Leonardo Fibonacci is one of the most famous mathematicians of the Middle Ages. One of his most important achievements is the number series, which determines golden ratio and can be traced throughout the nature of our planet.
An amazing property of these numbers is that the sum of all previous numbers is equal to the subsequent number (check for yourself):
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 ... - Fibonacci series
It turns out that this sequence has many interesting properties from the mathematical point of view. Here's an example: you can split a line in two. The ratio of the smaller part of the line to the larger will be equal to the ratio of the larger part to the entire line. This aspect ratio, approximately 1.618, is known as the golden ratio.
The Fibonacci series could remain only a mathematical incident, if not for the fact that all researchers of the golden ratio find this sequence throughout the plant and animal kingdom. Here are some amazing examples:
The arrangement of leaves on a branch, sunflower seeds, pine cones manifests itself as the golden ratio. If you look at the leaves of such a plant from above, you will notice that they bloom in a spiral. The angles between adjacent leaves form a regular mathematical series known as the Fibonacci sequence. Thanks to this, each individual leaf growing on the tree receives the maximum amount of heat and light available.
In a lizard, at first glance, proportions pleasant to our eyes are captured - the length of its tail is as much related to the length of the rest of the body as 62 to 38.
Scientist Zeising has done a tremendous job to discover the golden ratio in the human body. He measured about two thousand human bodies. The division of the body by the navel point is the most important indicator of the golden ratio. Proportions male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. The proportions of the golden ratio are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
During the Renaissance, it was believed that it was this proportion from the Fibonacci series observed in architectural structures and other forms of art that most pleases the eye. Here are some examples of the use of the golden ratio in art:
Mona Lisa portrait
The portrait of Monna Lisa has attracted the attention of researchers for many years, who discovered that the composition of the drawing is based on golden triangles, which are parts of a regular star-shaped pentagon, which is built on the principles of the golden ratio.
Parferon
Golden proportions are present in the dimensions of the facade of the ancient Greek temple of the Parthenon. it ancient structure with its harmonious proportions, it gives us the same aesthetic pleasure as our ancestors. Many art critics, who sought to uncover the secret of the powerful emotional impact that this building has on the viewer, sought and found the golden proportion in the ratios of its parts.
Raphael - "Beating the Babies"
The picture is built on a spiral that observes the proportions of the golden ratio. We do not know if Raphael actually painted the golden spiral when creating the composition "Beating the Babies" or only "felt" it.
Our world is wonderful and full of great surprises. An amazing thread of interconnection connects a lot of everyday things for us. The Golden Ratio is legendary in that it united, it would seem, two completely different branches of knowledge - mathematics, the queen of precision and order, and humanitarian aesthetics.
Sacred geometry. Energy codes of harmony Prokopenko Iolanta
Phi = 1.618
Phi = 1.618
To connect two parts with the third in a perfect way, a proportion is needed that would hold them together into a single whole. In this case, one part of the whole should relate to the other as the whole to the greater part.
The Phi number is considered the most beautiful number in the world, the basis of the foundations of all living things. One of sacred places Ancient egypt hides in its name this number - Thebes. This number has many names, it has been known to mankind for over 2500 years.
The first mention of this number is found in the work of the ancient Greek mathematician Euclid "Beginnings" (about 300 BC). There, this number is used to build a regular pentagon, which is the basis of the ideal "Platonic Solid" - the dodecahedron, the symbol of the perfect Universe.
Phi is a transcendental number and is expressed as infinite decimal... Leonardo of Pisa, a contemporary of Leonardo da Vinci, better known as Fibonacci, called this number "divine proportion." Later on the value of the constant "phi" was based on the "golden ratio". The term "golden ratio" was coined in 1835 by Martin Ohm.
The proportion of "phi" in the statue of the spearman Dorifor
The Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.) in ancient times was considered a unique key to the laws of the universe. You can find the quotient between two adjacent numbers and approach the number "phi", but you cannot reach it.
