Hippasus of Metapontum in an engraving made by Raffaello Morghen

The first definition of “golden ratio” dates back to the 6th century b.C. by the Pythagoreans. They identified it through the study of the regular pentagon, expressing it as the ratio between the side of the pentagon and any one of the five diagonals.

The number 5 had a particular meaning for the Pythagoreans; in fact, they associated it, in the sum of 2 with 3, to the male and female union and they considered it the number of universal harmony. It is believed that it was a student of Pythagoras, Hippasus of Metapontum, philosopher and mathematician of ancient Greece, who correctly identified the mathematical characteristics of the golden ratio, associating it with the concept of incommensurability.

It is not clear whether, before the Greeks, the golden section was known and if it was consciously used by Babylonians and Egyptians.

Statue of Euclid placed in the Natural History Museum of the University of Oxford

Around 300 b.C., the Greek mathematician and philosopher Euclid, in the work "Elements", provided the explanation of the division of a segment into "extreme and mean ratio", giving a rigorous definition of golden section. In this work, composed of 13 books and whose content concerned the principles of geometry known at that time, he wrote:

"It is said that a straight line has been divided into an extreme and mean ratio when the whole straight line is to its major segment as major segment is to its minor segment".

Euclid, Elements - Book VI

Leonardo Pisano known as Fibonacci

In 1202, the mathematician Leonardo Pisano (about 1170 - 1242), better known as Fibonacci, whose name derives from "filius Bonacci" (son of Bonacci), wrote the book "Liber abaci" in medieval Latin; this book gave a strong impetus to the rebirth of mathematical studies. In the "Liber abaci", Fibonacci proposed the use of Arabic numerals and he made use of a series of integers for the solution of a problem, which had the growth of a rabbit population as topic. This series, called "Fibonacci sequence", was characterized by having each number equal to the sum of the previous two numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... At a later time, the Fibonacci sequence was correlated with the golden ratio.

Portrait of Luca Pacioli (1495), attributed to Jacopo de' Barbari; National Museum of Capodimonte (Naples)

In 1509, Luca Pacioli (1445 - 1517), an Italian Franciscan friar, mathematician and economist, friend of Leonardo da Vinci and his consultant for mathematics, published the treatise "De Divina Proportione". The central topic of the treatise was the study of the "Divine Proportion" (i.e. the golden ratio). In the treatise, containing illustrations made by Leonardo da Vinci, Pacioli investigated every possible application of golden ratio in various fields, such as: "Philosophia, Perspectiva, Pictura, Sculptura, Architectura, Musica et altre Matematiche". Pacioli was convinced that the Divine Proportion was the secret of beauty and that, for this reason, it could transmit a magical overall harmony to architectural works.

Johannes Kepler in 1620

In 1611, the German astronomer, astrologer, mathematician, cosmologist, music theorist, natural philosopher and Lutheran theologian, Johannes Kepler (1571 - 1630), related the Divine Proportion with the Fibonacci sequence. Kepler, making the ratio between consecutive numbers of the Fibonacci series, noticed that this ratio approximated the golden ratio with increasing precision as one proceeded in the series. It was Kepler who correlated the golden ratio to mathematics, physics and cosmography, identifying the periodic presence of the golden number in the motion of the stars and in nature, particularly in the development of plants. His statement is famous:

"Geometry has two great treasures: one is the Pythagorean theorem; the other the division of a line according to the extreme and mean ratio. We can compare the first to a certain quantity of gold, and define the second a precious stone".

Johannes Kepler

Robert Simson (1687 - 1768)

A century later it was demonstrated exactly what Kepler had guessed. The Scottish mathematician Robert Simson discovered that, in the Fibonacci sequence, the ratio between a term and its previous one oscillated around a number, which was getting closer and closer to the value 1.618033... (resulting sometimes in excess, sometimes in defect); this number was the golden ratio:

3/2 = 1.5

5/3 = 1.66...

8/5 = 1.6

13/8 = 1.625

34/21 = 1.619...

.......................

144/89 = 1.61797...

The golden ratio has found innumerable applications in the scientific field, in the artistic field, in painting, in architecture and it is also recognizable in many aspects of nature.