“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 Kepler1
The golden ratio is a number with unique mathematical properties that can be expressed as the ratio of the whole to a part. Euclid defined it as the “extreme and mean ratio”2, a characteristic that has led to profound symbolic and philosophical meanings being attributed to it throughout history.
Given a segment AB and an intermediate point S, the golden ratio is such that AS is the mean proportional between AB and SB. The value is always constant, whatever the length of the segments:
AB/AS=AS/SB=1.618033988…
Symbolic meaning of the golden ratio
This number is irrational, meaning it has an infinite number of non-repeating digits after the decimal point. This has fascinated humanity throughout history. In fact, infinity is one of the philosophical attributes of the transcendent. Symbolically, 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. This analogy conveys beauty and harmony, which is why the golden ratio is recognised in art as a golden or divine figurative ideal.
Historical hints
Disciples of the Pythagorean school in the 6th century BC were the first to identify the numerical value of the golden ratio. The philosopher Iamblichus attributed the discovery to Hippasus of Metapontum3, but it was not until Euclid (c. 300 BC) that the actual definition of it was obtained4.
The Ancient Egyptians and the Pyramid of Cheops
However, there are artifacts from the ancient world that also attest to its use by the Babylonians and 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. This occurs 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 merely a consequence of the technique used for construction.
The Pythagoreans
The Pythagoreans identified the golden section through their study of the pentagon. Indeed, it derives from each of the five diagonals. They associated it with the meaning of union between man and woman, and therefore of totality. The Pythagoreans discovered that the planet Venus, associated with love, beauty and fertility, forms a pentagon during its approach from the sun to the earth.
Interestingly, it was the golden ratio that triggered the philosophical and moral crisis of the Pythagorean school. This was because Hippasus of Metapontum had correctly identified its mathematical characteristics, including irrationality. Irrational numbers cannot be written as a ratio of two integers. This implies the existence of incommensurable physical quantities. However 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 hidden5.
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 Venice6. 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.
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.
This is perhaps the reason why the golden section appears so frequently in nature. For example, it underlies the harmony of some shells (golden spiral) and the arrangement of petals of numerous plants. Furthermore, 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
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 rectangle7.
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
- K. Fink, W. W. Beman, D. E. Smith, A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink’s Geschichte der Elementar-Mathematik, Chicago, 1903. ↩︎
- Euclide, Elements, VI book. ↩︎
- M. Livio, La sezione aurea, Segrate, Rizzoli, 2003. ↩︎
- Euclide, Elements, XIII book. ↩︎
- C. Smorynski, The Discovery of Irrational Numbers, in History of Mathematics. A Supplement, Dodrecht, 2008. ↩︎
- Luca Pacioli, De divina proportione, Venice, 1509. ↩︎
- G. Fechner, Vorschule der Aesthetik, 1879. ↩︎