The constant phi was used in the construction of the Cheops pyramid, as well as to create bas-reliefs, household items and decorations from the tomb of Tutankhamun. The proportion of the "golden ratio" is used everywhere to this day in the works of artists, sculptors, architects and even choreographers and musicians.
The French architect Le Corbusier found the meaning of the constant "phi" in the relief from the temple at Abydos, the relief of the pharaoh Ramses, and the facade of the Greek Parthenon. Golden proportions are also hidden in the compass of the ancient Roman city of Pompeii. The phi proportion is also present in the architecture of the human body. (For details, see the Golden Ratio section.)
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From the book How to name a child so that he is happy the author Stephanie SisterNumber 13 At the energies of the Earth channel, the number 13, like a four, has green color- the level of sound and information. This is the fourth digit of the new reality, its double sign. The number 13 adds up to the number 4, the fourth point of reality. In the Natural sense, it is a flower awaiting pollination
From the book Eternal Horoscope the author Kuchin VladimirNumber 14 At the energies of the Earth channel, the number 14 appears in the representatives of the new, not yet mastered by our civilization, the first intellectual level of the Sky-blue color. Under the code number 14 people come who were born on the last day of the year. These people are not
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From the author's bookNumber 12 On the energies of the Cosmic Channel, the number 12 personifies the harmony and completeness of space at a new level of reality, which includes three basic concepts of life: past, present and future. Number 12 contains one - the sign of the leader and two - the sign of the owner
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From the author's bookNumber 14 On the energies of the Cosmic channel, the number 14 is the messenger of the Cosmos. The royal number 13 is not the last in the levels of development of our civilization. There is one more day in the year when missionaries come from the Cosmos itself, these people do not have a clear body code (Earth channel), they do not have
From the author's bookStep one. We calculate the number of birth, or the number of personality The number of birth reveals natural characteristic person, it, as we have already said, remains unchanged for life. Unless we are talking about the numbers 11 and 22, which can "simplify" to 2 and 4
From the author's book5th number. "Bor" Bor is often lucky at birth, and he inherits certain capitals, "factories" and "ships". Perhaps he will not squander the inheritance, and pass it on to his heirs. His personal preferences are vague - whether he loves harmony and feels, or whether he loves power and
There are many more in the universe unsolved mysteries, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for solving the world around, constructing its shape and optimal visual perception a person with the help of which he can feel beauty and harmony.
Golden ratio
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 ratio was founded as a result of studies by ancient scientists of the nature of numbers.
It is based on the theory of the proportions and ratios of divisions 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 (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (the entire segment):
The result is the equation: x 2 - x - 1 = 0, which is solved as x = (1 ± √5) / 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 Euclid's book "Beginnings", written back 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 asymmetrical. The pentagram symbolized life and health.
Fibonacci numbers
The famous book Liber abaci by a mathematician from Italy, Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time cites the regularity of numbers, in a row of 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 we move from the beginning of the sequence, this ratio will become more and more accurate.
- If we divide the number from the series by the previous one, then the result will rush to 1.618.
- One number divided by the next one after one will show a value tending to 0.382.
The application of the connection 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, which are very common in nature, were investigated by Archimedes, who even derived its equation. The spiral shape is based on the laws of the golden ratio. When it is untwisted, the length is obtained, to which the 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 “golden rectangle” with sides proportional to 1.618: 1. It is constructed, passing from a large rectangle to small ones so that the lengths of the sides will be equal to the numbers from the row. Its construction can be done in reverse order, starting with the box "1". When the corners of this rectangle are connected by lines in the center of their intersection, a Fibonacci spiral or logarithmic spiral is obtained.
The history of the use of golden proportions
Many ancient architectural monuments of Egypt were erected using golden proportions: famous pyramids Cheops and other architects Ancient Greece they were widely used 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, (Athens) and other objects that have become masterpieces of ancient architecture, demonstrating harmony based on mathematical laws.
