I have always had a fascination with numbers. In spite of the fact that I
generally consider myself to be “right-brained,” that is to say, more
linguistically skilled than the otherwise “left-brained” mathematically
skilled, the inherent order of numbers has always held an attraction for me. Possibly,
it is the organization that draws me; possibly it is the objectivity of the
discipline. Whatever the reason, I have always been someone who finds—comfort—in the patterns of mathematics.
Like a drummer who hears rhythm in
everything, for reasons that defy my explanation (and trend towards
compulsivity), I have always added numbers together. I saw patterns where others simply saw
numbers, and I put value to the sums that I came up with. For example, I might
see a sequence of numbers on a license plate that read: 453-8767. I would then
quickly add those numbers… 4+5 = 9+3 = 12, 1+2 = 3+8 = 11, 1+1 = 2+7 = 9+6 =
15, 1+5 = 6+7 = 13, 1+3 = 4… so that the numerical value, as far as my mind
made it, was 4, and that number would become special in that it would from that
point on symbolically represent that license plate or car.
It meant nothing, as far as I could
tell. It was just a mental exercise to me that seemed to keep my mind
relatively sharp, which I believe is one of the underlying purposes of mathematics.
Throughout my education, I have discovered that during the times when I studied
mathematics, for example, when I was studying computer science, my mind seemed
to me to be much sharper (at least, I suppose, from a left-brained perspective)
than it was when I was involved in purely linguistic pursuits, such as
literature or writing. Of course,
mathematics plays a fundamental role in music, and music is, at least in part,
comprised of lyrics, which are of course at their heart poetry, which is for
the most part bursting with meter, which is nothing but repetitive counting or,
mathematics.
In spite of my strange mathematical
fascination, I somehow made it through most of my adult life without ever
hearing of Fibonacci or the Golden mean or the value “phi.” Ironically, when I
left the computer science industry to return to teaching, and I became an
English teacher, I discovered through the help of a very right-brained
colleague Leonardo “Fibonacci” Pisano.
Leonardo “Fibonacci” Pisano
Leonardo “Fibonacci” Pisano was
born in Italy in 1170, and educated in North Africa, where he is said to have
learned various mathematical systems. He
is considered to be one of the most talented mathematicians of the middle Ages.
It was he who gave us our decimal number system, which replaced the Roman numeral
system. When he was studying mathematics, he used the Hindu-Arabic (0-9)
symbols instead of Roman symbols that didn't have 0's and lacked place value.
In fact, when using the Roman numeral system, an abacus was usually required.[1]
Fibonacci demonstrates how to use
our current numbering system in his book, Liber Abaci. It was in this book that Fibonacci introduces
us to the following problem:
“A certain man put a pair of rabbits in a
place surrounded on all sides by a wall. How many pairs of rabbits can be
produced from that pair in a year if it is supposed that every month each pair
begets a new pair, which from the second month on becomes productive?”[2]
He goes on to state the solution,
which becomes the infamous “Fibonacci Sequence:”
“Because the above written pair in the first
month bore, you will double it; there will be two pairs in one month. One
of these, namely the first, bears in the second month, and thus there are in
the second month 3 pairs; of these in one month two are pregnant and in the
third month 2 pairs of rabbits are born, and thus there are 5 pairs in the
month; there will be 144 pairs in this [the tenth] month;
to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month. To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above-written pair in the mentioned place at the end of the one year.”[3]
to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month. To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above-written pair in the mentioned place at the end of the one year.”[3]
Looking at it in words, the
sequence is difficult to grasp. If we
write it out in numbers, however, the pattern becomes clear: 1 2 3 5 8 13 21 34
55 89 144 233 377
The numbers are the sum of the
previous two numbers; thus, 1+2=3,
3+2=5, 5+3=8, 8+5=13, 13+8=21, 21+13=34, 34+21=55, 55+34=89, 89+55=144, 144+89=233,
233+144=377.
The Golden Ratio
Throughout history, the ratio for
length to width of rectangles of 1.61803 39887 49894 84820 has been considered
the most pleasing to the eye. This ratio was named the golden ratio by the
Greeks. In the world of mathematics, the numeric value is called “phi” (φ).[4]
If you divide a line into two parts
so that the longer part divided by the smaller part is also equal to the whole
length divided by the longer part:
In example, if a Fibonacci number
is divided by its immediate predecessor in the sequence, the quotient
approximates “φ.”
For example:
233/144=1.61805555555556; 377/233=1.61802575107296.
We can make another picture showing
the Fibonacci numbers 1,2,3,5,8,13,21, if we start with two small squares of
size 1 next to each other. On top of both of these draw a square of size 2
(=1+1). We can now draw a new square,
touching both a unit square and the latest square of side 2, so having sides 3
units long; and then another touching both the 2-square and the 3-square (which
has sides of 5 units).
We can continue adding squares
around the picture, each new square
having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two
successive Fibonacci numbers in length and which are composed of squares with
sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.
Fibonacci in Architecture
A perfect example of this is found
in the Parthenon in Athens, which was built in 440bc. It is a perfect Golden Rectangle:
Not to be left out, here is Notre
Dame in Paris, France, designed in 1163 and 1250 using the Golden Rectangle,
also known as the Golden Section:
During the time of the Renaissance, Leonardo Da Vinci knew these
equations as the Divine Proportion. Around the same time period in
India, it was used in the construction of the Taj Mahal, which was completed in
1648:
We can also see modern examples,
such as found in the United Nations building in New York, and the CN Tower in
Toronto:
Fibonacci in Nature
The next step for the above
equations would be to impose a spiral over the squares. Here is a spiral design
found quite frequently in nature. While
this spiral is not in itself a mathematical spiral, since it is made up of
quarter circles in each square and so does not get smaller and smaller but in
fact, it gets larger.
Fibonacci is well represented in
many forms found in nature, such as shells, seed patterns, and even galaxies. On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris
have 3 petals; some delphiniums have 8; corn marigolds have 13 petals;
some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals. Pinecones show the Fibonacci spirals clearly,
as well as many plants in the arrangements of the leaves around their stems,
and the patterning of the stems themselves:
The eyes, beak, wing and
key All the
key facial features
body markings of the penguin of the tiger fall
at golden sections
all fall at golden sections
of its height. of the
lines defining the length and
width of its face.[12]
Every key body feature of the angelfish falls at golden sections
of its width and length:
The nose, tail section, and centers of the fins of the angelfish
fall at first (blue) golden sections. The second golden section (yellow)
defines the indents on the dorsal and tail finds as well as the top of the
body. The green section defines the marking around the eye and the magenta
section defines the eye.
Fibonacci and the
Human Form
A look at the human hand structure reveals that it has:
- 2 hands each of which has—
- 5 fingers, each of which
has—
- 3 parts separated by—
- 2 knuckles
Further, if you measure the lengths of the bones in your
finger, the ratio of the longest bone in a finger to the middle bone is φ. The ratio of the
middle bone to the shortest bone is φ.[14]
It has long been said that beauty
is in the eye of the beholder and thought that beauty varies by race, culture
or era. The evidence, however, shows that
our perception of physical beauty is hard wired into our being and based on how
closely one's features reflect phi in their proportions. Take another look at beauty through the eyes
of medical science.
Dr. Stephen Marquardt has studied
human beauty for years in his practice of oral and maxillofacial surgery. Dr. Marquardt performed cross-cultural
surveys on beauty and found that all groups had the same perceptions of facial
beauty. He also analyzed the human face
from ancient times to the modern day.
Through his research, he discovered that beauty is not only related to
phi, but can be defined for both genders and for all races, cultures and eras
with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as
its foundation, which embody phi in all their dimensions:
The human head forms a golden rectangle with the eyes at its
midpoint. The mouth and nose are each placed at golden sections of the
distance between the eyes and the bottom of the chin. The beauty unfolds
as you look further:
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 eyebrows 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
eyelash to the eyebrow 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 of the eye.
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 eyebrows 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
eyelash to the eyebrow 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 of the eye.
The human ear perfectly reflects the
shape of a Fibonacci spiral:
Our teeth are based on phi as well:
The front two incisor teeth form a
golden rectangle, with a phi ratio in the height 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. [16]
There is divine proportion in the
body overall:
The white line (hard to see here)
is the body's height. The blue line, a
golden section of the white line, defines the distance from the head to the
fingertips. 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 shinbone. 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.
Another interesting relationship of
golden section to the design of the human body is that there are:
- 5
appendages to the torso, in the arms, leg and head;
- 5
appendages on each of these, in the fingers and toes and 5 openings on the
face;
- 5 senses
in sight, sound, touch, taste and smell.
