Thursday, June 20, 2019

XII Nature`s Numbers and Counting Systems




XII. Nature’s Numbers and Counting System
[Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri]
        Marvellous beauty of Nature is an aesthetic marvel of numbers.
Have you ever noticed how the plants grow their branches over a considerable period of time after germination from seeds? Or have you think of the number of petals and their arrangements in flower of plants,those blossom everyday here and there in front of our eyes? Likewise ,do you know the numbers of seeds a multi seed plant pack in their fruits and their arrangements inside it?  
If you observe carefully, you will find in each of the above cases an astonishing pattern in which numbers play a very important role in an orderly manner. We will explore those mysteries shortly.
In fact, these rules of numbers are found in innumerable things of nature around us.
Look at the amazing structure of shells of different snails, such as Chambered Nautilus or observe the body of the star fish which has 5 arms with beautiful five-point symmetry. Inside the fruits of many plants we also find this rule of numbers : the banana has 3 sections while an apple has 5, for example.
Have you ever thought of the number of bones in your fingers? You know that you have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in a thumb and one thumb in each hand.
So, if we arrange these numbers in an ascending order, it will be
             1,  1,  2,  3,  5,  8
That means      1,   1,   1+1,   1+2,   2+3,   3+5
Is it not striking? For, each number beginning from the third, is the sum of the previous two numbers. Following this rule, if we extend this array, we get
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ----- to as far as we please.
This type of array of numbers obeying definite rule or rules is called a “Sequence” or “Series” in mathematical term. We shall see shortly the great importance and amazing role of these numbers in nature – in plants, leaves, flowers, seeds, creatures big or small, in our very human body, in art, music, sculpture, architecture such as in the great Egyptian pyramids and many other famous ancient monuments.
Often, especially in modern usage, this sequence is extended by one more initial term :
0, 1, 1, 2, 3, 5, 13, 21, 34 ----
This sequence is named after its inventor, a famous Italian mathematician Leonardo of Pisa, better known as Fibonacci*, The “Fibonacci Sequence” and each number comprising the sequence is also called “Fibonacci numbers” or may be called nature’s numbers for their wide spread presence in nature.
[* Leonardo Fibonacci (1170 – 1250 AD)
(Pronounced as : fib – on – arch – ee  or  fee – bur – narch – ee)
Fibonacci was an Italian mathematician, considered to be the most talented mathematician of the Middle Ages. Not much is known about the life of Fibonacci.
He was born around 1170 in Pisa in a wealthy merchant family. As a young boy, he travelled with his father to North-Africa on a business trip of his father and it was in Bugia, now Algeria that he learned about Hindu-Arabic numeral system.
Fibonacci travelled extensively around Mediterranean Coast, meeting many merchants and learning about their systems of doing arithmetic. He soon realized the many advantages of the Hindu-Arabic system. In 1202 he completed his famous book “Liber Abaci” essentially a book of calculations which popularized Hindu-Arabic numerals in Europe.
The date of Fibonacci’s death is not known, but has been estimated to be around 1240 and 1250, most likely in Pisa.]
One thing is to be remembered that Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!
The so called Fibonacci sequence appears in Indian mathematics in connection with Sanskrit Prosody. There are clear evidences of applications of Fibonacci numbers in the writings of Pingala (200 BC), Virhanka (700 AD) and Hemchandra (1150 AD). Outside India, Fibonacci sequence first appeared in Liber Abaci by Fibonacci.
A closer inspection of the numbers making up Fibonacci Sequence brings light a good number of fascinating patterns and mathematical properties.
The most important and interesting of them is the idea of “Golden Ratio”. The very name suggests that it is the name of a ratio. In fact, it is called by many other names, such as “Golden Mean”, “Golden Section”, “Divine Proportion” and also a “Golden Number”. So, it is amply clear that this ratio is not an ordinary one but of great fame and importance. Now it is popularly known by the upper case Greek letter Փ , phi.
This Golden Ratio has been claimed to have held a special fascination for at least 2400 years, though without proof. Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa, Fibonacci and the Renaissance astronomer Johannes Kepler, to present day scientific figures such as Oxford physicist Roger Penrose,  have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematics. Biologists, artists, musicians, historians, architects, psychologists and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact it probably fair to say that Golden Ratio has inspire thinkers of all discipline like no other number in the history of mathematics.
I think, now you are very much curious to know what this extraordinary ratio is!

If we go further and further and take ratios of two Fibonacci numbers, obeying the above rule, we will get more and more accurate values of the golden ratio but never arrive at exact value.
We have already calculated the approximate value of the golden ratio as 1.618033988. So it has two parts – the integral part I and the decimal or fractional part 0.6180339887. One interesting thing is that the nature itself has adopted this fractional part for the spirals in seed heads and arrangement of leaves in many plants. So, considering its importance, it is separately denoted by the lowercase Greek letterφ. In fact Φ and φ play amazing role in nature and in many aspects of our lives.
This symbol phi for the golden ratio was first introduced by an American mathematician Mark Barr in 1900’s. It has some interesting aspects as well as inner meaning. The character for phi also has some interesting theological implications.
The description of this proportion as Golden and Divine is fitting because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for a single number to play but again this one number has played an incredible role in human history and the universe at large.
The meaning hidden in phi from the theological aspect is as :
The message from scripture of all the major monotheistic religions such as in the belief of the religions Judaism, Christianity and Islam that God is one and only one, he who created the universe from nothing, splitting, nothingness to offsetting energies and elements. This view has been substantiated by our modern science. Today we understand the universe to consist of positive and negative atomic and subatomic particles and charges, matter and anti-matter, all coming from a singularity in what we term the “Big Bang”.
The essence of the Big Bang theory is that, this universe was originated from the explosion of a very very minute particle, like a grain of sand, under unimaginably high gravitational pressure. There was nothing, what so ever, of today’s universe. So it may be conjectured that the baby universe was born from nothing or started life from zero.

