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“.

Johannes Kepler [1]

The golden ratio is a special number: it has had multiple uses in the history of mathematics, geometry and sciences, and even esoteric implications have occasionally been attributed to it. This is due to the inherent characteristics of this value, which derives from the ratio between the whole and a part of it.

Given a segment AB with an interior point S, the golden section is performed by finding a point S so that the segment AS is the mean proportional between AB and SB:

AB/AS=AS/SB=1.618033988…

The ratio is an irrational number, with an infinite number of digits to the right of the decimal point, without repetition

Symbolic meaning of the golden ratio

Initially the interest on the golden section was purely mathematical, but throughout history it has taken on a symbolic meaning. It contains the concept of infinity, one of the philosophical attributes of the transcendent, and at the same time represents an immanent equality because it is a proportion between two tangible and real ratios. The relationship between the totality and its part has become a metaphor for the cosmological connection between the Universe and the known world, God and creation, the infinite and the finite. The golden ratio was consequently identified as a figurative ideal of harmonious and divine beauty.

Historical hints

Disciples of the Pythagorean school in the 6th century B.C. were the first to identify the numerical value of the golden ratio – the philosopher Iamblichus attributed the discovery to Hippasus of Metapontum [2] – but it was not until Euclid (c. 300 B.C.) that the actual definition of it was obtained [3].

The Ancient Egyptians and the Pyramid of Cheops

However, there are artifacts from the ancient world that would attest to its use even among the Babylonians and the Egyptians. Perhaps the Egyptians used the golden section for the construction of the Pyramids. The Pyramid of Cheops, particularly, conceals a value very similar to the golden section if the apothem of a lateral face is related to the half-side of the base. Nevertheless, it is unclear whether obtaining this value was in the real intentions of the builders, or it is merely a consequence of the technique used for construction. 

Pyramids of Giza

The Pythagoreans

The Pythagoreans identified the golden section through the study of the pentagon, where it is derived from each of the five diagonals. They associated it with the meaning of union between man and woman and totality. The Pythagoreans observed that the planet Venus, the symbol of love, draws a pentagon during its path from the sun to the earth.

A curious fact is that the golden section generated a philosophical and moral crisis at the Pythagorean school. This was because Hippasus of Metapontum had correctly identified its mathematical characteristics, including irrationality. Now, irrational numbers cannot be written in fraction form; they cannot be expressed as the ratio of two integers. This implies the existence of incommensurable physical quantities, but for the Pythagoreans, who based their worldview on measurement through multiples and submultiples, this was unacceptable. For this reason, the discovery of the golden section was long kept hidden [4].

The golden section in the Renaissance

Interest in the golden section grew in the 15th century, when mathematician Luca Pacioli published De divina proportione in Venice (1509). The text, which used illustrations by Leonardo da Vinci, described the “divine proportion” in many of its many aspects. Pacioli was convinced that it was essentially the secret of beauty. The ratio, according to Pacioli, is the basis of the most beautiful architectural and natural works. The golden section would contain a divine harmony.

golden ratio
Leonardo da Vinci‘s Annunciation. The painter would use the aura section to paint the Virgin of the Rocks, the Annunciation, and the Vitruvian Man.

The golden section, according to a popular opinion at the time, represented the harmonic connection between macrocosm (the universe, the whole) and microcosm (man, the part of the whole). This idea has persisted for centuries, at least until the 19th century, when it took the name “golden section,” and perhaps even to the present day.

The golden ratio and the Fibonacci series

In 1611 Kepler demonstrated an interesting property of the divine proportion by relating it to the Fibonacci series. The astronomer discovered that the ratio of consecutive numbers in the Fibonacci series approximated the golden section with increasing accuracy as the numbers increased.

Fibonacci spiral approximates a golden spiral

This is perhaps the reason why the golden section is found so frequently in nature. For example, it underlies the harmony of some shells (golden spiral); the arrangement of petals of numerous plants; and again: the height of a man in ratio to the height of his navel returns exactly 1.618033988…

“The image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio”

Johannes Kepler [2]
golden ratio
The Vitruvian Man by Leonardo da Vinci: the ratio of the side of the square to the radius of the circle is a golden section

Recent studies about the golden ratio

Human attraction to the aesthetics of the golden ratio has been the subject of numerous psychological studies, the pioneer of which was Fechner in the 19th century. The German scholar sought to demonstrate an unconscious preference toward geometric solids constructed from the golden ratio, such as the golden rectangle. He asked a statistical sample, consisting of several people, to indicate their preference toward certain rectangles they were shown. According to Fechner, thirty percent of the sample indicated the golden rectangle.

Fechner’s studies, however have been repeated several times and revised, obtaining often conflicting results. Since then, no scientific work has been able with certainty to prove a real correlation between the properties of the golden ratio and the perception of beauty. Is it possible that the magical proportion represents one of the greatest illusions in human history?

Samuele Corrente Naso

Notes

[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

[2] Mario Livio, La sezione aurea, Segrate, Rizzoli, 2003

[3] Euclide, Elementi, XIII libro

[4] Craig Smorynski. History of Mathematics. A Supplement, Dodrecht, Springer 2008, “The Discovery of Irrational Numbers”

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