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Magic Numbers: the Golden Ratio

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel [1]”

Johannes Kepler
sezione aurea

The golden ratio is a particular number, whose geometric, mathematic, psychological, archaeological, biological and esoteric implications, have encouraged the discussion for more than three thousand years. The first who have identified it were the disciples of the Pythagorean school (Ippaso from Metaponto, according to Giamblico), but only Euclid (around 300 BC) has provided the current definition. Nonetheless, there are handworks of the ancient world which attest the use of the golden ratio among the Babylonians and Egyptians.

The golden ratio

As a number, the golden ration expresses the ratio between two unequal lenghts. It is irrational, since it has infinite non-periodic numbers after the comma. Its value is1,618033988…

Why did this bizarre number generate centuries and centuries of studies?

Initially, the interest on the “extreme and average proportion”, as defined by Euclid, was purely of a mathematical nature. Taken a segment AB, it is simply the ratio (f) such that, defined an intermediate point C:


Historical hints

In the 15th century the interest for this number changed, when the mathematician Luca Pacioli published De divina proportione in Venice (1509). The text, which was enriched with the illustrations by Leonardo da Vinci, described the “divine proportion” in many aspects. The scholar was convinced that it was essentially the secret of beauty. According to Pacioli, the ratio f was at the base of the most beautiful architectural or natural works. In fact, the proportion gives an almost magical overall harmony. This conception has lasted for centuries, until the 19th century, when it took the name of “golden ratio” until today.

An interesting property of the divine proportion was discovered in 1611 by Kepler, when it was associated to the Fibonacci series. The astronomer discovered that the ratio among the consecutive numbers of the Fibonacci series approximated with a better precision the golden ratio as the quantities increase.

The Fibonacci spiral approximated the golden ratio

Esoteric interpretations

From a philosophical point of view, the golden ratio well represents the ratio between the whole and a part of the whole. Figuratively, it can be associated to the relation between man and nature, or between God and the creation. The Pythagoreans identified it by the study of the pentagon, which can be derived from each of the five diagonals, and associated to it the female and male union that leads to the whole. Moreover, the planet Venus is symbol of love, and it depitcts a pentagon during its approach path from the sun to the earth.

This concept was well expressed in the hermetic and and esoteric context of the Renaissance: according to the spread opinion, the golden ratio represented the harmonic connection between the macrocosm (the universe, the whole) and the microcosm (the man, the part of the whole). These are conceptions dating back to the Pythagorean period.

Construction and the golden ratio

However, it was used for the construction of big architectural buildings, like the Pyramids of Egypt and the Parthenon.

In particular, the Pyramid of Cheops presents the golden ratio among each semilated and the height of the triangular facade built on it. Nonetheless, there is a big debate about the real intentions of the constructors to obtain the indicated ratio, or if it is the consequence of the tecnique used by its edification.

Leonardo da Vinci, that Pacioli cites in his treaty, used the golden ratio to depict the Vergine delle Rocce, the Annunciazione and the Uomo di Vitruvio.

golden ratio

Annunciazione by Leonardo da Vinci

Additionally, the magic proportion is frequent in nature. It is at the base of the harmonic step of certain shells (golden spiral) or several plants and seeds; the height of a man, divided by the height of his navel, exactly provides 1,618033988…

golden ratio
The “Uomo di Vitruvio” by Leonardo da Vinci: the ratio between the side of the square and the radius of the circle is a golden ratio

Recent studies

The human attraction to the aesthetics of the golden ratio was the object of several psychological studies, whose pioneer was Fechner in the 19th century. The German scholar tried to demonstrate an unconscious preference towards the geometric solids built starting from the golden ratio, like the golden rectangle. He asked to a statistic sample, composed by several people, to indicate their preference to some rectangles that were shown to them. According to Fechner, the thirty percent of the tested subjects indicated the golden rectangle.

Fechner’s studies were repeated and revisioned several times, often obtaining contrasting results. From that moment on, no scientific work reached with certainty to demonstrate a relation between the propriety of the golden ratio and the beauty perception.

Is it possible that the magic proportion represents one of the big suggestions of the human history?

Samuele Corrente Naso

(Translation by Daniela Campus)


[1] Fink, Karl; Beman, Wooster Woodruff; Smith, David Eugene. A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink’s Geschichte der Elementar-Mathematik, Chicago, 1903

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