The Fibonacci numbers may be called the Nature’s numbering system. For, they appear everywhere in Nature, from the leaf arrangements in plants, to the pattern of florets of a flower, the bracts of pinecone, the scales of a pineapple, in vegetables and fruits, seed heads, animal bodies including our human bodies. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees and even all mankind.
Without knowing this sequence plants grow in the most efficient ways. Many plants show the Fibonacci numbers in arrangements of the leaves around the stem. Many flowers like daisies, sunflower, rose also show these numbers. Many other plants with more than normally thickened and fleshy leaves, like Aloe Vera, fall in the category of “Succulents” manifest these numbers. Some plants having needle- shaped or scale-leaved, fall in the class of “Coniferous”, also show these numbers in the bumps on their trunks. Palm trees show the number in the rings on their trunks.
But the question is : why do these arrangements in mathematical order occur? In the case of leaf arrangements in some plants, one of the causes may be related to the maximizing the space for each leaf or average amount of light falling on each one. In case of close packed leaves in cabbages and succulents, the correct arrangement may be crucial for availability of space.
If we look closely the natural world around us, we can find many instances of mathematical order involving Fibonacci numbers themselves and closely related Golden elements.
Many plants during their growth show Fibonacci numbers in its growing points. Suppose, when a plant puts out a new shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing points we get the picture as shown below [ Pic. No.4 : Fibonacci numbers in growth of plants]



Pic. No.4 : Fibonacci numbers in growth of plants
Also many plants show Fibonacci numbers in the arrangements of the leaves around their stems. In an estimate, it is revealed that 90 percent of all plants exhibit this pattern of leaves involving these numbers.
It we look down on a plant we will find, the leaves are often arranged so that leaves above do not hide leaves below. The reason is obvious that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.
Botanists have noticed spiral growth of plants in many aspects such as in the arrangement of leaves, florets, seeds of flowers and fruits and other structures. It is also revealed by close observations that plants grow new cells in spirals. This spiral happens naturally because each new cell is formed after a turn. The seeds of a sunflower, the spines of a cactus and bracts of a pinecone all grow in whirling spirals patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand. A large number of plants have spiral patterns in which each leaf, seed or other structures follow the next at a particular angle, called the “golden angle” which is about 137.50. This golden angle is closely related to the golden ratio which ancient Greeks studied extensively and some had believed to have divine, aesthetic or mystical properties.
Scientists have puzzled over this pattern of plant growth for hundreds of years. Why do plants prefer the golden angle to any other? And how can plants possibly “know” anything about Fibonacci numbers?
Early researchers thought these patterns provide an evolution advantage and survival for plants. But most recently they have realized that the answer lies in biochemistry of plants as they develop new leaves, flowers or other structures. The mystery has not been solved entirely but a basic understanding of the process seems to be in sight. For searching these answers, botanists have gone back to their electron microscope to re-examine plants they thought that they had already understood.
The arrangements of leaves in plants obey mathematical diktat such as Fibonacci sequence, golden angle, golden ratio etc. The pattern of arrangements of leaves on the stem is called “Phyllotaxy”. As a stem grow at its apex new leaf buds form along the stem by a highly controlled development process. Leaf arrangement along the plant stem depends on the species and is an important identifying characteristic.
If we look around the beautiful botanic garden of nature we will find the basic phyllotactic patterns in the plant kingdom are either opposite, whorled or alternate.
When two leaves arise at each node opposite to each other the phyllotaxy is “opposite”. When each pair at each node placed at right angles to one another it is termed as “opposite decussate” such as in latex producing plants, guava etc. But if all the pairs occur in same line, the phyllotaxy is “opposite superposed” Jamun is an example.
If more than two leaves arise at a each node in the shape of whorl, it is called “whorled”, phyllotaxy as in oleander.
More than 80% of the 250,000 higher plant species have an “alternate” phyllotaxis, as in the case of potato, araucaria, yucca or sunflower for instances. Alternate phyllotaxis again manifests in two classes – “spiral” and “multijugate”.
In the natural process, leaves grow in the plants from their leaf primodia, the embryo of leaves, at the nodes of the plants.
In spiral phyllotaxis, leaf primodia grow on per node, each at a constant divergence angle of 137.5 degrees from the previous node.
Very often multijugate patterns look very similar to spiral patterns. The only way to detect them is to count number of spirals, called the parastiches, visible in the pattern. If the parastiches numbers do not have a common divisor other than 1, the pattern is spiral phyllotaxy. If these numbers, on the other hand, have a common divisor, say K, the pattern is called multijugate or more precisely K – Jugate and there are K elements or leaves at each node.
Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one. If we count in other direction, we get a different number of turns for the same number of leaves. We will be surprised to know that the number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!
If we have 3 clockwise rotations before we meet a leaf directly above the first passing 5 leaves where as if we go anti-clockwise, we need only two turns, than it is written as 3/5 or 2/5    respectively.
The sunflower here when viewed from the top shows the same pattern. Starting at the leaf marked x, we find the next lower leaf turning clockwise. [Pic. No.5 : Leaf arrangements in Sunflower plants] Numbering the leaves produces the patterns shown here in the picture.

 [Pic.No.5 Leaf arrangements in Sunflower plants]
We see that the third leaf and the fifth leaves are nearest below our starting leaf but the next nearest below is the eighth then the thirteenth. If we make a list of how many turns did it take to each leaf will be as
Leaf number
Turns clockwise
3
1
5
2
8
3

All the numbers in each column are Fibonacci numbers!
We find Fibonacci leaf arrangement in many common trees, such as 1/2 in elm, linden, lime, grasses 1/3 in beech, hazel, grasses, blackberry 2/5 in oak, cherry, apple, holly, plum 3/8 in poplar, rose, pear, willow 5/13 in pussy, willow, almond. [Pic. No. 6 : Spiral leaf arrangements]
Pic. No.6 

 Spiral leaf arrangements
Perhaps there is none of us who but loves flowers. Everyone has more or less, attractions to flowers for their exotic beauties, eye-soothing colours of wonderful variety of shades and sometimes for their mind-boggling fragrances.
But 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, we would be surprised to discover that the number of petals on a flower is one of the Fibonacci numbers.
Lilies and iris have 3 petals, buttercups have 5, 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.
One petalled and two petalled flowers are not common, yet we find white calla lily [Pic. No.7(a)] as an one- petalled and euphorbia as a two- petalled flower [Pic. No.7(b)]. Three- petalled [Pic. No.7(c)] are more common and we see it in flowers like lily, iris. But lilies appear to have six petals. In fact, 3 are petals and 3 are sepals. Sepals form the outer protection of the flowers when in bud. But almost in all flowers difference between sepals and petals is clearly visible.
Pic.No.7a.

                                                                    White calla lily.
                   Pic. No.7b                                                            
                                                         Euphorbia.

Pic. No. 7c


                                                              Trillium.


Very few plants show 4 petals or sepals, but some, such as fuchsia,
 Gentian do. [PicNo.7(d)] Since 4 is not a Fibonacci number, then is it a break of natural law? We shall try to find an explanation later.
There are hundreds of species, both wild and cultivated, with five petals. Such as buttercup, wild rose, larkspur, columbine, Adenium [Pic. No.7(e)] pinks etc. The humble buttercup has been bred into a multi-petalled form
                                                     
              Pic. No.7d
Purple Gentian.
                                                  
                                                                 
    Pic. No7e
                                          


                                                       Adenium.
                            
