He will talk about the concept of Fibonacci series and how it is related to the theory of waves, and will also lead to a refutation of the applicability of the series to natural processes.
, which the master developed in the 30s of the last century, is one of the most exciting sections. By itself, it has been highlighted in a new chapter of science that studies graphics. It is based on the developments of other specialists in the field of theory (I advise you to read - a book under the authorship).
So, for example, the great Italian mathematician Leonardo Fibonacci is ranked among the scientists (about whom I have already spoken in articles -,), who created the basis for Eliot's theory.
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The digital series of Fibonacci numbers - the golden ratio and coefficients or levels of correction + video. Fibonacci numbers in nature.The specialist lived in the 13th century. The scientist published a work called "The Book of Calculations". This book introduced Europe to an important discovery for those times and not only - the decimal number system. This system introduced the usual numbers from zero to nine into circulation.
The emergence of this system was the first major achievement of Europe since the fall of Rome. Fibonacci preserved numerical science for the Middle Ages. And also laid deep foundations for the development of other sciences, such as higher mathematics, physics, astronomy, mechanical engineering.
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How numbers and their derivatives appeared
Solving an applied problem, Leonardo stumbled upon curious series of Fibonacci numbers, at the beginning of which there are two units.
Each subsequent term is the sum of the previous two. The most curious thing is that the Fibonacci number series is a remarkable sequence in that if any term is divided by the previous one, you get a number that is close to 0.618. This number was given the name " Golden ratio».
It turned out that this number was known to mankind for a very long time. For example, in ancient Egypt, pyramids were built using it, and the ancient Greeks built their temples on it. Leonardo da Vinci showed how the structure of the human body obeys this number.
Nature uses Fibonacci numbers in its most intimate and advanced areas. From atomic structures and other small forms like DNA molecules and microcapillaries of the brain to huge ones like planetary orbits and structures of galaxies. The number of examples is so great that it should be argued that there is indeed a certain basic law of proportions in nature.
Therefore, it is not surprising that the Fibonacci series and the golden ratio have made their way to stock charts. And not just one number 0.618, but also its derivatives.
If the number of the golden ratio is raised to the first, second, third and fourth powers and the result is subtracted from unity, then a new series will be obtained, which is called “ Fibonacci correction factors". It remains only to add a mark of five tenths - this is fifty percent.
However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then we get 1.618, if we square it, then we get 2.618, if we square it, we get the number 4.236. These are Fibonacci expansion ratios. The only thing missing is the number 3.236, which was proposed by John Murphy.
What do experts think of sequencing
Someone might say that these numbers are already familiar because they are used in technical analysis programs to determine the magnitude of retracements and expansions. In addition, these same series play an important role in Eliot's wave theory. They are its numerical basis.
Our expert Nikolay Verified portfolio manager of the Vostok investment company.
- - Nikolay, do you think the appearance of Fibonacci numbers and its derivatives on the charts of various instruments is accidental? And can we say: "Fibonacci series practical application" takes place?
- - I have a bad attitude towards mysticism. And even more so on the charts of the exchange. Everything has its reasons. in the book "Fibonacci Levels" he beautifully told where the golden ratio appears, so he was not surprised that it appeared on the stock exchange quotes charts. But in vain! In many of the examples he gave, pi often appears. But for some reason it is not in the price ratios.
- - So you do not believe in the effectiveness of the Eliot wave principle?
- - No, that's not the point. The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
- - What, in your opinion, are the reasons for the appearance of the golden ratio on stock charts?
- - The correct answer to this question may be able to earn the Nobel Prize in economics. While we can guess about the true reasons. They are clearly not in harmony with nature. There are many exchange pricing models. They do not explain the indicated phenomenon. But not understanding the nature of a phenomenon should not deny the phenomenon as such.
- - And if ever this law is opened, will it be able to destroy the exchange process?
- - As the same theory of waves shows, the law of changes in stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and will not be able to destroy the exchange.
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The overlap of the foundations of the principles of mathematics in a variety of theories seems incredible. Maybe it’s a fantasy or a fit for the final result. Wait and see. Much of what was previously considered unusual or not possible: space exploration, for example, has become commonplace and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will become more accessible and understandable over time. What was previously unnecessary, in the hands of an experienced analyst, will become a powerful tool for predicting future behavior.
