We explain what the golden ratio is, its history and the golden number. Also, the golden ratio in nature and in art.
What is the golden ratio?
Is named proportion golden ratio, divine proportion, golden section or golden ratio, but also the golden number or the golden rectangle, among other names, to a mathematical element whose presence in artistic works, architectural and even in objects of the nature, supposedly explains its beauty.
To understand what the golden ratio is, it is first necessary to understand the golden number, an irrational algebraic number, represented by the Greek letter phi (ϕ) in honor of the Greek sculptor Phidias (500-431 BC), although sometimes also with tau (Τ) or even with lowercase alpha (α), equivalent to 1.618033988749894… and (1 + √5) / 2.
This number has interesting mathematical properties and was discovered in the Antiquity, but not as an arithmetic expression, but a geometric one: it is the relationship or proportion between two segments of a line a and b, which comply with the algebraic equation:
(a + b) / a = a / b.
This ratio is called the golden ratio.
Since then the human being has found that ratio in many different objects in nature, from the leaves of trees to the shells of turtles. It is also seen in various artistic and architectural works. It has even been given a certain mystical importance throughout history.
History of the golden ratio
According to some interpretations of the archaeological discoveries, in the Mesopotamian cultures of 2000 a. C. there is already evidence of the use of the golden ratio, although there is no documentation prior to the Ancient Greece in which it is discussed.
The first formal studies of the golden number belong to the philosopher Euclides (c. 300-265 BC), in his book The elements, where it is shown that it is an irrational number, and some others are attributed to Plato himself (c. 428-347 BC).
In 1509, the Italian theologian and mathematician Luca Pacioli (c. 1445-1517) suggested the divine connection of said number in his By divina proportione ("On the divine proportion"). Pacioli claimed that it was defined by three line segments as the Divine Trinity, that it was unattainable in its totality as God, and presented other interpretable characteristics such as metaphor of the sacred.
Undoubtedly influenced by this idea, the German Renaissance artist Albrecht Dürer (1471-1528) designed in 1525 the golden spiral, later called "Dürer's spiral": the artist described how to draw a golden spiral based on the proportion with a ruler and compass. divine.
There are other references to the golden ratio in the works of Johannes Kepler (1571-1630) and Martin Ohm (1792-1872), the latter being the one who coined the name of the “golden section” in 1835. However, there is evidence that the name was in common use already at that time.
Since then he has been represented with the Greek letter tau, until in 1900 the mathematician Mark Barr replaced it by phi, as a tribute to the Greek sculptor Phidias.
Golden ratio in nature
Some examples of the finding of the golden section in nature include:
- The logarithmic spiral inside the shells of the Marine animals called nautiluses.
- The arrangement of the petals of many flowers, according to Ludwig's Law.
- The relationship between the veins of the leaves of most trees.
- The number of spirals present in the bark of a pineapple.
- The distance from the navel to the feet of any person, with respect to their total height.
- The arrangement of the artichoke leaves.
Golden ratio in art
According to some scholars, the closer a work approaches the golden section, the more beautiful it will be or the closer it will be to the ultimate beauty. There is no scientific evidence of this, but it is true that the golden ratio can be found in the following artistic, sculptural or architectural works:
- In the relationships between the forms of the Great Pyramid of Giza, according to the theses of Herodotus in his History.
- The relationship between the parts, the columns and the roof of the ancient Greek temple known as the Parthenon in Athens.
- In the formal structures of Wolfgang Amadeus Mozart's sonatas, as well as in Beethoven's Fifth Symphony, and later in works by Schubert and Debussy.
- In the frame Atomic leda of the painter surreal Salvador Dali.
- In the structure of time of the films The battleship Potemkin and Ivan the terrible by Soviet filmmaker Sergei Eisenstein.
- The Italian pictorial movement of the Povera art he based his pictures on the succession of Fibonacci numbers, which embody the golden ratio.