                                        
                                               

                                                      
Again, six is not a Fibonacci number yet there are six- petalled flowers, such as Blue – Eyed Grass [Pic. No.7(f)]. Eight- petalled flowers are not common as five- petalled, but there are quite a number of well-known species with eight. Such as delphiniums blood root and dahlia [Pic. No.7(g)]
Some of the flowers with petals having the Fibonacci number 13 are ragwort, corn marigold, cineraria, some daisies, black-eyed Susan.
The black-eyed Susan belongs to the sunflower family as we see in the adjoining picture. [Pic. No.7(h)]

Pic.No.7f


                                                           Blue-eyed grass.

Pic.No.7g                                                    
                                                        


                                                                                Dahlia.


            Pic. No7h

Black-eyed Susan.

                                                               

                                                                 
Examples of flowers with
21 petals- aster [Pic. No.7(i)], chicory, Shasta daisy etc.
34 petals – plantain, pyrethrum, some daises, chrysanthemum [Pic. No.7(j)]
55 and 89 petals – michaelmas daises, the asteraceae family
   Pic. No7i
Aster.


                                                                .
 Pic. No7j

Chrysanthemum.


                                                           
                                                                            
Twenty one and thirty four petals are quite common. The outer ring of ray florets in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13, 21, 34, 55 or 89 petals are quite common.
In the pictures we see the pyrethrum best known as chrysanthemum with 34 petals and michaelmas daises, a close relative of asters with 55 and as much as 89 petals.
The Fibonacci numbers pattern encountered herein occur so frequently that very often we term the phenomenon as a “law of nature”.
But since 4 is not a Fibonacci number, yet four- petalled flowers or four-leaf clover, a genus of about 300 annual and perennial species in the pea family, are not rare in the nature. Even large deviations from the Fibonacci patterns may also be found. If it is at all disturbing to the modern botanists, it is not at all so to the Fibonacci devotees, for whom the whole phenomenon, if not a law, is at least a fascinating prevalent tendency!
The Fibonacci numbers can also be found in the arrangement of seeds on flower heads. The number of seeds and their spiral arrangements in the coneflowers and sunflowers are worthy to mention for fitting well with Fibonacci numbers and the related Golden Mean.
The coneflower is a native North American sporting daisy like flower with raised centres. The flower, plant and root of some types are used in herbal medicines.
Here in the picture [Pic. No. 8a] of a coneflower, we see that petals seem to form spirals running both to the left and to the right. At the edge of the picture, if we count those spiralling to the right as we go outwards, there are 55 spirals. A little further towards the centre, the count is 34. Are these two numbers not the two consecutive Fibonacci numbers? How many spirals we may expect at these places, if we go the other way? We will find that the pair of numbers, curving left and curving right are neighbours in the Fibonacci Series.

Pic. No. 8a

                                                          Spirals in a Coneflower.

Sunflowers are beautiful and iconic for the way their giant yellow heads against the back drop of the blue sky. But have you ever stopped to look at the pattern of seeds held within the centre of these special flowers? They are also a mathematical marvel.
The seed heads of sunflowers optimize the packing of seeds by arranging them in spirals of Fibonacci numbers. This patterning helps to pack seeds uniformly with no crowding at the centre and no bald patches at the edges.
The Fibonacci sequence fits so well for the sunflower because of its key characteristic – growth. On a sunflower seed head, as an individual seed grows, the centre of the seed head continues to add new seeds pushing those at the periphery outwards. As a result, the seeds will always be packed uniformly and with maximum compactness. [Pic. No. 8b]

Pic. No. 8b


                                                   Spirals of the Sunflower seed head.
    
In the spirals of the sunflower seed heads, there are two series of curves winding in opposite directions, starting at the centre and stretching out to the petals, with each seed sitting at a certain angle from the neighbouring seed to create the spiral.
On close observations, it is found that in order to optimize in filling of the seeds in the flower’s centre, it is necessary to choose a number that well approximated by a fraction. This number is exactly the golden mean or golden ratio ( φ), an irrational number which has already been mentioned. We may recall the value of this number  Φ  = 1.6180339887 ---, the fractional part  φ = 0. 6180339887 --- plays an important role in filling the seeds.
The corresponding angle, the golden angle is 137.5 degrees. This angle is chosen by the plants very precisely. Otherwise variations of one tenth of a degree destroy completely the optimization. When the angle is exactly this golden angle, two families of spirals, one in each direction are then visible.
Some fruits like pineapple, banana, sharon fruit, apple and more, exhibit patterns following Fibonacci sequence. When a banana or an apple is cut properly, it is found that banana has 3 sections where as an apple has 5, both are Fibonacci numbers.
Beautifully grown pineapples with spirally arranged fruitlets follow a simple rule that illustrates perfection in natural Fibonacci sequence.
On many vegetables, we find the presence of Fibonacci numbers and golden means in their formations, spirals, sections, seed heads etc.
Take for example a cauliflower or a broccoli. If you count the number of florets in spirals on a cauliflower, the number in one direction and in the other will be Fibonacci numbers. Then to take a closer look at a single floret, break one off near the base of the cauliflower. It is also a mini cauliflower with its own little florets, all arranged in spirals around a centre. If possible, count with patience, the spirals in both directions. Surprisingly counting them again shows the Fibonacci numbers.
Philosophers, scientists and artists believe that “Nature has designed the human body so that its members are duly proportioned to the frame as a whole”. It is widely believed that the human body follows the Golden Ratio.
An important relationship of golden section to design human body is that there are :    (i)  5 appendages to the torso, the trunk of the human body, which are two arms, two legs and head.
            (ii) 5 appendages on each of these, the fingers and toes,
            (iii) 5 openings on the face,
(iv) 5 sense organs for sight, sound, touch, taste and smell.

Pic. No. 9a Golden Ratio in human body
Let us have a close look at our hands. [Pic. No. – 9b] Human hand has 5 fingers, each finger has 3 phalanx (bones of a finger or toe) joints separated by two average sizes of falangelor, the lengths of bones of fingers, are 2 cm, 3 cm, 5 cm. In continuation is a bone of the hand which has an average length of 8 cm. Surprisingly, all these numbers belong to Fibonacci sequence 
   Pic.No.9b
 
 Golden Ratio in human hand.

