In 12th century a man named Leonardo, Fibonacci took an active interest in the number of rabbits that could be raised in a year if one begins with a single pair. Fibonacci reasoned that if the rabbits reached maturity two months after birth and produced an additional player every month thereafter, the population of the rabbits would increase monthly according to the series


This deceptively simple looking series is obtained by sequentially adding the numbers where each number is the sum of its preceding two numbers.

The ratio of each successive pairs of numbers approximates Phi (where phi =1.1618) as 5 divided by 3 is 1.666 and 8 divided by 5 is 1.60. The ratio of successive numbers in the Fibonacci series quickly converges on Phi where is called the Golden ratio. After the 40th number in the series the ratio is accurate to 15 decimal places.

The Golden ratio consists of two numbers 1.618034 and .618034 each of which is reciprocal of the other. Rectangles with sides proportioned to .618034 to 1 (1.618034 to 1) are often the shape taken by commonplace items as picture frames and playing cards. Both of these are remarkable numbers. Not only are the figures after the decimal point identical in both, but each is reciprocal of the other ( i.e 1 divided by either of the number will yield the other one). These are the only two numbers in the entire universe that demonstrate this unique property.

Thus the shape, appears subliminally pleasing to the human eye , as witnessed by many by the ways it is used in various forms of art and architecture. Evidences of it are abundant in nature, reflected in essentially every form of a snail shell to the arms of a galaxy.


Golden Ratio In Nature And Civilization

The Golden ratio has an ubiquitous presence in nature and in various civilizations. Some are so mundane in our everyday life, that they get ignored. The number of spiral arms seen in two individual florets of several flowers were found to be 21 and 34 ; two consecutive Fibonacci numbers. There are two spirals of respectively 5 and 8 arms on a pine cone whereas a pineapple has spirals of 5, 8 and 13 – all are Fibonacci numbers. Such examples are so commonplace that they remain largely unnoticed.


The earliest instance of human beings fascination with the Golden Ratio or the Divine proportion was seen in the symmetry of the great Pyramid of Gaza which seemed to have been built with a height and base ratio of 5 is to 8 (.625) very close to the Golden Ratio. Whether or not the Egyptians were actually cognizant of this ratio and its ramifications, remain debatable.  


However this ratio was extremely prevalent in ancient Greece . Evidences of this ratio are abundant in their texts and writings, where it is referred to as the Golden Ratio. Though the broader implication of this ratio remained unknown to them, the ratio’s subliminal attraction compelled the Greeks to use it freely in various arts and architectural pieces (including the iconic Pantheon, widely regarded as the finest structure of ancient times


However after the collapse of Hellenic culture, the Fibonacci numbers lost its significance for a long time, till it was made famous again by the maverick genius Leonardo Da Vinci. Da Vinci liberally used this ratio in many of his paintings and sculptures and very soon several other masters followed suit and the ratio again gained prominence in artistic circles. However it should be noted that the Golden ratio’s charm is not merely confined to the artistic eye, musical harmonies are also derived from the Golden ratio.


The human anatomy also displays clear evidence of this perfect ratio. The human hand is an example of a perfect golden proportion. Another example of the Golden ratio is spiral arm of galaxies – the well known galaxy M51.

Since the entire Fibonacci concept can be easily comprehended by non-mathematicians also, the numbers have been embraced by popular culture. The numbers have cropped up  in books ( Da Vinci Code ) in films (Darren Arnofosky’s  Phi) and in various television shows like (NUM3RS , Fringe) . A curious correlation with the financial markets as regards to stock price movements, price and demand elasticity has also been seen, substantiating the omnipresence of these numbers.

The Golden ratio or the Divine proportion are patterns found in nature over and over again. The curves of a wave , spirals of a shell , segments of a pineapple or in our own DNA , the buildings blocks of our lives – the entire universe is made up of precise ratios and patterns. They are all around us . We do not see them, but if we could life would  be magical beyond comprehension, a quantum entanglement of cause and effect where everything and everyone would reflect on each other.