In later centuries, interest in the Golden Ratio subsided, and the patterns were forgotten, but again resumed in the Renaissance, together with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). It contained illustrations by Leonardo da Vinci, who consolidated the new name "golden ratio". Also, 12 properties of the golden ratio were scientifically proven, and the author talked about how it manifests itself in nature, in art and called it "the principle of building the world and nature."
Vitruvian Man Leonardo
The drawing, with which Leonardo da Vinci illustrated the book of Vitruvius in 1492, depicts a human figure in 2 positions with arms spread apart. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions. human body(male), described by Leonardo based on 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 a 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 drawing has been used as a symbol to show the internal symmetry of the human body.
Leonardo used the term "Golden Ratio" 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 height to the first length (from the waist down). This calculation is done similarly to the ratio of the segments when calculating the golden ratio and tends to 1.618.
All of these harmonious proportions are often used by artists to create beautiful and impressive pieces.
Studies of the Golden Ratio in the 16th-19th Centuries
Using the golden ratio and Fibonacci numbers, research work on the question of proportions have been going on for more than one century. In parallel with Leonardo da Vinci, the German artist Albrecht Durer was also developing the theory of the correct proportions of the human body. For this, he even created a special compass.
In the 16th century. the question of the connection between the Fibonacci number and the golden ratio was the subject of the works of the astronomer I. Kepler, who was the first to apply these rules to botany.
A new "discovery" 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 everyone. natural phenomena... He conducted research huge amount people, or rather their bodily proportions (about 2 thousand), based on the results of which conclusions were drawn about the statistical confirmed patterns in the ratios different parts body: the length of the shoulders, forearms, hands, fingers, etc.
Objects of art (vases, architectural structures), musical tones, sizes when writing poems - Zeisig reflected all this through the lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that a Fibonacci series is obtained.
Fibonacci number and the golden ratio in nature
In the plant and animal world, there is a tendency to form formation in the form of symmetry, which is observed in the direction of growth and movement. Division into symmetrical parts, in which the golden proportions are observed, is 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, are related to the number of given numbers 1, 1, 2, 3, 5, 8, 13 and further;
- 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 whole body of the lizard;
- the shape of the egg, if you draw a line conditionally through its wide part;
- the ratio of the size of the fingers on a person's hand.
And, of course, the most interesting shapes They represent spiraling snail shells, patterns on spider webs, wind movement within a hurricane, a double helix in DNA and the structure of galaxies - all of which include a sequence of Fibonacci numbers.
The use of the golden ratio in art
Researchers looking for examples of the use of the golden ratio in art are examining various architectural objects and paintings in detail. Famous sculptural works, the creators of which adhered to the golden proportions, are known - statues of Olympian Zeus, Apollo Belvedere and
One of the creations of Leonardo da Vinci - "Portrait of Mona Lisa" - has been the subject of research by scientists for many years. They found that the composition of the work entirely consists of "golden triangles" combined together to form a regular pentagon-star. All da Vinci's works 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 La Gioconda.
Golden ratio in architecture
As an example, scientists have studied the masterpieces of architecture, created according to the rules of the "golden section": Egyptian pyramids, 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 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 ratio" (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 "modulator") 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.
Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and Golitsyn Hospital(now the 1st Clinical named after N.I. Pirogov), - was one of the architects who used the laws on the golden section in design and construction.
Applying proportions in designs
In clothing 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 by nature not all people have ideal proportions.
When planning landscape design and the creation of volumetric park compositions with the help of 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 the elements of the park are in such proportions that with the help of the geometric structure, mutual arrangement, illumination and light, to give a person the impression of harmony and perfection.
Application of the Golden Ratio in Cybernetics and Engineering
The patterns of the golden ratio and Fibonacci numbers are also manifested in energy transitions, in processes occurring with elementary particles constituting chemical compounds, v space systems, in the gene structure of DNA.
Similar processes occur in the human body, manifesting themselves 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 beginner programmers are given to solve is to write a formula and determine the sum of the Fibonacci numbers up to a certain number using programming languages.