Fibonacci numbers in nature.
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Now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.
Take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next term in the sequence is equal to the sum of the two previous ones. For example, let's take two numbers: 6 and 51. Now let's build a sequence, which we end with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.
At the same time, the numbers that were obtained by dividing other pairs decreased from the first to the last, which allows us to assert that if this series is continued infinitely, then we will receive a number equal to the golden ratio.
Thus, Fibonacci numbers do not stand out by themselves. There are other sequences of numbers, of which there are infinitely many, which give, as a result of the same operations, the golden number phi.
Fibonacci was not esoteric. He didn’t want to put any mysticism into numbers, he was just solving an ordinary problem about rabbits. And he wrote a sequence of numbers that followed from his problem, in the first, second and other months, how many rabbits there will be after breeding. Within a year, he received that very sequence. And I didn't make a relationship. There was no golden proportion, the Divine attitude was out of the question. All this was invented after him during the Renaissance.
Before mathematics, the merits of Fibonacci are enormous. He adopted the system of numbers from the Arabs and proved its validity. It was a hard and long struggle. From the Roman numeral system: heavy and inconvenient for counting. She disappeared after the French Revolution. Fibonacci has nothing to do with the golden ratio.
There are infinitely many spirals, the most popular are: natural logarithm spiral, Archimedes spiral, hyperbolic spiral.
Fibonacci lived a long, especially for his time, life, which he devoted to solving a number of mathematical problems, formulating them in the voluminous work "The Book of Accounts" (early 13th century). He was always interested in the mysticism of numbers - he was probably no less brilliant than Archimedes or Euclid. Problems related to quadratic equations were posed and partially solved earlier, for example, by the famous Omar Khayyam, a scientist and poet; however, Fibonacci formulated the problem of rabbit breeding, the conclusions from which did not allow his name to be lost for centuries.
In short, the task is as follows. A pair of rabbits was placed in a place fenced off on all sides by a wall, and each pair gives birth to another every month, starting from the second month of its existence. The reproduction of rabbits in time will be described by the following series: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. This series is called the Fibonacci sequence, also called the formula or Fibonacci numbers. From a mathematical point of view, the sequence turned out to be simply unique, since it had a number of outstanding properties:
the sum of any two consecutive numbers is the next number in the sequence
the ratio of each number in the sequence, starting from the fifth, to the previous one, is 1.618
the difference between the square of any number and the square of a number two positions to the left will be the Fibonacci number
the sum of the squares of the adjacent numbers will be the Fibonacci number, which is two positions after the larger of the squared numbers
Fibonacci Golden Ratio
Of these conclusions, the second is the most interesting, as it uses the number 1.618, known as the Golden Ratio. This number was known to the ancient Greeks, who used it in the construction of the Parthenon (by the way, according to some sources, it served as the Central Bank). No less interesting is the fact that the number 1.618 can be found in nature both at micro- and macroscales - from coils on the shell of a snail to large spirals of cosmic galaxies.
The pyramids at Giza, created by the ancient Egyptians, during their construction also contained several parameters of the Fibonacci series at once. A rectangle, one side of which is 1.618 times larger than the other, looks the most pleasing to the eye - this ratio was used by Leonardo da Vinci for his paintings, and in a more everyday sense, it was intuitively used when creating windows or doorways. Even a wave can be thought of as a Fibonacci spiral.
In living nature, the Fibonacci sequence manifests itself no less often - it can be found in claws, teeth, sunflowers, cobwebs, and even the reproduction of bacteria. If desired, consistency is found in almost everything, including the human face and body. And nevertheless, many statements that find the golden ratio of Fibonacci in natural and historical phenomena are clearly incorrect - this is a common myth that turns out to be an inaccurate adjustment to the desired result. There are comic drawings that inscribe the Fibonacci spiral in scoliosis or the hairstyles of famous people.
Fibonacci numbers in financial markets
One of the first who was most closely involved in the application of Fibonacci numbers to the financial market was R. