FIBONACCI NUMBER AND GOLDEN RATIO IN
NATURE
ABSTRACT
In
mathematics , the Fibonacci numbers or Fibonacci sequence are the numbers in
the following integer sequence.1,1,2,3,5,8,13,21,34,55,89,144,…….. Or (often in modern usage) 0,1,1,2,3,5,8,13,
21,34,55,89,144,…….. In mathematical terms ,the sequence F
n of
Fibonacci numbers is defined by the recurrence relation
with seed values
Or
Fibonacci
number is visible in petals of flowers and seed arrangement. We can see Golden
ratio in animals ,plants ,fruits, galaxy and in human body also.
INTRODUCTION
Fibonacci an
Italian mathematician described the Fibonacci number in 1902 AD , although it
had been described by Indian mathematicians earlier . The Fibonacci numbers are
a sequence of numbers . You start with zero and one . Then the next number is
the sum of the previous two . Zero plus one is one , the next number ,one and
one are two , the next number . one and two are three , two and three are five
, three and five are eight , and so on. Leonardo Fibonacci (circa
1175-1240),who discovered the sequence named after him and its properties, was
an Italian mathematician who helped introduce the Hindu-Arabic
numerals(0,1,2,3, etc.) into Western Europe. He is also better known as the
originator of the special sequence of numbers , now called the Fibonacci
Sequence or Fibonacci Numbers . Fibonacci was born in Pisa , Italy and he
sometimes known as Leonardo of Pisa . In his youth he travelled in the Middle
East , where he learned the Hindu – Arabic numeral system. The original problem the Fibonacci
investigated (in the year 1202) was about how fast rabbit could breed in ideal
circumstances.
NAMES
FOR ALL FIBONACCI NUMBERS
The inveterate Fibonacci addicts tends to attribute a certain individuality
to each Fibonacci number . Mention 13 and he thinks F7 ;55 and F10
flashes through his mind. But regardless of this psychological quirk , it is
convenient to give the Fibonacci numbers identification tags and since they are
infinitely numerous , these tags takes the form of subscripts attached to the
letter F . Thus 0 is denoted by F0 ; the first 1 in the series is F1
; the second 1 is F2 ; 2 is F3 ; 3 is F4 ;
5 is F5 ; etc. The following table for Fn shows a few of
the Fibonacci numbers and then provides additional landmarks so that it will be
convenient for each Fibonacci explorer to make up his own table.
N
|
Fn
|
N
|
Fn
|
0
|
0
|
11
|
89
|
1
|
1
|
12
|
144
|
2
|
1
|
13
|
233
|
3
|
2
|
14
|
377
|
4
|
3
|
20
|
6765
|
5
|
5
|
30
|
832040
|
6
|
8
|
40
|
102334155
|
7
|
13
|
50
|
12586269025
|
8
|
21
|
60
|
1548008755920
|
9
|
34
|
70
|
190392490709135
|
10
|
55
|
80
|
23416728348467685
|
SUMMATION
PROBLEM
The first
question we might ask is ; what is the sum of the first n terms of the series ?
A simple procedure for answering this question is to make up a table in which
we list the Fibonacci numbers in one column and their sum up to a point in
another.
N
|
Fn
|
Sum
|
1
|
1
|
1
|
2
|
1
|
2
|
3
|
2
|
4
|
4
|
3
|
7
|
5
|
5
|
12
|
6
|
8
|
20
|
7
|
13
|
33
|
8
|
21
|
54
|
What does the sum look like ? It is not a Fibonacci number , but
if we add 1 to the sum , it is a Fibonacci number two steps ahead . Thus we could write
1+2+3+…+34+55(F10)
= 143 = 144-1 = F12-1,
where we have indicated the names of the key Fibonacci numbers in
parenthesis. It is convenient at this
point to introduce the summation notation . The above can be written and
concisely :
k=F
12-1
It appears that the sum of any number of consecutive Fibonacci
numbers starting with F1 is found by taking the Fibonacci number two
steps beyond the last one in the sum and subtracting 1 .
Let us go back then to sum of the first ten Fibonacci numbers , we
have seen that this sum is F12-1. Now suppose that we add 89 or (F11)
to both sides of the equation . Then on the left hand side we have the sum of
the first eleven Fibonacci numbers and on the right we have,
144-1+89=F12-1+F11=233-1=F13-1 .