Modern research on the theory of the golden ratio
Since the middle of the 20th century, interest in the problems and the influence of the laws of golden proportions on human life has been sharply increasing, and on the part of many scientists different professions: mathematicians, researchers of ethnos, biologists, philosophers, medical professionals, economists, musicians, etc.
In the USA, since the 1970s, the production of The magazine Fibonacci Quarterly, where works on this topic are published. In the press there are works in which the generalized rules of the golden ratio and the Fibonacci series are used in various industries knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that golden proportion is multilaterally connected with fundamental issues of science and manifests itself in the symmetry of many creations and phenomena of the world around us.
The development of humanity is delineated certain periods in the oldest and modern history... Can the elements of a series of Fibonacci numbers correspond to the chronological boundaries of periods in the most ancient and modern history of mankind, that is, do the boundaries of periods obey mathematical laws? Is there such a pattern in other periods: periods of world history, periods of reign of famous Russian statesmen, and in the dates of modern events having historical meaning? The purpose of our work is to draw an analogy between mathematics and history, that is, to establish some connection. To achieve this goal, it was necessary to solve the following tasks:
- Get acquainted with Fibonacci numbers and the golden ratio, which is the most harmonious attitude;
- Check if the boundaries of the periods of ancient, modern and world history correspond to the numbers of the Fibonacci series;
- Calculate the years of reign of famous Russian statesmen and find their attitude;
- Consider dates of historical significance in the time periods of modern history;
- Check if the obtained relationships between the given objects are known mathematical relationships.
The objects of research will be archaeological epochs, periods of world history, periods of reign of famous Russian statesmen, dates of events of historical significance. The results of the research of the sociologist - analyst V.V. Dudikhin and the method of the poet and translator A. Chernov, which confirm the mathematical regularities of the Fibonacci numbers corresponding to chronological boundaries, turned out to be very useful for us. ancient history humanity. The work belongs to applied research, its results, expressed in mathematics, will show the connection between mathematics and history, which obeys mathematical laws.
Fibonacci numbers and the golden ratio
A numerical sequence in which the sum of two adjacent numbers gives the value of the next one is the Fibonacci sequence (for example, 1 + 1 = 2; 2 + 3 = 5 (1,1,2,3,5,8,13,21,34 , 55, etc.)). The properties of the various members of the sequence, the so-called Fibonacci ratios, (i.e. constant ratios) are defined as follows:
- The ratio of each number to the next more and more tends to 0.618 in increasing serial number... The ratio of each number to the previous one tends to 1.618 (inverse to 0.618);
- When dividing each number by the next one after it, we get the number 0.382, on the contrary - 2.618, respectively;
- Choosing the ratio in this way, we get the main set of Fibonacci coefficients: ... 4.235; 2.618; 1.618; 0.618; 0.382; 0.236; we also mention 0.5. All of them play a special role in nature, and in particular in technical analysis.
Fibonacci, as it were, reminded his sequence to humanity. She was known even to the ancient Greeks and Egyptians. And indeed, since then in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other areas, patterns were found described by the Fibonacci ratios.
Let's turn to the number 0.618, we have already met it (Fibonacci coefficient). This is the numerical value of the golden ratio.
One of the proportions most often found in art is called the golden ratio - the division of a segment, in which one part of it is as many times larger than the other, as much as it itself is less than the whole. Proportional relationships close to the golden ratio give the impression of the development of forms, their dynamics, proportional complement to each other.
Research scientists
Let us turn to modern research: sociologist - analyst V.V. Dudikhin, poet and translator A. Chernov.
Sociologist and analyst V.V. Dudikhin considered the chronology of eras, as a tool of chronology, he chose a harmonic system of numerical ratios, the so-called Fibonacci series. V.V. Dudikhin compared the numbers of the Fibonacci series and the archaeological era. His research showed that some elements of this sequence, indeed, correspond to chronological lines in the ancient history of mankind, especially if we add to the numbers the name “thousand years BC”, or “thousand years ago,” or simply “thousand . years". Chronology and periodization historical development using the Fibonacci series, it is divided into 18 time steps: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, which is confirmed 60% matches verified.
Also, the method of A. Chernov will be useful to us, which is based on finding the relations of the parts of one whole, i.e. proportional relationship.
Chernov's attention was drawn to speculations about the golden ratio and the number PI, which go back to Pythagoras. Research by Andrey Chernov made it possible to conclude that the construction of poetry ancient author The words about Igor's regiment, consisting of nine songs, obey mathematical laws. Namely, if the number of verses in all three parts (there are 804) is divided by the number of verses in the first and last parts (256), we get 3.14, i.e. PI number up to the third decimal place.
The above studies are of interest, not only in terms of the methods used, but also in terms of the results obtained. Relying on data modern research it can be assumed that not only these archaeological eras, but also other historical periods obey mathematical laws.
The relationship between historical periods and the laws of mathematics
Let's draw an analogy between the boundaries of historical periods, Fibonacci numbers and the golden ratio, based on the data of scientists and our own research. To do this, consider some of the boundaries of historical periods, in chronology with ancient and modern history.
Let us check the research of the sociologist V.V. Dudikhin of the boundaries of historical periods in chronology with an ancient history. Let's compare the boundaries of historical periods with Fibonacci numbers, i.e. let's carry out their correspondence. To do this, consider the boundaries of the periods of ancient history:
The Iron Age dates back to the 2nd millennium AD. In the Middle East, Egypt, Greece - from the beginning of the 1st millennium AD, in Africa - from the 1st millennium AD;
The Bronze Age dates back to South America from the middle of the 1st millennium AD, in Tropical Africa from the 1st millennium BC, in Europe from the middle of the 3rd millennium BC, in India from the end of the 3rd millennium BC, in Egypt from the beginning of the 2nd millennium BC, in the Front Asia from the end of the 4th millennium BC;
The Copper Age (Eneolithic) dates back to the 8th - 4th millennium BC;
Stone Age(Paleolithic) early dates back to 35 thousand years ago, late 35 - 13 thousand years ago;
The Stone Age (Mesolithic) dates from the beginning of the XX - VIII millennium BC. by V - IV millennium AD;
The Stone Age (Neolithic) dates back to the VIII - III millennium AD;
If we consider the origin of man, then the following period boundaries are distinguished: Australopithecus anfmensis, 4 - 3.7 million years ago, Australopithecus africanus, 3-2 million years, Australopithecus boisei, 2.4 - 1.1 million years ago, Homo rudolfensis , 2.5 - 1.8 million years, Homo erectus, 1.8 - 400 thousand years, Homo neandertalensis, 220 - 27 thousand years The results obtained correspond to the Fibonacci numbers (1, 3, 8, 13, 21, 33 , 233, 1597, 2584, 4181) or close to them.
Let's conduct a study of the boundaries of the periods of world history and prehistory: The era of primitive communal relations 2.5 mil. years ago - III millennium BC; Ancient world III millennium BC - V millennium AD; History of the Middle Ages of the 5th century - the end of the 15th century; History of modern times of the 16th - 20th centuries; Modern era XX - XXI centuries. The results obtained correspond to the Fibonacci numbers (3, 5, 13, 21) or are close to them.
Let's conduct a study of the periods of reign of famous Russian statesmen from 862 AD.
Let's recalculate the years of their reign:
Rurik (862 - 879) - 17 years old; Vasily III (1505 - 1533) - 28 years old; Ivan the Terrible (1533 - 1584) - 51 years old; Romanov M.F. (1613 - 1676) - 63 years old; Peter I (1682 - 1725) - 43 years old; Catherine II (1762 - 1796) - 34 years old; Alexander II (1855 - 1981) - 26 years old; Nicholas II (1894 - 1917); the fall of the Romanov monarchy 1917 to 1931 - 14 years; Stalin I.V. (1931 -1953) - 22 years old; Khrushchev N.S. (1953 - 1964) - 11 years old; Brezhnev L.I. (1964 - 1982) - 18 years old; Gorbachev M.S. (1985 - 1991) - 6 years old; Boris Yeltsin (1991 - 1999) - 8 years old; V.V. Putin (2000 - 2008) - 8 years old.
