The golden ratio is the relationship or proportion between two segments of a line. It was discovered in antiquity and can be found not only in geometric figures but also in nature. Objects that incorporate this number are often attributed a special aesthetic quality, and this relationship appears in numerous works of architecture and art. For example, the Vitruvian Man, drawn by Leonardo Da Vinci and considered an ideal of beauty, is proportioned according to the golden ratio. What is the origin and importance of this mathematical value?
A Number With Many Names
Some numbers have intrigued humanity for centuries. Values like PI — the mathematical ratio between a circle's circumference and its diameter — or e — the base of natural logarithms — often appear as results of disparate equations or in the proportions of different natural objects. The golden ratio — often called the golden number, the golden ratio, the golden mean, the divine proportion, or the golden section — also possesses many interesting properties and appears, hidden and enigmatic, in the most disparate places.
A Brief History
The first to make a formal study of the golden ratio was Euclid, about three centuries before Christ, in his work The Elements. Euclid defined its value by saying that "a straight line is divided in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the lesser." In other words, two positive numbers a and b are in golden ratio if and only if (a+b)/a = a/b. The value of this ratio is a number that, as Euclid also demonstrated, cannot be described as the ratio of two integers (that is, it is irrational and has infinite decimals) whose approximate value is 1.6180339887498...
Almost two thousand years later, Albrecht Dürer published his "Instrucción sobre la medida con regla y compás de figuras planas y sólidas", in which he describes how to draw with ruler and compass the spiral based on the golden section, the same one we know today as the "Dürer spiral". A few decades later, the astronomer Johannes Kepler developed his model of the Solar System, explained in Mysterium Cosmographicum (The Cosmic Mystery). To get an idea of the importance this number had for Kepler, it suffices to quote a passage from that work: "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 measure of gold; the second we should name a precious jewel."
It is possible that the first to use the adjective golden, or of gold, to refer to this number was the German mathematician Martin Ohm (brother of the physicist Georg Simon Ohm). In fact, in the second edition of his book "Die Reine Elementar Matematik" (Pure Elementary Mathematics), Ohm writes in a footnote: "One also usually calls this division of an arbitrary line into two such parts the golden section." The fact that this note was not included in the first edition is a strong indication that the term may have gained popularity around that time.
The Golden Ratio and Fibonacci
The golden ratio is also "related" to the Fibonacci series. If we call Fn the n-th Fibonacci number and Fn+1 the next one, we can see that as n grows larger, the ratio between Fn+1 and Fn oscillates, being alternately less and greater than the golden ratio. This connects it in a very special way with nature, since as we have seen before, the Fibonacci series appears constantly in the structure of living beings. The golden ratio, for example, relates the number of male and female bees in a hive, or the arrangement of flower petals.
The Golden Ratio in Nature
In fact, the role the golden ratio plays in botany is so great that it is known as "Ludwig's Law". Perhaps one of the most well-known examples is the relationship in the distance between the spirals of the inner shell of snails like the nautilus. In reality, almost all spirals that appear in nature, such as in the sunflower or pinecones, possess this golden ratio, since their number is usually a term of the Fibonacci sequence.
The Golden Ratio in Art and Architecture
This number also appears very frequently in art and architecture. For some reason, figures that are "proportioned" according to the golden ratio seem more pleasing to us. Although research reveals that there is no proof connecting this proportion with Greek aesthetics, the truth is that throughout history it has been used to "beautify" many works. For example, the use of the golden section can be found in the main works of Leonardo Da Vinci. It is well known Leonardo's interest in the mathematics of art and nature, and this proportion was not indifferent to him. In fact, in his study of the human figure, embodied in the Vitruvian Man, it can be seen how all parts of the human body relate to the golden section. Some experts believe that Leonardo's great unfinished painting, San Jerónimo, which shows this saint with a lion at his feet, was painted on purpose so that a rectangle with these proportions would fit perfectly around the central figure. Also, the face of the Mona Lisa contains a perfect "golden rectangle". Obviously, Leonardo was not the only one to use this proportion in his work. Michelangelo, for example, made use of the golden ratio in the impressive sculpture David, from the position of the navel with respect to height to the placement of the joints of the fingers.
Architecture is not alien to this mathematical value. The relationship between the parts, the roof and the columns of the Parthenon in Athens, for example, is also related by the golden ratio. Many mass consumer products are designed following this relationship, since they result more pleasant or comfortable. Credit cards or cigarette boxes have dimensions that maintain this proportion. The golden ratio can be found everywhere, and often we are not even conscious that it is there. But in general, when something attracts us, it hides this relationship among its parts. Isn't it amazing?
For more information, see the Golden ratio. You can also read more about the Fibonacci sequence in nature at https://old.neoteo.com/la-sucesion-de-fibonacci-en-la-naturaleza/.
