12/05/2023
It all starts with Fibonacci
Leonardo Fibonacci, also known as Leonardo Pisano or Leonard of Pisa, was the leading mathematician in Europe during the Middle Ages. Fibonacci was born around 1170 to a wealthy family in Pisa. In his youth, Fibonacci spent time in Algeria, where he learned computation from a Muslim schoolmaster. This marked Fibonacci's exposure to the Indo-Arabic numeration system and computational techniques, which formed the foundation for Fibonacci's groundbreaking contributions to mathematics. The Indo-Arabic numeration system is a base 10 system that included zero and also allowed for decimal representation. This system was far more robust than the Roman numeral system which was used well into the Late Middle Ages.
In 1202, what would come to be known as the Fibonacci sequence was born when the book "Liber Abaci" was published. The contents of Fibonacci's book consisted of arithmetic and elementary algebra. Notably, this work introduced the Indo-Arabic numeration system and arithmetic algorithms to Europe. One of the problems that Fibonacci included in "Liber Abaci" posed a particular problem about the population of rabbits, which is as follows:
Suppose there are two newborn rabbits, one male and the other female. Find the number of pairs of rabbits
produced in a year if: 1) each pair takes one month to become mature; 2) each pair produces a mixed pair every month, from the second month on; and
3) no rabbits die during the course of the year.
The progression of the total number of pairs of rabbits throughout the first year correspond to the first 12 Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. Notice that each term of the sequence, except for the first two, is the sum of the two previous terms. We call this type of relationship a recurrence. The first two terms 0 and 1 are assumed, and these are called the initial conditions. So we can generalize the Fibonacci sequence with a linear recurrence defined by F(n) = F(n-1) + F(n-2) where n denotes the index of the sequence term.
It remains unknown whether or not Fibonacci was aware of the recursive nature of this sequence. The sequence, however, transcends Fibonacci's era, as evidenced by its appearance as a special case in a formula established by Narayana Pandita in 1356 A.D., and possibly even much earlier work by other Indian mathematicians. The enduring legacy of Fibonacci's work continues to captivate mathematicians and other creatives with its inextricable link to natural phenomena.
A sister sequence is born
The Fibonacci sequence was given its name in 1876 by world famous French mathematician Francois-Edouard-Anatole-Lucas. He studied the Fibonacci sequence extensively, in addition to other sequences with the same general recursive formula S(n) = S(n-1) + S(n-2). His own particular sequence of this form is called the Lucas sequence, and it has initial conditions L(0) = 2, L(1) = 1. What was particularly exciting about Lucas' new sequence was the wealth of relationships that link these sequences together as "sister" sequences. For example, it is well known that the ratio of successive terms of the Fibonacci sequence tends towards the golden ratio, which is an irrational number approximated by 1.618033988748. What developed from Lucas' work has been over a century of continued study of the Fibonacci and Lucas sequences in addition to many other related sequences, and the study of the relationships between them. This is the work that has inspired my own McNair research with Dr. aBa Mbirika. As a new McNair scholar, the opportunity to contribute to such a historically robust body of mathematical literature François Édouard Anatole Lucas (4 April 1842-3 October 1891)
is exciting and inspiring. Both aBa and I are excited to see
where this research leads us.
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