 
Moreover, each section of our index finger, from the tip to the base of the wrist is longer than the preceding one by about the Fibonacci ratio of 1.618, also fitting the Fibonacci numbers 2, 3, 5 and 8. By this scale our fingernail is taken as a unit.
Curiously enough we have two hands, each with 5 digits and 8 fingers are each comprised of 3 sections. All are Fibonacci numbers!
Also our hand creates a golden section in relation of our arm, as the ratio of our forearm to our hand is also about 1.618, the Divine Proportions!
Even on our feet, we may notice the presence of the golden section. Our foot has several proportions based on phi lines, including
i) The middle of the arch of the foot
ii) The widest part of the foot.
iii) The base of the toe line and big toe
iv) The top of the toe line and base of the index toe. 
The most prominent, beautiful and important part of our human body is face [pic. No.9c ]            considered as the index of the mind.
pic.No.9c
                                                          Golden Ratio in human face.
Human face is characterized in terms of aesthetic through several dimensions : The distance between the eyes, distance between eyes and mouth, between eyes and nose, mouth size. In terms of aesthetic in science that a girl appears more pleasant to our eyes when these dimensions maintain the Fibonacci sequence better.
In this context, I may ask, ‘Are you not amazed to look at the beauty of Mona Lisa, a half portrait of a woman by the Italian artist Leonardo Da Vinci, which acclaimed as the best known, most visited, the most written about, the most parodied work of art in the world?’
The portrait includes lot of Golden Rectangle, related to Golden Ratio. It is believed that Leonardo as a mathematician tried to incorporate mathematics into art. This painting seems to be made purposefully line up with golden rectangle.
But one thing is to be kept in our mind that not every individual has body dimensions in exact proportion but averaged across the population tend towards phi and the phi proportions are perceived as being the most natural or beautiful.
We have already seen the spiral arrangements in leaves, arrangements of florets in flowers and on fruits. But have you ever looked at an interactive weather map depicting hurricane activity? If you have observed, you will be convinced to believe that a hurricane takes a spiral shape and most probably you will draw an analogy with the shape of a seashell or a winding stair case. Yes, they have the same shape and what you are observing is called Fibonacci spiral. This spiral shows up in many areas of nature, art, architecture, astronomy, oceanography and many other places.
You will be surprised to know that the shape of this spiral obeys some definite mathematical rules and the answer of this puzzle lies in golden rectangle.
All of us are familiar with this geometrical figure of a rectangle. But do you have a rectangle size that is your favourite? Is it an odd question? No, not always, for there is a certain type that is found to be aesthetically pleasing to the eye. That is the golden rectangle.

i)  Construct a simple square of your choice.
ii) Draw a line from the midpoint of one side of the square to an opposite corner.
iii) Taking that line as the radius, draw an arc that defines the height of the rectangle.
iv) Complete the rectangle.
This will be a golden rectangle.]

Pic. No. 10a
                                                           A Golden Rectangle.
Now the question is, how is the Fibonacci spiral related to the golden rectangle? The answer is : the golden rectangle has a unique property that when a square is removed, a smaller rectangle of the same shape, and of course a golden rectangle, remains. Thus a smaller square can be removed and so on and on, resulting with a spiral pattern. This is exactly the Fibonacci spiral. It will be clear from the adjoining picture. [Pic. No. 10b]
  Pic.No 10b.

                                                The Fibonacci spiral.
If we look closely at the animal world around us, we will find not only beauties in them but also the presence of the golden section or Divine Proportions in their bodies.
Here are some examples which will illustrate this.
Dolphin : The eye, fins and tail all fall at golden sections of the length of the dolphin’s body. The dimensions of the dorsal fin are golden sections (yellow and green). The thickness of the dolphin’s tail section corresponds to same golden section of the line from head to tail. [Pic. No. – 11a]
        Pic.No1 11a.
                                                           Golden Ratio on Dolphin.
                            .                           
Moth : The eye-like markings of this moth fall at golden sections of the lines that mark its width and length. [Pic. No. 11b]
     Pic.No11b.
                                                              Golden Ratio on Moth.
Angel fish : Every key body feature of the angel fish falls at golden sections of its width and length, such as nose, tail sections and centres of the fins firstly and indents of the dorsal and tail find as well as the top of the body secondly. It is also observed around the eye. [Pic. No. 11c]
            Pic.No11c.
                                                            Golden Ratio on Angel fish.
Penguin : The eyes, beak, wing and key body markings of the penguin all fall at golden sections of its height. [Pic. No. 11d]
                   Pic.No11d.

                                                                      Golden Ratio on Penguin.

Tiger : All key facial features of the tiger fall at golden sections of the lines defining the length and width of its face. [Pic. No. 11e]
     Pic.No11e.
                                                      Golden Ratio on Tiger face.

Ant : The body sections of an ant are defined by the golden sections of its length, its leg sections are also golden sections of its length. [Pic. No. 11f]
  Pic.No.11f.                        
                                                                  Golden Ratio on Ant.
Sea-shell : The spiral growth of Sea-shells provide a simple but beautiful example. [Pic. No. 11g]
        Pic.No11g.
                                                              Golden Ratio on Seashell.
Luca Pacioli, a contemporary of the legendary genius Leonardo Da Vinci once said, ‘Without mathematics there is no art.’
We have already found the presence of golden section in the design and beauty of nature. It can also be used to achieve beauty, balance and harmony in art and design. It’s a tool not a rule, for composition. But learning how to use it can be great art on laying out a painting on a canvas.
For those with greater understanding yet, the golden ratio can be used in more elegant ways to create aesthetics and visual harmony, in any branch of the design arts as it is found to be used by the greatest artists world has known.
But oddly enough some staunch critics may be found who say that the golden ratio cannot be found in art at all, for it has an infinite number of digits ( we may recall that (Φ  = 1.61803398875 ---- ). By a similar reasoning, a pi (π) the ratio of the circumference of a circle to its diameter, does too (π =3.14159 ---), so there can be no circle in the real world. Whatever those critics may say, for the rest of us, practical applications of mathematical concepts are a simple and everyday occurrence in arts, engineering and applied sciences.
Now we may find below the presence of golden rectangle and golden ratio in some unforgettable paintings of some legendary artists, the world has ever seen and also in some wonders of architectures strewn around the world.

Mona Lisa: By Leonardo Da Vinci (1452 – 1519)
This picture includes lots of Golden Rectangles. In this figure, we can draw a rectangle whose base extends from the woman’s right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head, then we will have golden rectangle. [Pi c. No. 12a]

Pic.No12a.

                                                   Golden Ratio in Mona Lisa.

Also if we draw squares inside this golden rectangle, we will discover that the edges of these new squares come to all the important focal points of the woman : her chin, eye, nose and the upturned corner of her mysterious mouth.

