## The Numbers That Follow You Everywhere

Leonardo Fibonacci, also known as Leonardo Bonacci or Leonardo of Pisa, was an Italian mathematician and considered one of the most talented in the middle ages. He is most known for creating the Fibonacci sequence and is often overlooked for the creation of the Liber Abaci, a book of calculations published in 1202. This book not only revoluntionised mathematics and is the reason why we have a standardised and efficient numeral system, but also provided us with the Fibonacci Sequence which is abundant in our world such as in art, cells and animals. By digging into this mathematician’s work, we can discover one of the most unique and universal series of numbers.

Born to a customs officer, Fibonacci spent his childhood in North Africa and was sent to an Arab master to learn calculations. Later, he frequently travelled to countries such as Algeria, Egypt, and Syria, where he studied their domestic numeral systems. Upon his return, he decided to write the Liber Abaci which was initially intended to help merchants in his small town. However, this book grew to be one of the most important books in the history of European mathematics! It introduced the western world to the logical and efficient systems of calculation and the Hindu- Arabic numeral system  (1, 2, 3, 4, 5, 6, 7, 8, 9, 0)  and we still use these concepts today. The book starts with an explanation of how to use this positional decimal numeral system and is followed by an explanation of calculations such as multiplication, addition, simultaneous equations and quadratics using it. Although the Liber Abaci was a considerable success, Fibonacci is better known for one example from the book, the Fibonacci Sequence.

While writing his book, Leonardo pondered the question: Given ideal conditions, how many pairs of rabbits could be produced from a single pair of rabbits in one year? The conditions were that the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. Using this information he started to construct his series in this way: At the end of the first month, they mate, but there is still one only 1 pair. ; At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field; At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field; At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs and so on.

As he continued this pattern he serendipitously discovered the Fibonacci sequence, a series of numbers starting with two 1s, and then each preceding number is the sum of the previous two, giving the the following recursive definition for the nth Fibonacci term:

F₀= F₁= 1    AND    Fₙ = Fₙ₋₁ + Fₙ₋₂

What started as an unrealistic and hypothetical scenario created one of the most common naturally seen pattern in the world. Using this series, we can create the Fibonacci spiral (formed when squares with lengths corresponding to the series are constructed adjacently) and the golden ratio (the quotient between each successive pair of Fibonacci numbers approximating to 1.618). All three forms of the Fibonacci sequence are abundant in nature; For example, have you ever wondered why finding a four-leaf clover is lucky? It’s because leaves follow the Fibonacci sequence and often arrange in a Fibonacci spiral in large trees, and four does not appear in the sequence hence a clover with four leaves is extremely rare! Similarly, petals also appear in Fibonacci numbers as seen in buttercups (5 petals) and diasies (34, 55 or 89 petals). Apart from the environment, we also have these numbers inside us! The bronchi in our lungs follow this sequence to maximise airflow at different angles and even our DNA measures 34 angstroms long by 21 angstroms wide for each complete cycle of its double helix spiral. If you noticed, 34 and 21 appear consecutively in the sequence and their quotient is the golden ratio. Additionally, Humans have been unknowingly using this sequence in art for thousands of years. Some of the world’s most famous buildings, like the Notre Dame, owe this ratio to their organic, balanced, and aesthetically pleasing composition. Similarly, artists like Michael Jackson and Vijay Iyer produce asymmetrical compositions consisting of a short cord and then a long chord, three beats plus five beats, totaling eight beats, a rhythm which dates back to the Vedic times (5000BCE) in India and is found in Indian classical music and dances like Bharatnatyam.

What started as an example from a book for merchants in a small town in Italy grew into a revolutionary concept in mathematics with branches in every sector of our life. Nobody is free of this remarkable sequence of numbers. From DNA molecules to the universe, the Fibonacci sequence is one of the most distinct connections between the life sciences and mathematics. The legacy of Leonardo Bonacci lies in the heart of every cell, organism and the modern numeral system and will never be forgotten.

Illustrated by Eric Lua
Written by Sakshi Nitin Deshpande