The Golden Mean & Fibonacci

The Golden Mean or the Golden Ratio or the Divine Proportion or just call it Phi has fascinated intellectuals throughout recorded history. The golden ratio can be obtained by dividing a line into 2 parts, such that the ratio of length of the original line to the larger section is the same as the ratio of the larger section to the smaller section.

A/ B = B / C = 1.618…

This ratio is the number Phi (1.618…). It can also be arrived at by dividing a large Fibonacci series number by its previous number in the series.

The beauty of this number however lies in its applicability. It provides a beautiful link between symmetry and asymmetry. The ancient Greeks used it to lend beauty of proportion to their architecture. Leonardo Da Vinci found the golden mean in the human body through his anatomical studies – the ratio of length of human body below and above the navel is the golden ratio. It is also found in the length of bones as one travels from the tip of the finger up the length of a human hand. The renaissance painters are supposed to have used it extensively. The ratio of adjacent sides of a credit card works out within 2% of the golden mean. The horizontal member of most christian crosses splits the vertical member according to the golden mean. The General Assembly building of United Nations in New York also makes extensive use of this ratio.

This brings us to Fibonacci. How exactly did he discover the golden mean? Well, he was tackling a problem related to breeding of rabbits when he chanced upon the now famous Fibonacci series and subsequently the golden mean. The problem concerned the number of rabbits that would be born in a year from an original pair of rabbits, assuming that every month each pair produces another pair and that rabbits begin to breed when they are 2 months old. While solving this he found that the number of pairs every month after the fourth month would be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 243 and so on. The beauty of this series is that each successive number is the sum of the two preceeding numbers. If the rabbits kept up the pace for a hundred months, the total number of pairs would be a whopping 354,224,848,179,261,915,075!!!

But more than anything else, Fibonacci probably established the significance and utility of the Arab numeric system in the eyes of the Europeans. His publication of the Liber Abaci not only intoduced the new numeral system to Europeans, it also demonstrated ways of using this system to solve practical issues like bookkeeping, conversion of weights, etc.

Many people believe that the Fibonacci numbers can be used to make a variety of predictions, especially about the stock market. Although the concept of risk and its measurement evolved in the fifteenth century, Fibonacci in the 12th century perhaps enabled the first step in making measurement a key factor in the taming of risk.

And by the way, Fibonacci means Blockhead. The real name of the mathematician is Leanorado Pisano, and he was anything but a blockhead.

Post inspired by the first few chapters of “Against the Gods” by Peter L. Bernstein.