Fibonacci numbers and the golden ratio: the relationship

In the universe there are still many unsolved mysteries, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for unraveling the surrounding world, constructing its shape and optimal visual perception by a person, through which he can feel beauty and harmony.

Golden Section

The principle of determining the size of the golden section lies at the basis of the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden proportion was laid as a result of research by ancient scientists of the nature of numbers.

It is based on the theory of the proportions and ratios of division of segments, which was made even by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (the entire length):

X: Y = Y: X + Y.

The result is an equation: x ² - x - 1 = 0, which is solved as x = (1 ± √5) / 2.

If we consider the ratio 1 / x, then it is 1.618 ...

Evidence of the use of the golden thinkers by the ancient thinkers is given in Euclid's book The Beginning, written back in the 3rd century. BC, who applied this rule for the construction of regular 5-gons. In Pythagoreans this figure is considered sacred, because it is both symmetrical and asymmetric. The pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber abaci mathematics from Italy by Leonardo of Pisa, which later became known as Fibonacci, was published in 1202. In it, the scientist first gives the regularity of numbers, in the series of which each number is the sum of the two previous digits. The sequence of Fibonacci numbers is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.

Also, the scientist cited a number of regularities:

• Any number from the series, divided by the subsequent one, will be equal to the value that tends to 0.618. And the first Fibonacci numbers do not give such a number, but as we move from the beginning of the sequence this ratio will be more accurate.
• If we divide the number from the series into the previous one, the result will go to 1.618.
• A single number, divided by the next through one, will show a value tending to .382.

The application of connection and regularities of the golden section, Fibonacci numbers (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction and in many other sciences.

Archimedes' spiral and golden rectangle

Spirals, very common in nature, were investigated by Archimedes, who even derived its equation. The shape of the spiral is based on the laws of the golden section. When it is untwisted, the length is obtained, to which proportions and Fibonacci numbers can be applied, the step increment occurs evenly.

The parallel between the Fibonacci numbers and the golden section can be seen and constructed a "golden rectangle", in which the sides are proportional, like 1.618: 1. It is constructed by moving from a larger rectangle to a smaller one so that the sides are equal to the numbers in the row. You can build it in the reverse order, starting with the square "1". When connecting the angles of this rectangle at the center of their intersection, a Fibonacci spiral or a logarithmic spiral is obtained.

History of the application of gold proportions

Many ancient monuments of Egypt's architecture are built using golden proportions: the famous pyramids of Cheops and others. The architects of ancient Greece used them extensively in the construction of architectural objects such as temples, amphitheaters, stadiums. For example, such proportions were used in the construction of the ancient temple of the Parthenon, the theater of Dionysus (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony, based on mathematical regularity.

In later centuries, interest in the golden section subsided, and patterns were forgotten, but again resumed in the Renaissance, along with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). In it, illustrations by Leonardo da Vinci were given, which also reinforced the new name "golden section". 12 properties of the golden proportion were also scientifically proven, and the author talked about how it manifests itself in nature and art and called it "the principle of building the world and nature."

Vitruvian man of Leonardo

The drawing by which Leonardo da Vinci in 1492 illustrated the book of Vitruvius, depicts a figure of a man in two positions with his hands divorced in the sides. The figure is inscribed in a circle and a square. This figure is considered to be the canonical proportions of the human body (male), described by Leonardo on the basis of their study in the treatises of the Roman architect Vitruvius.

The center of the body as an equidistant point from the end of the arms and legs is the navel, the length of the arms is equal to the height of the person, the maximum width of the shoulders = 1/8 of growth, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.

Since then, the figure is used as a symbol showing the internal symmetry of the human body.

The term "Golden Section" Leonardo used to refer to proportional relationships in the figure of a person. For example, the distance from the waist to the feet is related to the same distance from the navel to the crown as well as the growth to the first length (from the belt down). This calculation is done analogously to the ratio of the segments in calculating the golden proportion and tends to 1.618.

All these harmonious proportions are often used by artists to create beautiful and impressive works.

Studies of the golden section in the 16th-19th centuries

Using the golden ratio and Fibonacci numbers, research work on the issue of proportions continues for more than one century. In parallel with Leonardo da Vinci, the German artist Albrecht Durer also worked on the theory of the correct proportions of the human body. To do this, they even created a special compass.

In the 16th century. The connection between the Fibonacci number and the golden section was devoted to the work of the astronomer I. Kepler, who first applied these rules to botany.

