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About Fibonacci

The Fibonacci numbers first appeared in a work written in ancient India sometime between 450 and 200 BC. The 13th century Italian mathematician Leonardo Pisano Bigollo described one origin of the series in his work Liber Abbaci.


Leonardo's father's name was Guglielmo Bonaccio. Leonardo was given the invented name Fibonacci in the nineteenth century. Fibonacci is short for  "filius Bonacci"  latin for "son of Bonacci". 


In Leonardo's mathematical treatise Liber Abbaci he described a hypothetical problem and its solution as follows. A pair of rabbits produce another pair in a single month and at the end of the second month the new pair also produce a pair of rabbits.  This continues each month for each pair of rabbits. At the end of the first month there are two pairs of rabbits, the second month three pair, the third month five pair, the fourth month 8 pair, the fifth month 13 pair and so on to create the Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.. 



 In the Fibonacci series each number is equal to the sum of the previous two numbers. In the Fibonacci game the two rabbits on a base card illustrate the possibility of the start of a set.


The ratio of one number to the prior number as the series becomes longer approaches an irrational number known as Phi, which is close to 1.618. For example 13/8 = 1.625 and 144/89 = 1.618.


Phi is also known as the Golden Ratio or the Divine Proportion. The Golden Ratio occurs in nature and across the universe and is believed to create order in the relationship of one thing to another. It has been used by architects, sculptors, painters and designers to create pleasing proportions in their work for many centuries.


The Fibonacci numbers appear in nature to an extent not likely due to chance. For example, the number of petals on different flowers is often a Fibonacci number. 


The Golden Ratio first appeared in Euclid's Elements written around 350 BC. The connection between the Golden Ratio and the Fibonacci series was first verified in the nineteenth century.


The mathematical relationship defined by the Golden Ratio is shown in the diagram.



A great deal of information is available on the internet about the Fibonacci series, the Golden Ratio and related matters. Search with the word Fibonacci. The Fibonacci Association, incorporated in 1963, focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas. Their web site is



2011 by Robin Andrews

Patent Pending