There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio are the basis for solving the world around us, building its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.
Golden ratio
The principle of determining the dimensions of the golden section underlies 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 ratio was laid as a result of research by ancient scientists on the nature of numbers.
It is based on the theory of the proportions and ratios of the division of segments, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (large), the ratio of the larger to the smaller will be equal to the ratio of their sum (the entire segment):
X: Y = Y: X + Y.
The result is the equation: x 2 - x - 1 = 0, which is solved as x = (1 Ā± ā5) / 2.
If we consider the ratio 1 / x, then it is equal to 1,618 ...
Evidence of the use of the golden ratio by ancient thinkers is given in the book of Euclid's "Beginnings", written in the 3rd century. BC, who applied this rule to construct regular 5-gons. Among the Pythagoreans, this figure is considered sacred, since it is both symmetrical and asymmetric. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci mathematician from Italy Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time gives a pattern of numbers, among which each number is the sum of 2 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.
The scientist also cited a number of patterns:
- Any number from the series divided by the next will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will become more accurate.
- If you divide the number from the series to the previous one, the result will rush to 1.618.
- One number divided by the next through one will show a value tending to 0.382.
The application of the relationship and the laws of the golden ratio, the Fibonacci number (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 deduced its equation. The shape of the spiral is based on the laws of the golden ratio. When it is untwisted, a length is obtained, to which proportions and Fibonacci numbers can be applied, the step increases evenly.
The parallel between the Fibonacci numbers and the golden ratio can be seen by constructing a āgolden rectangleā, in which the sides are proportional, like 1.618: 1. It is built, moving from a large rectangle to a small one so that the lengths of the sides will be equal to the numbers in a row. Its construction can be done in the reverse order, starting with the "1" box. When connecting the lines of the corners of this rectangle in the center of their intersection, a Fibonacci spiral or logarithmic is obtained.
The history of the use of gold proportions
Many ancient architectural monuments of Egypt were erected using golden proportions: the famous pyramids of Cheops and others. Architects of Ancient Greece widely used them in the construction of architectural objects such as temples, amphitheaters, stadiums. For example, such proportions were applied in the construction of the ancient Parthenon temple, 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 Ratio subsided, and patterns were forgotten, but resumed again in the Renaissance with the book of the Franciscan monk L. Pacioli di Borgo āDivine Proportionā (1509). It contained illustrations by Leonardo da Vinci, who enshrined the new name "Golden Section". 12 properties of the golden ratio were also scientifically proven, and the author talked about how it manifests itself in nature, in art and called it āthe principle of building the world and natureā.
Vitruvian Man Leonardo
The drawing by which Leonardo da Vinci illustrated the book of Vitruvius in 1492 depicts a figure of a man in 2 positions with his arms spread apart. The figure is inscribed in a circle and 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 navel is considered the center of the body as an equidistant point from the end of the arms and legs, the length of the arms is equal to the height of the person, the maximum shoulder width = 1/8 of the height, 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 has been used as a symbol showing the internal symmetry of the human body.
Leonardo used the term āGolden Ratioā to mean proportional relationships in a human figure. For example, the distance from the waist to the feet corresponds to the same distance from the navel to the top of the head as does the growth to the first length (from the waist down). This calculation is done similarly to the ratio of the segments in the calculation of the golden ratio and tends to 1.618.
All these harmonious proportions are often used by artists to create beautiful and impressive works.
Golden Section Studies in the 16-19th Centuries
Using the golden ratio and Fibonacci numbers, research on the issue of proportions has been going on for more than a century. In parallel with Leonardo da Vinci, the German artist Albrecht Durer was also developing a theory of the correct proportions of the human body. For this, he even created a special compass.
In the 16th century the connection between the Fibonacci number and the golden ratio was devoted to the work of astronomer I. Kepler, who first applied these rules to botany.
A new ādiscoveryā awaited the golden ratio in the 19th century. with the publication of Aesthetic Research by the German scientist Professor Zeisig. He elevated these proportions to absolute and announced that they are universal for all natural phenomena. He conducted research on a huge number of people, or rather their bodily proportions (about 2 thousand), based on which conclusions were made about statistical confirmed patterns in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.
Objects of art (vases, architectural constructions), musical tones, sizes when writing poems were also investigated - Zeisig displayed all this through the lengths of segments and numbers, he also introduced the term āmathematical aestheticsā. After receiving the results, it turned out that it turns out the Fibonacci series.
Fibonacci number and 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 the golden proportions are observed, is such a pattern inherent in many plants and animals.
The nature around us can be described using Fibonacci numbers, for example:
- the location of the leaves or branches of any plants, as well as the distances correspond with a number of the given numbers 1, 1, 2, 3, 5, 8, 13 and further;
- sunflower seeds (scales on cones, pineapple cells), located in two rows along twisted spirals in different directions;
- the ratio of the length of the tail and the entire body of the lizard;
- the shape of the egg, if we draw the line conditionally through its wide part;
- aspect ratio of fingers on a personās hand.
And, of course, the most interesting forms are spiral snail shells, patterns on the web, the movement of the wind inside the hurricane, the double helix in DNA and the structure of galaxies - all of them include a sequence of Fibonacci numbers.
The use of the golden ratio in art
Researchers in the art of searching for examples of the use of the Golden Ratio examine in detail various architectural objects and paintings. Famous sculptural works are known, the creators of which adhered to golden proportions - statues of Olympian Zeus, Apollo Belvedere and Athena Parthenos.
One of Leonardo da Vinciās creations - āPortrait of Mona Lisaā - has been the subject of research by scientists for many years. They found that the composition of the work consists entirely of "golden triangles" combined together into a regular pentagon-star. All the works of da Vinci are evidence of how deep his knowledge was in the structure and proportions of the human body, thanks to which he was able to catch the incredibly mysterious smile of Gioconda.
Golden ratio in architecture
As an example, scientists studied architectural masterpieces created by the rules of the "golden section": Egyptian pyramids, the Pantheon, Parthenon, Notre Dame de Paris Cathedral, 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 protrusions 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 āmodulatorā) was the French architect Le Corbusier. The modulator is based on a measuring system associated with the conditional division into parts of the human body.
The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the Senate building in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N. I. Pirogov), was one of the architects who used the laws in the design and construction about the golden ratio.
Applying proportions in design
In fashion design, all fashion designers make new images and models, taking into account the proportions of the human body and the rules of the golden ratio, although not all people have ideal proportions by nature.
When planning landscape design and creating volumetric park compositions using plants (trees and shrubs), fountains, and small architectural objects, the laws 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 compositional center.
All elements of the park are in such proportions that with the help of geometric structure, relative positioning, lighting and light, to impress a person with harmony and perfection.
The use of the golden ratio in cybernetics and technology
The laws of the golden section and Fibonacci numbers are also manifested in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, in the DNA genetic structure.
Similar processes occur in the human body, manifesting itself in the biorhythms of his life, in the action of organs, for example, the brain or vision.
Algorithms and patterns of golden proportions are widely used in modern cybernetics and computer science. One of the simple tasks that novice programmers can solve is to write a formula and determine the sum of the Fibonacci numbers to a certain number using programming languages.
Modern research on the golden ratio theory
Since the mid-20th century, interest in the problems and the influence of the laws of golden proportions on human life has been growing sharply, and on the part of many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, etc.
In the USA, from the 1970s, The Fibonacci Quarterly magazine began to be published, where works on this topic are published. Works appear in the press in which the generalized rules of the golden section and Fibonacci series are used in various branches of knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that the golden proportion is multilaterally connected with the fundamental issues of science and is manifested in the symmetry of many creations and phenomena of the world around us.