A new "discovery" was expecting a golden section in the 19th century. With the publication of the "Aesthetic Study" of the German scientist Professor Tseizig. He elevated these proportions to the absolute and declared that they are universal for all natural phenomena. He carried out research on a large number of people, or rather their body proportions (about 2 thousand), which led to conclusions about statistical confirmed regularities in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.

Art objects (vases, architectural constructions), musical tones, sizes when writing poems were also studied - all this Zeisig displayed through the lengths of the segments and figures, he also introduced the term "mathematical aesthetics". After receiving the results it turned out that the Fibonacci series is obtained.

The Fibonacci number and the golden ratio in nature

In the plant and animal world, there is a tendency to form in the form of symmetry, which is observed in the direction of growth and movement. The division into symmetrical parts, in which gold proportions are observed, is such a regularity that is inherent in many plants and animals.

The nature around us can be described with the help of Fibonacci numbers, for example:

• The arrangement of leaves or branches of any plants, and also the distances are correlated with a number of reduced numbers 1, 1, 2, 3, 5, 8, 13 and on;
• Sunflower seeds (scales on cones, pineapple cells), arranged in two rows along twisted spirals in different directions;
• The ratio of the length of the tail and the entire body of the lizard;
• Shape of the egg, if you draw a line conditionally through a large part of it;
• The ratio of the size of the fingers on the person's arm.

And, of course, the most interesting forms are spirally twisted snail shells, web patterns, wind movement inside a hurricane, a double helix in DNA and the structure of galaxies - all of which include a sequence of Fibonacci numbers.

Use of the golden section in art

Researchers who search the art of examples of the use of the golden section, explore in detail various architectural objects and paintings. Famous sculptural works are known, the creators of which adhered to golden proportions - statues of Zeus of Olympia, Apollo of Belvedere and Athena Parthenos.

One of the creations of Leonardo da Vinci - "The Portrait of the Mona Lisa" - has been the subject of research for many years. They found that the composition of the work consists entirely of "golden triangles", combined together in a regular pentagon-star. All the works of da Vinci are evidence of how deep his knowledge of the structure and proportions of the human body, thanks to which he was able to catch the incredibly mysterious smile of the Mona Lisa.

Golden Section in Architecture

As an example, scientists studied masterpieces of architecture created by the rules of the "golden section": the Egyptian pyramids, the Pantheon, the Parthenon, the Notre-Dame de Paris Cathedral, the St. Basil's Cathedral, etc.

The Parthenon - one of the most beautiful buildings in ancient Greece (5th century BC) - has 8 columns and 17 on different sides, the ratio of its height to the length of the sides is 0.618. The projections on its facades are made according to the "golden section" (photo below).

One of the scientists who invented and successfully applied the improvement of the modular system of proportions for architectural objects (the so-called "modulor") was the French architect Le Corbusier. The modulor is based on a measuring system connected with the conditional division into parts of the human body.

The Russian architect M. Kazakov, who built several apartment buildings in Moscow, as well as the Senate building in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after NI Pirogov), was one of the architects who used to design and build laws About the golden section.

Applying proportions in design

In the design of clothing, all the designers make new images and models with regard to the proportions of the human body and the rules of the golden section, although by nature not all people have ideal proportions.

When planning landscape design and creating voluminous park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the patterns of "divine proportions" can also be applied. After all, the composition of the park should be focused on creating an impression on the visitor who can freely navigate in it and find a composition center.

All the elements of the park are in such proportions that, with the help of a geometric structure, interposition, illumination and light, impress the person with harmony and perfection.

The use of the golden section in cybernetics and technology

The patterns of the golden section and the Fibonacci numbers are also manifested in energy transitions, in processes that occur with elementary particles that make up chemical compounds, in space systems, in the gene structure of DNA.

Similar processes occur in the human body, manifested in biorhythms of his life, in the action of organs, for example, the brain or vision.

Algorithms and regularities of golden proportions are widely used in modern cybernetics and computer science. One of the simple tasks that beginners can do is write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.

Modern studies of the theory of the golden proportion

Since the middle of the 20th century, interest in problems and the influence of the regularities of golden proportions on human life has increased dramatically, and from many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, and others.

In the USA, since the 1970s, the journal The Fibonacci Quarterly has been published, where work on this topic is published. In the press there are works in which the generalized rules of the golden section and the Fibonacci series are used in various branches of knowledge. For example, for encoding information, chemical research, biological, etc.

All this confirms the conclusions of ancient and modern scholars that the golden proportion is multilaterally related to the fundamental issues of science and manifests itself in the symmetry of many creations and phenomena of the world around us.