pinecone, where spirals from the center
have 5 and 8 arms, respectively (or of 8 and
13, depending on the size)- again, two
Fibonacci numbers:
A pineapple has three arms of 5, 8, and 13 even more evidence that this is not a
coincidence. Now is Nature playing some
kind of cruel game with us? No one knows
for sure, but scientists speculate that plants
that grow in spiral formation do so in
Fibonacci numbers because this
arrangement makes for the perfect spacing
for growth. So for some reason, these
numbers provide the perfect arrangement
for maximum growth potential and survival
of the plant.
Do these faces seem attractive to you?
Many people seem to think so. But why? Is
there something specific in each of their
faces that attracts us to them, or is our
attraction governed by one of Nature's
rules? Does this have anything to do with
the Golden Ratio? I think you already know
the answer to that question. Let's try to
analyze these faces to see if the Golden
Ratio is present or not. Here's how we are
going to conduct our search for the Golden
Ratio: we will measure certain aspects of
each person's face. Then we will compare
their ratios. Let's begin. We will need the
following measurements, to the nearest
tenth of a centimeter:
a = Top-of-head to chin = cm
b = Top-of-head to pupil = cm
c = Pupil to nosetip = cm
d = Pupil to lip = cm
e = Width of nose = cm
f = Outside distance between eyes = cm
g = Width of head = cm
h = Hairline to pupil = cm
i = Nosetip to chin = cm
j = Lips to chin = cm
k = Length of lips = cm
l = Nosetip to lips = cm
Now find the following ratios:
a/g = cm
b/d = cm
i/j = cm
i/c = cm
e/l = cm
f/h = cm
k/e = cm
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
from the side, the
human head
illustrates the
Divine Proportion.
The first golden section
(blue) from the front of the
head defines the position of
the ear opening. The
successive golden sections
define the neck (yellow), the
back of the eye (green) and
the front of the eye and back
of the nose and mouth
(magenta). The dimensions
of the face from top to bottom
also exhibit the Divine
Proportion, in the positions of
the eye brow (blue), nose
(yellow) and mouth (green
and magenta).
The ear reflects the shape of
a Fibonacci spiral.
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.
Visit the site of Dr. Eddy Levin for more
on the Golden 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 5
The human body illustrates the Golden Section. We'll use the same building blocks again:
The Proportions in the 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.