Thus , proceeding from the sum of the first ten Fibonacci numbers,
we have shown that the same type of relation must hold. Is it not evident that
we could now go on from eleven to twelve ; then from twelve to thirteen ;
etc.,so that the relation must hold in general ?
This is the type of reasoning that is used in the general proof by
mathematical induction . We suppose that the sum of the first n Fibonacci
numbers in Fn+2-1. In symbols:
n=F
n+2-1
We add Fn+1 to both sides and obtain
k=F
n+2-1 + F
n+1 =
F
n+3-1
By reason of the fundamental property of Fibonacci series that the
sum of any two consecutive Fibonacci number is the next Fibonacci number , we
have now shown that if the summation hold for n , it holds also for n+1 . All
that remain to be done is to go back to the beginning of the series and draw a
complete conclusion. Let us suppose , as can readily be done that the formula
for the sum of the first n terms of the Fibonacci sequence hold for n≤7. Since
the formula hold for seven , it hold for eight, since it is hold for eight, it
hold for nine; etc. Thus the formula is true for all integral positive values
of n.
We have seen from this example that there are two parts to our
mathematical exploration. In the second we prove that the formula is true in
general.
GOLDEN
RATIO
If we take the ratio of two successive
numbers in Fibonacci´s series ,(1,1,2,3,5,8,…) and we divide each by the number
before it , we will find the following series of numbers :
1/1=1
2/1=2 3/2=1.5
5/3=1.666…
8/5=1.6 13/8=1.625 21/13=1.61538…
It is easier to see what is happening
if we plot the ratios on a graph
The ratio seems to be setting down to a particular value , which
we call the golden ratio or the golden numbers . It has a value of
approximately 1.618034, although we shall find even more accurate value.
The golden ratio 1.618034 is also called the golden section or the
golden mean or just the golden number . It is often represented by a Greek
letter Phi Ø . The closely related value
which we write as phi with a small “p” is just the decimal part of phi ,namely
0.618034.
FIBONACCI RECTANGLE
We can make
picture showing the Fibonacci numbers 1,1,2,3,5,8,13,… if we start with two
smaller squares of size one next to each other . On top of these draw a square
of size two.
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 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 have 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
SPIRAL
The Fibonacci spiral : an approximation of the golden spiral
created by drawing circular arcs connecting the opposite corners of squares in
the Fibonacci tiling, this one uses squares of sizes 1,1,2,3,5,8,13,21, and 34
.
Here is a spiral drawn in the squares , a quarter of a circle in
each square. The spiral is not a true mathematical spiral( since it is made up
of fragments which are parts of circles and does not go on getting smaller and
smaller) but it is a good approximation to a kind of spiral that does appear
often in nature. Such spirals are seen in the shape of shells of snails and sea
shells and, as we later , in arrangement of seeds on flowering plants too. The
spiral-in-the-squares makes a line from the centre of the spiral increase by a
factor of the golden number in each squares . So points on the spiral are 1.618
times as far from the centre after a quarter-turn. In a whole turn the points
on a radius out from the centre are 1.618⁴=6.854 times further out than when
the curve last crossed the same radial line.
GOLDEN
SPIRAL
The Golden Spiral is a type of logarithmic spiral that is made up
of a numbers of Fibonacci relationships , or more specifically , a number of
Golden ratios , for example , if we take a specific arc and divide it by its
diameter , that will also give as the Golden Ratio 1.618 . We can take , for example
, arc WY and divide it by its diameter of WY . That produces the multiple 1.618
. Certain arcs are also related by the ratio of 1.618 . If we take the arc XY
and divide that by are WX , we get 1.618 . If we take radius 1 (r1)
, compare it with the next radius of an arc that’s at a 900 angle with r1 ,which is r2
, and divide r2 by r1 , we also get 1.618.
FIBONACCI NUMBER AND GOLDEN RATIO IN NATURE
NAUTILUS
SHELL
The shell of the chambered nautilus has Golden proportions . It is
a logarithmic spiral. The shell has grown by a factor of the golden ratio.
BODY
PROPORTION OF DOLPHIN
The eyes , fins and tail of dolphin fall at golden section along
body. Other fishes also exhibit golden ratio in their body.