Holy Family: By Michelangelo (1475 – 1564)
We can notice that this picture is positioned to the principal figures in alignment with a Pentagram or Golden Star. [Pic. No.12b]
Pic.No12b.
                                                     Holy Family in a Golden star.
Pentagram, a star shaped symbol enclosed in a circle has a deeper meaning. Of the five points, one always pointing upward is representative of spirit and the other four points represent the elements of earth, air, fire and water. All these things are parts of life of each of us. 
Moreover, the number 5 is always been regarded as mystical and magical yet essentially human; for we have many aspects of our body consisting of 5 parts, such as fingers, toes, senses etc.

A fresco in Vatican Palace: By Raphael (1483 – 1520)                                                            
The picture is from the Stanza Della (The school of Athens) Segnatura, The room of the Segnatura is decorated by Raphael’s most famous frescoes, besides being the first work executed by the great artist in Vatican.
Raphael’s ‘School of Athens’ provides another wonderful example of the application of the golden ratio in composition. A small golden rectangle at the front and centre of the painting signals the artist’s intent in the use of this proportion. It gives the paintings a wonderful visual harmony. [Pic. No.- 12c] 
           Pic.No12c.
                                                                A fresco in a Golden Rectangle.

Self – Portrait : By Rembrandt (1606 – 1669)
We can draw three straight lines in to this picture. Then, the image of the feature is included into a triangle. Moreover, if a perpendicular line would be dropped from the apex of the triangle to the base, the line would cut the base in golden ratio. [Pic. No.12d]
       Pic.No12d.
                                                     

                                                        Self -Portrait by Rembrandt.
The Sacrament of the Last Supper: By Salvador Dali (1904 -1989)
This picture is painted inside a golden rectangle. Also, we can find part of an enormous dodecahedron above the table. Since the polyhedron consists of 12 regular pentagons, it is closely connected to the golden section. [Pic. No.12e]

Pic. No.12e


                                                      The Sacrament of the Last Supper.

Bathers: By Georges Seurat (1859-1891)
Seurat attached most of Canvas by the golden section. This picture has several golden subdivisions. [Pic. No.- 12f]

Pic. No.- 12f

                                                                 Bathers.

The Golden Stairs : By Edward Burne Jones (1833 – 1898) [Pic. No. 12 g]

        Pic.No12g.

                                                      The Golden Stairs.
The artist meticulously planned the minute of details using golden section. If we cast a close look at the painting, we may notice that golden section appear in the stairs and the ring of the trumpet carried by the fourth woman from the top. The lengths of the gowns from sash below the breast to the bottom helm hits the phi point at their knees. The width of the interior door at the back of the top of the stairs is a golden section of the width of the top of the opening of the sky light. So, there are a lot of things of the golden ratio in the picture!
The above examples are only a very few. There are a pretty large number of applications of this golden ratio in paintings of many legendary artists.
We may also be surprised to find the traces of this golden ratio in construction marvels of many famous monuments, wonders of the world, temples, chapels, basilicas, places of worship, even in some modern buildings and complexes, all over the world.
We may cite some examples of these marvels.
The great pyramid of Giza in Egypt is the most substantial ancient structure in the world and most mysterious also. The three pyramids on the Giza plateau are funerary structures of three Egyptian kings of the fourth dynasty during 2575 to 2465 B.C.
The great pyramid was originally 481 feet 5 inches tall and measured 755 feet along its sides, covering an area of 13 acres, constructed from approximately 2.5 million lime stone blocks weighing on an average of 2.6 tons each.
The ancient Egyptians were first to use mathematics in art. It seems almost certain that they ascribed magical proportion of the golden section and used in the design of their great pyramids.
If we take a cross section of the Great pyramid, we get a right-angled triangle, the so called Egyptian triangle. The ratio of the slant height of the pyramid, here the hypotenuse of the triangle, to the distance from the ground centre, which is half the dimension, is 1.61804 and surprisingly it matches with the value of phi, the golden ratio.[Pic. No.13a]

Pic. No.13a
                                         The Great Pyramid of Giza.
The Egyptians thought the golden ratio is sacred. Therefore it was important in their religion. They used the golden ratio in building temples and places for the dead. They were aware that they were using the golden ratio but they called it ‘Sacred ratio’.
We have evidences that Greeks were also acquainted with golden ratio from ancient times. AS an example, they applied it in the architecture of Parthenon,Chief temple [Pic. No.13b] of the Greek goddess Athena, the virgin, on the hills of Acropolis at Athens. It was built in the mid. 5th century BC. The name Parthenon refers to the cult of Athena Parthenon. Work began in 447 BC and it was completed by 438 BC. The same year a great gold and ivory statue of Athena, made by Phidias for the interior, was dedicated. Work on the exterior decoration of the building continued until 432 BC. One interesting aspect is that, all temples in Greece were designed to be seen only from the outside. The viewers never entered a temple and could only glimpse the interior statues through the open doors. The Greek sculpture Phidias sculpted many things including the bands of sculpture that run above the column of the Parthenon. The Parthenon was perhaps the best example of mathematical approach to art. Once its ruined triangular pediment is restored, the ancient temple site almost perfectly in to a golden rectangle.
Pic.No13b.
                                                                     Parthenon.
Now let us explore a monument of immeasurable beauty, the Taj Mahal, one of the Seven Wonders of the World and now a UNESCO world heritage site, standing majestically on the bank of the River Yamuna flowing by the historically famous city of Agra in India. The visceral charisma it emanates is closely related to golden section. [Pic. No.-13c]
        
               Pic. No.-13c

                                 Golden Ratio in Taj Mahal.


