Golden section in nature and art

With the manifestation of such an effect as the "golden ratio", it seems, almost every person came across in childhood. If we are guided by this principle, cut a smaller rectangle from a rectangular sheet of paper, then we must get a figure with exactly the same aspect ratio. This, in fact, is the simplest content of the principle of the golden section. A more scientific definition is as follows: the golden section is the operation of dividing a continuous quantity into 2 parts in such a proportion that the smaller part refers to the larger one as large refers to the whole quantity.

The history of this pattern is quite large. Even in antiquity, the mention of it is first encountered in Euclid in his famous work "Elements". Euclid applies this rule to the construction of some geometric figures. The pattern was so common that, at the time of Leonardo, da Vinci was called the "divine proportion", and the term was introduced by Martin Om in 1835.

Indeed, the prevalence of this phenomenon is simply amazing, some people in general tend to "see" the signs of a golden section in the world around them and even in everyday events. There was even an ironic joke about this, which says that many people find it possible to detect the presence of signs of a golden section everywhere and in everything.

The effect is geometrically proportioned. The golden section, for example, used in a regular five-pointed star, manifests itself by the fact that each segment connecting the rays of the star to each other is divided by another crossing segment, according to the rule of this proportion.

The golden section is widely represented in works of art and sculpture, common in architecture. In these areas of activity, it is usually understood as an asymmetric construction or composition, which contains the proportions of the section. Moreover, it is not always possible to express the essence of this entire construction mathematically.

There is an opinion that objects and objects, which include elements containing a golden section, are perceived as the most harmonious and attractive. But this is not always the case, because it may just be a coincidence or a mistake of perception. For example, it is commonly believed that the proportions of the famous pyramid indicate that the Egyptians used these ratios. Signs of the presence of proportions of the cross section are found even in music. For example, some researchers argue that these principles are entirely built musical works of the Baroque era , including the works of the great Beethoven, Bach and Mozart.

There are many examples that demonstrate the evidence "for" and "against" the presence of signs of the golden proportion in this or that sphere of human activity. However, it should be recognized unambiguously that the golden section in nature is widely spread, it should only be "seen".

The astronomer Johannes Kepler called this proportion the treasure of geometry, and saw its manifestation in botany - on the example of the structure and growth of plants, keeping the prescribed proportions at growth. The German researcher Zeising made measurements of 2000 human bodies and established the rule of division of the body by the point of the navel in the proportion corresponding to the golden section. In the lizard, the proportions that are familiar to the eye are seen - the tail and the rest of the body are related in length in a proportion of 62 to 38. As we see, both in the plant and animal world, a tendency to form, containing the signs of a golden proportion, is clearly seen.

Nature, as it were, by itself produces division according to the rules of symmetry and the golden section.