FIBONACCI NUMBER AND GOLDEN RATIO IN NATURE

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 F_n 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=F₁₂-1

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 F_n+2-1. In symbols:

_n=F_n+2-1

We add F_n+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 r_{2 ,} 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

·         http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3578919/

·         http://www.nauticcus.org/exhibits/fibonacci_forecourt

·         http://goldenratio.wikidot.com/human-body

·         http://www.goldennumber.net/human-body/

·         http://www.goldennumber.net/human-hand-foot/

·         http://www.divinetemplatecreations.com/sacred_geometry/fibonacci.html

·         https://en.wikipedia.org/wiki/Patterns_in_nature

·         http://math.arizona.edu/~anewell/publications/Plants_and_Fibonacci.pdf

·         http://fcbs.org/articles/fibonacci.htm

·         http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html

·         http://www.goldennumber.net/spirals/

·         https://en.wikipedia.org/wiki/Golden_spiral

·         http://knowhowledge.com/website/staticpages/Edutainment/Goldratio/Goldratioanimal/goldratioanimal10ap.asp

·         http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

·         http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm

·         http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

http://www.slideshare.net/sagarian/maths-in-nature-complete