Fibonacci, his series, and φ

Leonardo Bonacci, born in Pisa around the year 1170, was also known in his lifetime as Leonardo Pisano or Leonardo Bigollo Pisano ('Leonardo of Pisa' or 'Leonardo the Traveller from Pisa.' It wasn't until around 600 years after his death that a Franco-Italian historian, Guillaume Libri, invented the name Fibonacci for him, short for "filius Bonacci" - 'son of Bonacci.' Why his claim to fame would have derived from that of his merchant father is a mystery; his last name "Bonacci" by itself indicates that he was well-born. His major contribution to science and society was the popularization of the Hindu-Arabic number system that, through his advocacy, replaced both Roman numerals and abacuses as a means of doing calculations. His book "Liber Abaci" 'Book of Calculation' is dedicated to this effort; it is full of sample problems illustrating the power of doing calculations using Hindu-Arabic numerals instead of Roman ones. 

One of those problems concerned the growth of a population of rabbits under a set of ideal (and unrealistic) conditions: If a pair of baby rabbits are put into a field, how many pairs will there be at the end of each month, and at the end of one year? Assume that the rabbits are fully grown at one month and have a pair of baby rabbits at two months, that each pair is a male and a female, and that the rabbits don't die. The solution to this problem generates the Fibonacci series; after the initial two months, the number of pairs at the end of each month is equal to the sum of pairs in the preceding two months, viz 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144 at the end of the year.

So Leonardo is famous for the answer to one of many examples given in his book on calculations and not for the major contribution of the book itself — the near-universal adoption of the Hindu-Arabic numeral system. And he is known by a name never used in his lifetime.

The Fibonacci series begins with the terms 0 and 1 and continues such that each successive term is equal to the sum of the two previous ones. The ratio of successive terms converges to φ, the golden mean (1.618033988749894848204586834...).  φ is a magic number like π and e; it is irrational and cannot be expressed as a ratio of integers. But like other mathematical constants, it is well-defined and can be determined to any required precision. Calculating the ratio of successive terms in the continuation of the series leads one ever closer but never reaches the exact value of φ.

The series and φ crop up repeatedly in nature, found in spirals of sunflower seeds and pine cones, branching of trees, arrangement of leaves, and etcetera. 

Fibonacci was not the first to describe the series; that distinction belongs to Indian mathematicians of the first millennium. He also did not observe that the ratio of successive terms rapidly converges to φ; that observation was made first by Simon Jacob and then by Johannes Kepler instead. 

Not only is φ implicit in the Fibonacci series, but the series crops up in the powers of  φ. Viz:

φ1  =                               1φ+0

φ2 = φ*(1φ+0) = 1φ2+0φ = 1φ+1+0φ = 1φ+1

φ3 = φ*(1φ+1) = 1φ2+1φ = 1φ+1+1φ = 2φ+1

φ4 = φ*(2φ+1) = 2φ2+1φ = 2φ+2+1φ = 3φ+2

φ5 = φ*(3φ+2) = 3φ2+2φ = 3φ+3+2φ = 5φ+3

φ6 = φ*(5φ+3) = 5φ2+3φ = 5φ+5+3φ = 8φ+5