The Taj Mahal is synonymous to love and romance and considered to be an “epitome of love”. The name ‘Taj Mahal’ was derived from the name of the Mughal emperor Shah Jahan’s wife, Mumtaz Mahal and means “Crown Palace”.
The purity of the white marble, the exquisite ornamentation, precious gemstones and its picturesque location, all make a visit to the Taj Mahal gain a place amongst the most sought after tours in the world.
Here one thing is to be kept in mind that it is not just an outstanding beautiful building monument, it is the love behind which has given a life to it.
But why are we so amazed with the beauty of Taj Mahal? For at the brink of dawn when the first rays of the sun hits the dome of this epic monument, it radiates like a heavenly abode, cloaked in bright golden and at dusk, basking in the glory of moon, it shines like a perfectly carved diamond, leaving the viewers awe struck by its sense of grandeur.
 Taj Mahal was built in 22 years (1631 - 1653). 20000 workers laboured and 32 crores of rupees spent, which may amount now to the budget of our one of the five-year plans of several lakh crores rupees.
The architectural of the Taj Mahal mausoleum amazes every visitor for more than three and a half hundred years. The complex consists of:
The mausoleum ensembles with four minarets stand on the quadrant podium approximately 120 x 120 m.
Two buildings aside of the ensemble standing symmetrical to it, the mosque and Jamaat Khana, meaning meeting place.
And the garden is approximately 320 x 320 m.
Huge scale of the whole complex is only an accompaniment to the refined architecture of the tomb ensemble.
Decoded geometrical structure of the Taj Mahal mausoleum’s architectural composition reveals that the idea of the Taj Mahal is rationally and intently based on golden section, as so as all non-Golden regularities are hierarchically dependent to the Golden ones. Main entrance door frame rectangle is accurately golden rectangle. Geometry of the arc inside it is as well built after golden regularities.
Geometry of the garden plan, places and sizes of the buildings are clear and nothing but rectangle. But it is hard even to compare the simplicity of the master plan with high complexity of the proportional hierarchy of the tomb ensemble’s composition.
Another example of application of golden ratio in architecture, is the Eiffel Tower, an engineering marvel at Champ-de-Mars, a public large green space by the side of the river Seine, located in central Paris, France. [Pic. No.- 13d]

Pic. No. 13d                 


                                       Golden Ratio in Eiffel Tower.
          
It was built as an entrance to “Exposition Universelle” – the World’s Fair, hosted in 1889 to mark 100 year anniversary of the French Revolution. The name ‘Eifel Tower’ derived from the name of its builder Alexandre Gustave Eiffel, an acclaimed bridge builder, architect and metal expert.
Of the plans submitted by more than 100 artists to build the monument for the fair, Alexandre Gustave Eiffel’s consulting and construction firm was finally entrusted with the job.
The final design called for more than 18000 pieces of a special type of wrought iron used in constructions and 2.5 million rivets assembling the framework of the iconic lattice tower, which at the time of inauguration in March 1889 stood nearly 1000 ft. high and the tallest structure of the world at that time. Later in 1957, an antenna was added that increased the structure’s height by 65 feet. Initially, only the Eiffel tower’s second platform was open to the public, latter all three levels, two of which now feature restaurants, would be reachable by stairway or one of the eight elevators.
Millions of visitors during and after the World’s Fair marveled at Paris’ newly erected architectural wonder. But many Parisians were of different opinion. They thought it was structurally unsound and even some considered it an eyesore. For example, the noted novelist Guy de Maupassant allegedly hated the tower so much that he often ate lunch in the restaurant at its base choosing a vantage point to sit, so that he could completely avoid glimpsing its looming silhouette.
The Eiffel Tower was originally intended as a temporary exhibit and so it was almost dismantled and scrapped in 1909. But it was realized by the civic authority later that this structure may be used to set up a radiotelegraph station on it. So it was restored. During World War – I, this tower intercepted enemy’s many radio communications and helped a lot in army operations. During World War – II Hitler ordered to destroy it but that order was never carried out and this iconic tower was saved for the second time.
Since then much water had flown through the river Seine, Eiffel Tower has been the site of numerous high profile stunts, ceremonial events and even scientific experiments. Now one of the most recognizable structures in the world, the Eiffel Tower underwent a major face-list in 1986 and is repainted in every seven years. It welcomes more visitors than any other monument in our globe – an estimated average of 7 million people per year The Eiffel Tower also inspired more than 30 replicas and similar structures in various cities around the world.
But keeping aside all praises, criticisms and apathies concerning the Eiffel Tower, we are curious to know whether the golden ratio has any role to play in the construction of this structure. Surprisingly yes! The taper of the tower at major subdivisions – the first stage, second stage, the intermediate platform and the third stage or top appears to follow multiples of golden ratio.
The ratio of the width of the base to the width of tower of the first stage is 1.62 (fairly equals to the value of the golden ratio, phi). At the second stage, it is 2 x 1.62, at the intermediate platform, 4 x 1.62 and at the top about 8 x 1.62.
There are much more examples of application of golden ratio in architecture. Some of them are worthy to be mentioned as:
The famous, historic Notre Dame Cathedral in Paris, the United Nation’s Secretariat building in New York, the CN Tower in Toronto the tallest and free standing structure in the world.
It is found surprisingly that the number of atheists – namely, the persons, who disbelieve in the existence of a creator like God and hence belong to no religion, is so insignificant that may be ignored altogether. Almost all people of the world population belong to one or other religion. Again there lies a creator, as they believe at the core of every religion who is behind this beautiful, fantastic yet perilous universe. Many scientist of world fame are no exceptions.
Some specific secret rules and surprising orders those exhibited in a multitude of shapes and patterns whose relationship can only be the result of an omnipotent, good and all-wise God of scripture. This secret is now attributed to the role of Divine Proportion, existing in the smallest to the largest parts, in living and also in non-living things --- reveals the awesome handiwork of God and His interest in beauty, function and order.
We find things of various shapes around us in the nature of which spiral shape is most common and familiar to us. It is visible as diverse as shells, hurricanes, spiral seeds, the cochlea of the human ear, ram’s horn, the curl of an elephant tusk, sea-horse tail, growing fern leaves, DNA molecules, waves breaking on the beach, tornado, galaxies, distribution of planets in the solar system, the tail of a comet as it winds around the sun, whirlpools, seeds patterns of many flowers and so many. These spirals follow a precise mathematical pattern.
We have already seen that there are many many evidences of the presence of golden ratio in the nature around us as well as in arts, architecture etc. But if we think on a larger perspective, we find that this divine ratio has a great role to play on the functioning of our universe also.
Albert Einstein (1879 – 1955), perhaps the greatest scientist of the twentieth century, has proved beyond doubt that the vast space of this universe in which lie about 170 billion galaxies containing two to four billion stars, with their satellites, six on an average including our solar system – our earth and of course we ourselves, the vast inter stellar space etc. behaves like a sheet of rubber and hence bend with the weight of the heavenly bodies lying on it.
His epoch making discovery that time is interwoven with this space and time to be regarded as the fourth dimension. The other three, as we know, are length, breadth and height of a material body. The space and time together termed as “Space Time”. Modern researchers claim that this universe is governed by the divine ratio, the space-time itself is defined by this mathematical constant ratio of 1.618, the golden ratio.
On the perspective of the “Multiverse” theory (many universes like that of ours), they suggested that our universe to have this ratio that allowed it to form as is observed now.
But can all the natural phenomena be bounded by the Fibonacci numbers only? The answer would be an emphatic ‘No’. The whole of the universe is definitely guided by some mathematical order but little of them are within our reach. Up till now, we are able to unravel out only a bit of the mysteries of the nature, which is nothing but a tip of an ice berg. The most are shrouded in darkness. There are 100s of what, why, when, where and how.
So, if you find the numbers 1, 2, 3 and 5 occurring some where it does not always mean the Fibonacci numbers although they could be.
There are many examples where Fibonacci numbers do not appear in plants. For example:
a 4 leaved clover, a Fuchsia has 4 petals and 4 sepals, some white sweet peppers have 4 chambers. 
There are many flowers with six petals, such as crocus, narcissus, amaryllis, blue-eyed grass. [
                                                                               
