March 4, 2024

## Leonardus Pisanus

## The Rabbit Reproduction Problem

## The Connection with the Golden Ratio

## Using Fibonacci Numbers

**The Fibonacci sequence is one of the most famous and astonishing mathematical discoveries, continuing to inspire scientists, engineers, artists, and researchers worldwide. It showcases the profound connection between mathematics and natural, cultural, and technological processes. This universal concept serves as a vivid example of how abstract mathematical ideas can find practical application in various fields of human activity, affirming the idea of the interconnectedness of all phenomena in the world. The Fibonacci sequence is actively used, including in trading on financial markets. In the MetaTrader 4 (MT4) platform, among the built-in graphic tools, one can find the Draw Fibonacci retracement option. By using it, a trader can identify potential ****support and resistance levels****, and calculate possible price reversal points. So, who was this mathematical genius, and what does his sequence entail?**

Leonardo of Pisa, better known as Fibonacci, was born around 1170 in Pisa (a city-state, now part of Italy). Leonardo himself never called himself "Fibonacci." The first known mention of "Leonardo Fibonacci" appears in the records of Perizolo da Pisa, a notary of the Holy Roman Empire, from the year 1506. The word Fibonacci is a contraction of two words, "filius Bonacci," which appeared on the cover of the "Book of Abacus," and it possibly meant "son of Bonacci." According to another theory, "Bonacci" could be interpreted as a nickname meaning "fortunate." The mathematician usually signed himself as "Bonacci," although he sometimes also used the name "Leonardo Bigollo" (the word "bigollo" in the Tuscan dialect meant "wanderer" as well as "idler").

Growing up in a merchant family, Leonardo was introduced to commerce and the practical aspects of mathematics from an early age, significantly shaping his scientific interests and achievements. His father frequently travelled to Algeria for trade matters, where Leonardo studied mathematics under Arab teachers. Later, Fibonacci visited Egypt, Syria, and Byzantium, acquainting himself with the works of ancient and Indian mathematicians in Arabic translation. Drawing on this knowledge, Fibonacci wrote several mathematical treatises, the most significant of which, the "Book of Abacus" (Latin: Liber abaci), was first published in 1202, with a revised edition following in 1228.

This book was dedicated to the exposition and promotion of decimal arithmetic and laid the foundation for the spread of Indo-Arabic numerals, including the concept of zero. In this work, Fibonacci explored the potential of these numbers, previously misunderstood, radically changing European mathematics. Importantly, the "Book of Abacus" was written in simple language, far clearer than its ancient and Islamic prototypes. The practical problems it presented, aimed primarily at merchants, facilitated its fame and popularity.

Fibonacci's most renowned contribution to mathematics comes from the numbers that bear his name. The problem of rabbit reproduction, outlined in the "Book of Abacus," serves as a classic example that led to the formulation of the famous sequence. This problem was proposed to illustrate the principle of population growth among rabbits. It is stated as follows: suppose there is a pair of newborn rabbits, one male and one female. The rabbits begin to reproduce upon reaching the age of one month. At the end of each month, every adult pair produces a new pair of rabbits (one male and one female). Assuming rabbits do not die and continue to reproduce according to these rules, how many pairs of rabbits will there be in a year?

The essence of solving the problem lies in the fact that the number of rabbit pairs in each subsequent month is equal to the sum of the number of pairs in the previous month and the number of pairs that were newborn in the month before that. This is because each adult pair contributes one more pair to the total count. Thus, the sequence looks as follows: (0), 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, and so on, where each number is the sum of the two preceding numbers. This sequence has become known as the Fibonacci sequence.

The Fibonacci sequence not only demonstrates a mathematical model of population growth but also showcases the interplay between mathematics and natural laws, closely linking it with the Golden Ratio.

The origins of the Golden Ratio extend deep into history. Some studies suggest that ancient Egyptians might have been aware of it, particularly in the construction of the pyramids, although evidence can be interpreted in various ways. The first known systematic exposition of the Golden Ratio principles is attributed to the ancient Greek mathematician Euclid, who in his work "Elements" described the division of a segment into "extreme and mean ratio." Euclid laid out the mathematical basis of this ratio but did not ascribe to it the aesthetic value that it holds today. Among the subsequent practical examples of this ratio in Ancient Greece is the famous Parthenon in Athens (447–438 BC), attributed to architects Ictinus and Callicrates.

During the Renaissance, interest in the Golden Ratio increased as artists and architects like Leonardo da Vinci and Le Corbusier began to actively use it in their works, striving to achieve harmony and perfection of form. Leonardo da Vinci explored the Golden Ratio and applied it in his famous works, including the "Mona Lisa" and "Vitruvian Man." He referred to the Golden Ratio as the "Divine Proportion," highlighting its profound significance for art and architecture.

So, what is this "Divine Proportion"? It is an irrational number, denoted by the Greek letter φ (phi), approximately equal to 1.618033988749895. This ratio arises when a line (or another object) can be divided in such a way that the ratio of the whole to the larger part is equal to the ratio of the larger part to the smaller.

The connection between the Fibonacci sequence and the Golden Ratio manifests in that the farther we progress through the sequence, the closer the ratio of two consecutive Fibonacci numbers comes to the Golden Ratio. For example, dividing the number 21 by the preceding number in the sequence, 13, yields approximately 1.615. As the numbers in the sequence increase, this ratio becomes closer to 1.618, or the "Divine Proportion."

This relationship is reflected not only in mathematics but also in nature, art, architecture, and other fields, where proportions close to the Golden Ratio are considered especially harmonious and aesthetically pleasing. Its unique properties and the embodiment of the idea of harmony make the Golden Ratio an eternal subject of study and application.

The close connection of the Fibonacci sequence with the Golden Ratio makes it a unique tool for analysing and understanding natural forms and phenomena. Fibonacci numbers are found in many aspects of various scientific fields, from the arrangement of leaves and flowers on plants to the spirals of galaxies. In music, some composers have structured their works by defining the lengths of melody or harmony segments with Fibonacci numbers.

In biology, these numbers explain the arrangement of leaves, branches, and even seeds in flowers, which helps to maximise exposure to sunlight and other resources. For instance, in sunflowers, the number of seed spirals in one direction and the other often corresponds to consecutive Fibonacci numbers. Just as in the original rabbit problem, this sequence can model realistic population growth scenarios for various biological species.

In quantum physics, sequences similar to Fibonacci can describe certain properties of quasicrystals and other complex structures. The arrangement of atoms in the molecules of some chemical compounds follows a sequence analogous to Fibonacci, affecting their physical and chemical properties. In programming, the Fibonacci sequence is widely used for teaching recursive and iterative algorithms. It has been applied in some artificial intelligence models to optimise learning processes and pattern recognition. It is also used in optimisation theory and the development of efficient algorithms, including for assessing problem complexity, optimising database queries, and improving system performance.

Research in psychology shows that people intuitively use principles similar to the Fibonacci sequence when making decisions under uncertainty, for example, when assessing probabilities. This phenomenon has found application in trading on financial markets, merging mathematics and market intuition, which we will discuss in detail in a separate article.