STAR
FISH
A star fish has five arms ( 5 is the fifth
Fibonacci number ). If a regular pentagon is drawn and diagonals are added , a
five sided star or pentagon is formed . Where the sides of the pentagon are one
unit in length , the ratio between the diagonals and the sides is Phi , or the
Golden ratio . This five-point symmetry with Golden proportion is found in
starfish.
SPIRAL
GALAXIES
Spiral galaxies also follow the familiar Fibonacci pattern . The
Milky Way has several spiral arms , each of them a logarithmic spiral of about
12 degrees . As an interesting aside , spiral galaxies appear to defy Newtonian
physics . As early as 1925 , astronomers realized that , since the angular
speed of rotation of the galactic disk varies with distance from the centre ,
the radial arms should become curved as galaxies rotate . Subsequently , after
a few rotations , spiral arms should start to wind around the galaxy . But they
don’t – hence the so called winding problem . The stars on the outside , it
would seem , move at a velocity higher than expected – a unique traits of the
cosmos that helps preserve its shape .
PLANTS AND FIBONACCI
Since the time of Johannes Kepler, natural
scientists have been fascinated and intrigued by the observation that the
phylla (elements such as leaves, florets, stickers or branch) on many plants
are arranged in such a manner that each phyllo lies on three families of
spirals. Moreover the numbers of arms in each of the spiral families are in
almost 95% of al cases sequential triads in the regular Fibonacci sequence,
1,1,2,3,5,8,…further the surface of these plants are tiled in polygonal shapes.
PETALS ARRANGEMENT IN PLANTS
The number of petals in a flower
consistently follows the Fibonacci sequence. Famous examples include the lily,
which has three petals, buttercups, which have five (pictured at left), the
chicory's 21, the daisy's 34, and so on. Phi appears in petals on account of
the ideal packing arrangement as selected by Darwinian processes; each petal is
placed at 0.618034 per turn (out of a 360° circle) allowing for the best
possible exposure to sunlight and other factors.
Probably most of us have never
taken the time to examine very carefully the number or arrangement of petals on
a flower. If we were to do so, several things would become apparent. First, we
would find that the number of petals on a flower is often one of the Fibonacci
numbers.
Scale patterns
of pinecones
The
seed pods on a pinecone are arranged in a spiral pattern. Each cone consists of
a pair of spirals, each one spiraling upwards in opposing directions. The
number of steps will almost always match a pair of consecutive Fibonacci
numbers. For example, a 3-5 cone is a cone which meets at the back after three
steps along the left spiral, and five steps along the right.
The seed-bearing scales of a
pinecone are really modified leaves, crowded together and in contact with a
short stem. Here we do not find phyllotaxis as it occurs with true leaves and
suchlike. However, we can detect two prominent arrangements of ascending spirals
growing outward from the point where it is attached to the branch.
Pineapple scales
Seed heads
The head of a flower
is also subject to Fibonaccian processes. Typically, seeds are produced at the
center, and then migrate towards the outside to fill all the space. Sunflowers
provide a great example of these spiraling patterns.
In some cases, the
seed heads are so tightly packed that total number can get quite high — as many
as 144 or more. And when counting these spirals, the total tends to match a
Fibonacci number. Interestingly, a highly irrational number is required to
optimize filling (namely one that will not be well represented by a fraction). Phi
fits the bill rather nicely.
HUMAN BODY
AND GOLDEN RATIO
Man
is structured divine proportion
All cultures and spiritual
traditions have always sought to discover the canon of human perfection. We are
heirs of many visions and concepts relating to the harmony of human being.
Sages have found but, above all, this model of perfection in beauty and harmony
of mind, those that allow man to become truly 'measure of all things
"after the famous phrase of Greek Protagoras.' And God created man his own
image, says Genesis, first book of Moses. Univer ¬ sal man in which this
utterance, the Christian doctrine of Christ, Adam Kadmon of the Kabbalistic
tradition, is even seen as a person divine macrocosm manifest, therefore, all
the qualities and heavenly Father. Good, truth and beauty are divine being
fully awakened to
In turn, the soul of every man
hiding in the potential of this face divine, whose lead in acquiring the full
revelation neasemuitei Frumuseti spiritual. Thus, man carries in his treasure
deep perfection, order, harmony, proportion and symmetry of the divine. It was
still looking and outward signs, body of these beauties, tangible evidence of
the omnipresence of the Divine Proportion, as an expression of Great Power
Cosmic Tripura Sundari.