Let's find the relationship of the years of government.
If we divide the years of reign of Rurik (17 years) by the years of reign Basil III(28 years old), their ratio is 0.607. If we divide the years of the reign of Vasily III (28 years) by the years of the reign of Ivan the Terrible (51), then their ratio is 0.549. If we divide the years of reign of Ivan the Terrible (51 years) by the sum of the years of reign of Vasily III and Ivan the Terrible (79 years), then their ratio is 0.646. The ratio of the years of the reign of M.F. Romanov. (63 years) to the years of the reign of Peter I (43 years) is 0.682. The ratio of the years of the reign of Catherine II (34 years) to the years of the reign of M.F. Romanov. (63 years) is equal to 0.54. If we divide the years of reign of Peter I (43 years) by the sum of the years of reign of Peter I and Catherine II (77 years), then their ratio is 0.55. The ratio of the years of Stalin's rule to I.V. (22 years) to the sum of years from 1917 to 1953 (36 years) is equal to 0.611 i.e. the numerical value of the golden section with an accuracy of the third decimal place;
The ratio of the years of Khrushchev's rule N.S. (11 years) to the sum of years from 1917 to 1964 (47 years) is equal to 0.234. Relations between the years of Khrushchev N.S. (11 years old) to the years of Brezhnev's reign L.I. (18 years old) and vice versa, equal to 0.611 and 1.636, respectively. These ratios are close to the Fibonacci coefficients (0.236; 0.618; 1.618) with an accuracy of the third and second digits, respectively. The ratio of the years of Stalin's rule to I.V. (22 years) to the sum of the years of Stalin's rule I.V. and Khrushcheva N.S. (33 years) is 0.666. The ratio of the years of the reign of M.S. Gorbachev (6 years) to the years of the reign of Khrushchev N.S. (11 years) is equal to 0.545. Relations between the years of Khrushchev N.S. (11 years) to the sum of the years of Khrushchev N.S. and Brezhnev L.I. (29 years old) and vice versa, equal to 0.379 and 0.620, respectively, i.e. Fibonacci coefficients (0.382; 0.618) up to the second decimal place.
Consider the time intervals, periods of reign of famous Russian statesmen, and the dates of some events in these periods of historical significance.
- The time interval is from 1984 to 1917, the years of the reign of Nicholas II. Historical event is 1904 - beginning Russo-Japanese War... Let us find the ratio of the years after this event (13 years), in the time interval, to the years of the entire time interval (23 years). The ratio of years is 0.565.
- The time interval from 1894 to 1931, from the beginning of the reign of Nicholas II to the beginning of the reign of Stalin I.V. The historical event is 1917 - the beginning of the revolution in Russia. Let us find the ratio of the years before the given event (23 years) to the years after the given event (14 years). The ratio of years is 1.64.
- Time span from 1917 to 1931, the fall of the Romanov monarchy. A historical event is 1922 - the formation of the Union of Soviet Socialist Republics. Let us find the ratio of the years before the given event (5 years) to the years after the given event (9 years). The ratio of years is 0.556.
- The time interval from 1931 to 1953, the years of Stalin's rule I.V. The historical event is 1941 - the German attack on the USSR.Let's find the ratio of the years before this event (10 years) to the years of this time interval (22 years). The ratio of years is 0.454.
- The time interval from 1985 to 2000, from the beginning of the reign of M.S. Gorbachev. at the beginning of the reign of Putin V.V. A historic event is 1991 - the collapse of the Union of Soviet Socialist Republics. Let us find the ratio of the years before the given event (6 years) to the years after the given event (9 years). The ratio of years is 0.666.
The results obtained correspond to the Fibonacci coefficients (0.618; 1.618) up to the second decimal place or are close to them.