Succulents with clear arrangements are of 4 spirals in one direction and 7 in the other or 11 and 18 spirals.
Many cacti do not show Fibonacci numbers.
So, it is clear that not all plants show Fibonacci number but another series of numbers, called Lucas numbers. But what is it?
  The French mathematician Edward Lucas (1842 – 1891), who gave the series 0, 1, 1, 2, 3, 5, 8, - - - , the name Fibonacci numbers found a similar series occurs often when he was investigating Fibonacci numbers patterns, which is as 2, 1, 3, 4, 7, 11, 18, 29 - - -
The Lucas numbers have lots of properties similar to those of Fibonacci numbers. In fact, for every series formed by adding the latest two values to get the next, and no matter what two positive numbers we start with, we will always end up having terms whose ratio is 1.6189339 - - -- our well known golden ratio!
In fact, no matter what two numbers we begin with, the ratio and two successive numbers in all these Fibonacci type sequences always approach a spiral value the golden mean and thus seems to be the secret behind the series.
Hence it is amply clear that some plants do not manifest Fibonacci numbers but sometimes Lucas numbers or even more anomalous sequence 5, 1,4, 5, 7 - - - - or 5, 2, 7, 9, 16 - - -
So, we should admit that phyllotaxis is really not a universal law but fascinatingly a prevalent tendency.

                                                                                                                   [To continue]
         Reference Internet: Images are downloaded from Public Domain (except otherwise stated)
         Pic.No.5 Leaf arrangements in Sunflower plant
         Dr.Ron Knott: ( http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
       Attribution:
       Pic.No 9a,b,c---Golden Ratio in human body,hand and face respectively
       International Journal of Arts 2011 (https://pdfs.semanticscholar.org/1f56/48a45169c011c82a23273f30cc297850484a.pdf )
      Image credit:Pic.No 11a,b,c,d,e,f,g---Golden Ratio on Dolphin, Moth,Angel fish,Penguin,Tiger face,Ant,Shell respectively: J.W. Wilson.
  (  http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Hekimoglu/emat6700/golden%20ratio/animalgold.html )
                           Pic.No 12 a,b,c,d,e,f,---Golden Ratio in Mona Lisa,Holy Family,Fresco in Vatican Palace,self portrait by Rembrandt,Last Supper,Bathers Pic.No 13a Great Pyramid in Giza,13b Parthenon: Samuel Obara (ref.J. Wilson )
( http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio /golden.html)
              
 Pic.No13c.  Goloden Ratio shown on Taj Mahal
 Image credit: Wikimedia Cosmos (https://en.wikipedia.org/wiki/Taj_Mahal )
  Pic.No13d.Golden Ratio shown on Eiffel Tower
 Image credit:Wikimedia Cosmos
( https://en.wikipedia.org/wiki/Eiffel_Tower#/media/File:Tour_Eiffel_Wikimedia_Commons_(cropped).jpg )




























































































































                                 Golden Ratio in Taj Mahal.
                                       
The Taj Mahal is synonymous to love and romance and considered to be an “epitome of love”. The name ‘Taj Mahal’ was derived from the name of the Mughal emperor Shah Jahan’s wife, Mumtaz Mahal and means “Crown Palace”.
The purity of the white marble, the exquisite ornamentation, precious gemstones and its picturesque location, all make a visit to the Taj Mahal gain a place amongst the most sought after tours in the world.
Here one thing is to be kept in mind that it is not just an outstanding beautiful building monument, it is the love behind which has given a life to it.
But why are we so amazed with the beauty of Taj Mahal? For at the brink of dawn when the first rays of the sun hits the dome of this epic monument, it radiates like a heavenly abode, cloaked in bright golden and at dusk, basking in the glory of moon, it shines like a perfectly carved diamond, leaving the viewers awe struck by its sense of grandeur.
 Taj Mahal was built in 22 years (1631 - 1653). 20000 workers laboured and 32 crores of rupees spent, which may amount now to the budget of our one of the five-year plans of several lakh crores rupees.
The architectural of the Taj Mahal mausoleum amazes every visitor for more than three and a half hundred years. The complex consists of:
The mausoleum ensembles with four minarets stand on the quadrant podium approximately 120 x 120 m.
Two buildings aside of the ensemble standing symmetrical to it, the mosque and Jamaat Khana, meaning meeting place.
And the garden is approximately 320 x 320 m.
Huge scale of the whole complex is only an accompaniment to the refined architecture of the tomb ensemble.
Decoded geometrical structure of the Taj Mahal mausoleum’s architectural composition reveals that the idea of the Taj Mahal is rationally and intently based on golden section, as so as all non-Golden regularities are hierarchically dependent to the Golden ones. Main entrance door frame rectangle is accurately golden rectangle. Geometry of the arc inside it is as well built after golden regularities.
Geometry of the garden plan, places and sizes of the buildings are clear and nothing but rectangle. But it is hard even to compare the simplicity of the master plan with high complexity of the proportional hierarchy of the tomb ensemble’s composition.
Another example of application of golden ratio in architecture, is the Eiffel Tower, an engineering marvel at Champ-de-Mars, a public large green space by the side of the river Seine, located in central Paris, France. [Pic. No.- 13d]