Not by chance, hermetistul
Agrippa von Netteseheim (1486-1535) said, 'Man, work and complete choice that
God created it, have a more harmonious body than all other creatures and in him
all the numbers ingemaneaza all measures, all weights, movements, elements […]
are touched by this sublime work perfectiuneaf …] There is not even a single
component of the human that does not have in the world of divine ideas, mail
or: a celestial sign, a star , an idea-force, a name of God. Human body shape
is an expression of completeness. "
Ancient eel known symbol of the
number of Gold in the human body is the navel, a microcosm reflecting
Omphalosului, called the new and old,, Navel of the Earth”.
Only after the Renaissance, when
oral transmission was about to die, this knowledge began to be revealed in a
required wider insa'din increasingly poor their metaphysical core. History has
preserved the names of some of connoisseurs ancient secret: Pythagoras, Plato,
Vitruvius, etc.. Canonical figures of Leonardo da Vinci and Albrecht Dürer were
but those who have submitted today to the ancient idea of body perfection.
They reveal us that,
semniflcativ, navel divides the human body maltimea Gold Report (a fact noted
by all curiosii statistical and a range satisfying). Meanwhile, the pelvic area,
connections with the Earth element, lower extremity octave symbolic 1-2,
divided maltimea half. Observe the symbolic connotations: Earth corresponds to
four, square, but his two, dejos end of the octave, the primary division
matrix. It is known that, at birth, the navel is placed at half height, and
then to move, with maturation, to the point of division as harmonic, is given
by the ratio of gold. This evokes the idea of moving from state primary duality
in a relationship, the proportion of Divine Unity, as the assimilation
experience of life, in order to proclaim then:
The Human Body is based on
patterns of 5, which is the basis for Phi as well
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
·
5
openings on the face.
·
5
sense organs for sight, sound, touch, taste and smell.
The golden section in turn, is also based on 5, as the number
phi, or 1.6180339…, is computed using 5’s, as follows:
5 ^ .5 * .5 + .5 = Phi
In this mathematical construction “5 ^ .5” means “5 raised to
the 1/2 power,” which is the square root of 5, which is then multiplied by .5
and to which .5 is then added.
HUMAN
FINGERS AND HAND
We have
·
2 hands each of which has…
·
5 fingers each of which has…
·
3 parts , separated by…
·
2 knuckles.
·
If we measure the length of the bones in
finger it looks as the ratio of the longest bone in a finger to the middle bone
with ratio is phi.
·
The ratio between the forearm and the hand is
in the Golden ratio .
It was found
statistically between the lengths of phalanges hand close and Fibonacci's
series, even if in practice the relationship is not always rise to the Golden
Number, the principle of growth identical to itself and preserves the shape is
very well ogliiidit decreasing length of the phalanges or skeletal segments.
FACES
Faces , both human and nonhuman , abound with examples of the
Golden ratio. The mouth and nose are
each positioned at Golden sections of the distance between the eyes and the
bottom of the chin . Similar proportions can been seen from the side , and even
the eye and ear itself ( which follows along spiral ).
It’s worth nothing that every person’s body is different , but
that averages across populations tends towards phi. It has also been said that
the more closely our proportions adhere to phi , the more attractive those
though traits are perceived . As an example , the most “beautiful” smiles are
those in which central incisors are 1.618 wider than the lateral incisors ,
which are 1.618 wider than canines , and so on . It’s quite possible that ,
from an evo-psych perspective , that we are primed to like physical forms that
adhere to the Golden ratio – a potential indicator of reproductive fitness and
health .
DNA
MOLECULES
Even the microscopic realm is not immune to Fibonacci . The DNA
molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of
its double helix spiral . These numbers , 34 and 21 , are numbers in the
Fibonacci series , and their ratio 1.6190476 closely approximates phi ,
1.6180339.
Research has shown that if you look at the height of the DNA
molecule relative to its length , it is in the proportion of .618 : 1 if we
look at the components of the DNA molecule
, there is a major groove in the left section and a minor groove in the
right section . The major groove is equal to .618 of the entire length of the
DNA molecule , and the minor groove is equal to .382 of the entire length .
UTERUS
According to Jasper Veguts , a gynecologist at the university
hospital Leuven in Belgium , doctors can tell whether a uterus looks normal and
healthy based on its relative dimensions that approximate the Golden ratio .
BIBLIOGRAPHY