The world around us, from the smallest invisible particles to distant galaxies of endless space, is fraught with many unsolved mysteries. However, the veil of mystery has already been lifted over some of them thanks to the inquisitive minds of a number of scientists.
One such example is "Golden ratio" and Fibonacci numbers that make up its foundation. This regularity was displayed in a mathematical form and is often found in surrounding man nature, once again excluding the possibility that it arose by chance.
Fibonacci numbers and their sequence
A sequence of Fibonacci numbers
called a series of numbers, each of which is the sum of the two previous ones:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 …
A feature of this sequence is numerical values, which are obtained by dividing the numbers of this series by each other.
A number of Fibonacci numbers have their own interesting patterns:
- In a series of Fibonacci numbers, each number divided by the next will show a value tending to 0,618 ... The further the numbers are from the beginning of the row, the more accurate the ratio will be. For example, the numbers taken at the beginning of the series 5 and 8 will show 0,625 (5/8=0,625 ). If we take the numbers 144 and 233 , then they will show the ratio 0.618 .
- In turn, if in a series of Fibonacci numbers the number is divided by the previous one, then the result of the division will tend to 1,618 ... For example, the same numbers are used as discussed above: 8/5=1,6 and 233/144=1,618 .
- The number divided by the next one after it, will show a value approaching 0,382 ... And the further from the beginning of the row the numbers are taken, the more precisely the value ratio: 5/13=0,385 and 144/377=0,382 ... Dividing the digits in reverse order will give the result 2,618 : 13/5=2,6 and 377/144=2,618 .
Using the above calculation methods and increasing the intervals between the numbers, the following series of values can be derived: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236, which is widely used in Fibonacci instruments in the forex market.
Golden ratio or Divine proportion
The "golden ratio" and Fibonacci numbers are very clearly represented by an analogy with a segment. If the segment AB is divided by point C in such a ratio that the condition is met:
AC / BC = BC / AB, then it will be the "golden ratio"
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Surprisingly, this is exactly the ratio that can be traced in the series of Fibonacci numbers. Taking a few digits from a series, you can check by calculation that this is so. For example, such a sequence of Fibonacci numbers ... 55, 89, 144 ... Let the number 144 be the whole segment AB mentioned above. Since 144 is the sum of the two previous numbers, 55 + 89 = AC + BC = 144.
Dividing the line segments will show the following results:
AC / BC = 55/89 = 0.618
BC / AB = 89/144 = 0.618
If we take the segment AB as a whole, or as a unit, then AC = 55 will be 0.382 of this whole, and BC = 89 will be equal to 0.618.
Where do Fibonacci numbers meet
The Greeks and Egyptians knew the natural sequence of Fibonacci numbers long before Leonardo Fibonacci himself. This number acquired this name after the famous mathematician ensured the wide distribution of this mathematical phenomenon in the scientific ranks.
It is important to note that the golden Fibonacci numbers are not just a science, but a mathematical representation of the world around us. Many natural phenomena, representatives of flora and fauna, have the "golden ratio" in their proportions. These are the spiral curls of the shell, and the arrangement of sunflower seeds, cacti, pineapples.
The spiral, the proportions of the branches of which are subject to the laws of the "golden section", underlies the formation of a hurricane, the spider's web weaving, the shape of many galaxies, the interweaving of DNA molecules and many other phenomena.
The length of the tail of the lizard to its body has a ratio of 62 to 38. The chicory shoot makes an ejection before releasing the leaf. After the first sheet is released, a second ejection occurs before the release of the second sheet, in force equal to 0.62 of the conventionally accepted unit of force of the first ejection. The third burst is 0.38 and the fourth is 0.24.
For a trader also great importance has the fact that price movement in the forex market is often subject to the pattern of golden Fibonacci numbers. Based on this sequence, a number of tools have been created that a trader can use in his arsenal.
The tool "", which is often used by traders, can show with high accuracy the targets of the price movement, as well as the levels of its correction.