Pic. No. 13d                 
                                      Golden Ratio in Eiffel Tower.          
It was built as an entrance to “Exposition Universelle” – the World’s Fair, hosted in 1889 to 
mark 100 year anniversary of the French Revolution. The name ‘Eifel Tower’ derived from the name of its builder Alexandre Gustave Eiffel, an acclaimed bridge builder, architect and metal expert.
Of the plans submitted by more than 100 artists to build the monument for the fair, Alexandre Gustave Eiffel’s consulting and construction firm was finally entrusted with the job.
The final design called for more than 18000 pieces of a special type of wrought iron used in constructions and 2.5 million rivets assembling the framework of the iconic lattice tower, which at the time of inauguration in March 1889 stood nearly 1000 ft. high and the tallest structure of the world at that time. Later in 1957, an antenna was added that increased the structure’s height by 65 feet. Initially, only the Eiffel tower’s second platform was open to the public, latter all three levels, two of which now feature restaurants, would be reachable by stairway or one of the eight elevators.
Millions of visitors during and after the World’s Fair marveled at Paris’ newly erected architectural wonder. But many Parisians were of different opinion. They thought it was structurally unsound and even some considered it an eyesore. For example, the noted novelist Guy de Maupassant allegedly hated the tower so much that he often ate lunch in the restaurant at its base choosing a vantage point to sit, so that he could completely avoid glimpsing its looming silhouette.
The Eifel Tower was originally intended as a temporary exhibit and so it was almost dismantled and scrapped in 1909. But it was realized by the civic authority later that this structure may be used to set up a radiotelegraph station on it. So it was restored. During World War – I, this tower intercepted enemy’s many radio communications and helped a lot in army operations. During World War – II Hitler ordered to destroy it but that order was never carried out and this iconic tower was saved for the second time.
Since then much water had flown through the river Seine, Eifel Tower has been the site of numerous high profile stunts, ceremonial events and even scientific experiments. Now one of the most recognizable structures in the world, the Eifel Tower underwent a major face-list in 1986 and is repainted in every seven years. It welcomes more visitors than any other monument in our globe – an estimated average of 7 million people per year The Eifel Tower also inspired more than 30 replicas and similar structures in various cities around the world.
But keeping aside all praises, criticisms and apathies concerning the Eifel Tower, we are curious to know whether the golden ratio has any role to play in the construction of this structure. Surprisingly yes! The taper of the tower at major subdivisions – the first stage, second stage, the intermediate platform and the third stage or top appears to follow multiples of golden ratio.
The ratio of the width of the base to the width of tower of the first stage is 1.62 (fairly equals to the value of the golden ratio, phi). At the second stage, it is 2 x 1.62, at the intermediate platform, 4 x 1.62 and at the top about 8 x 1.62.
There are much more examples of application of golden ratio in architecture. Some of them are worthy to be mentioned as:
The famous, historic Notre Dame Cathedral in Paris, the United Nation’s Secretariat building in New York, the CN Tower in Toronto the tallest and free standing structure in the world.
It is found surprisingly that the number of atheists – namely, the persons, who disbelieve in the existence of a creator like God and hence belong to no religion, is so insignificant that may be ignored altogether. Almost all people of the world population belong to one or other religion. Again there lies a creator, as they believe at the core of every religion who is behind this beautiful, fantastic yet perilous universe. Many scientist of world fame are no exceptions.
Some specific secret rules and surprising orders those exhibited in a multitude of shapes and patterns whose relationship can only be the result of an omnipotent, good and all-wise God of scripture. This secret is now attributed to the role of Divine Proportion, existing in the smallest to the largest parts, in living and also in non-living things --- reveals the awesome handiwork of God and His interest in beauty, function and order.
We find things of various shapes around us in the nature of which spiral shape is most common and familiar to us. It is visible as diverse as shells, hurricanes, spiral seeds, the cochlea of the human ear, ram’s horn, the curl of an elephant tusk, sea-horse tail, growing fern leaves, DNA molecules, waves breaking on the beach, tornado, galaxies, distribution of planets in the solar system, the tail of a comet as it winds around the sun, whirlpools, seeds patterns of many flowers and so many. These spirals follow a precise mathematical pattern.
We have already seen that there are many many evidences of the presence of golden ratio in the nature around us as well as in arts, architecture etc. But if we think on a larger perspective, we find that this divine ratio has a great role to play on the functioning of our universe also.
Albert Einstein (1879 – 1955), perhaps the greatest scientist of the twentieth century, has proved beyond doubt that the vast space of this universe in which lie about 170 billion galaxies containing two to four billion stars, with their satellites, six on an average including our solar system – our earth and of course we ourselves, the vast inter stellar space etc. behaves like a sheet of rubber and hence bend with the weight of the heavenly bodies lying on it.
His epoch making discovery that time is interwoven with this space and time to be regarded as the fourth dimension. The other three, as we know, are length, breadth and height of a material body. The space and time together termed as “Space Time”. Modern researchers claim that this universe is governed by the divine ratio, the space-time itself is defined by this mathematical constant ratio of 1.618, the golden ratio.
On the perspective of the “Multiverse” theory (many universes like that of ours), they suggested that our universe to have this ratio that allowed it to form as is observed now.
But can all the natural phenomena be bounded by the Fibonacci numbers only? The answer would be an emphatic ‘No’. The whole of the universe is definitely guided by some mathematical order but little of them are within our reach. Up till now, we are able to unravel out only a bit of the mysteries of the nature, which is nothing but a tip of an ice berg. The most are shrouded in darkness. There are 100s of what, why, when, where and how.
So, if you find the numbers 1, 2, 3 and 5 occurring some where it does not always mean the Fibonacci numbers although they could be.
There are many examples where Fibonacci numbers do not appear in plants. For example:
a 4 leaved clover, a Fuchsia has 4 petals and 4 sepals, some white sweet peppers have 4 chambers. [Pic. No.-7(d)]
There are many flowers with six petals, such as crocus, narcissus, amaryllis, blue-eyed grass. [Pic. No.-7(f)]



                                                                          
Succulents with clear arrangements are of 4 spirals in one direction and 7 in the other or 11 and 18 spirals.
Many cacti do not show Fibonacci numbers.
So, it is clear that not all plants show Fibonacci number but another series of numbers, called Lucas numbers. But what is it?
  The French mathematician Edward Lucas (1842 – 1891), who gave the series 0, 1, 1, 2, 3, 5, 8, - - - , the name Fibonacci numbers found a similar series occurs often when he was investigating Fibonacci numbers patterns, which is as 2, 1, 3, 4, 7, 11, 18, 29 - - -
The Lucas numbers have lots of properties similar to those of Fibonacci numbers. In fact, for every series formed by adding the latest two values to get the next, and no matter what two positive numbers we start with, we will always end up having terms whose ratio is 1.6189339 - - -- our well known golden ratio!
In fact, no matter what two numbers we begin with, the ratio and two successive numbers in all these Fibonacci type sequences always approach a spiral value the golden mean and thus seems to be the secret behind the series.
Hence it is amply clear that some plants do not manifest Fibonacci numbers but sometimes Lucas numbers or even more anomalous sequence 5, 1,4, 5, 7 - - - - or 5, 2, 7, 9, 16 - - -
So, we should admit that phyllotaxis is really not a universal law but fascinatingly a prevalent tendency.