Finally, Fibonacci numbers mark key
points in the human aging and development process, as illustrated in the
following table:
Age Dev. Stage Attributes
0 Gestation Conception
1 Newborn Birth
1 Infant Walking,
vocalizing
2 Toddler Talking,
expressing, imitating
3 Toddler Self
image and control, toilet training
5 Early child Formal
education begins
8 Mid child Age of
reason, knowing of right and wrong
13 Adolescent Thinking,
puberty, sexual maturation and drive
21 Young adult Full physical
growth, adult in society, education complete,
beginning
career, financial responsibility
34 Mid adult Refinement
of adult skills, parenting role
55 Elder adult Fulfillment
of adult skills, serving, retirement begins with
eligibility
for Medicare, Social Security and AARP
Fibonacci in Population
When Fibonacci, in his book Liber
Abaci, proposed his rabbit problem, “A
certain man put a pair of rabbits in a place surrounded on all sides by a wall.
How many pairs of rabbits can be produced from that pair in a year if it is
supposed that every month each pair begets a new pair, which from the second
month on becomes productive,” that gave rise to his infamous number set,
the problem could be graphically represented thus:
Month
|
Rabbits from A:
|
from B:
|
from C:
|
D:
|
B1:
|
Total
|
||||||||
0
|
A
|
1
|
||||||||||||
1
|
A
|
1
|
||||||||||||
2
|
A
|
B
|
2
|
|||||||||||
3
|
A
|
B
|
C
|
3
|
||||||||||
4
|
A
|
B
|
C
|
D
|
B1
|
5
|
||||||||
5
|
A
|
B
|
C
|
D
|
E
|
B1
|
B2
|
C1
|
8
|
|||||
6
|
A
|
B
|
C
|
D
|
E
|
F
|
B1
|
B2
|
B3
|
C1
|
C2
|
D1
|
B11
|
13
|
etc.
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
etc.
|
Taking the same impetus, we can use
the formula to show and predict population, as follows:
Area
|
Census
Rank |
Actual
Population |
Predicted Population
|
|
Method 1
|
Method 2
|
|||
New York, NE NJ
|
1
|
16,206,841
|
||
LA Long Beach CA
|
2
|
8,351,266
|
10,016,379
|
10,016,379
|
Chicago NW IN
|
3
|
6,714,578
|
6,190,462
|
5,161,366
|
Detroit, MI
|
5
|
3,970,584
|
3,825,916
|
4,149,837
|
Washington DC
|
8
|
2,481,459
|
2,364,546
|
2,453,956
|
Houston, TX
|
13
|
1,677,863
|
1,461,370
|
1,533,626
|
Cincinnati, OH
|
21
|
1,110,514
|
903,176
|
1,036,976
|
Dayton, OH
|
34
|
685,942
|
558,194
|
686,335
|
Richmond, VA
|
55
|
416,563
|
344,983
|
423,935
|
Las Vegas, NV
|
89
|
236,681
|
213,211
|
257,450
|
New London, CT
|
144
|
139,121
|
131,772
|
146,277
|
Great Falls, MT
|
233
|
70,905
|
81,439
|
85,982
|
Method 1 takes the population of the largest city and divides it again
and again by phi. Method 2 takes the population of each successive city
and divides it by phi, thus enabling the relatively active predicting of the
respective populations.[19]
Conclusion
I end, much like I began, with a
deep fascination for mathematics in general, which keeps my mind acute, and for
numbers in specific, which takes that acuteness and connects it, as this paper
demonstrates, to nearly everything in the known universe. Fibonacci proves that everything is
connected, from the largest spiral galaxy in the deepest parts of space, to the
cochlea inside the human ear, to the proportions of the human face, the shape
of a tiger’s head, the sweep of a thrush’s wing, the design of a piece of
classical architecture.
I find it remarkable that more
attention is not given to the Golden Ratio, the Golden Section, the Golden
Mean, the Divine Proportion, the Fibonacci sequence, and the golden number,
phi, in everyday life.
They are startling
propositions—gauntlets of potential evidence for intelligent design. At the very least, they suggest extraordinary
coincidence, are certainly awe-inspiring, and very, very humbling…
Contact the author for a record of all works cited.
© Ray Cattie
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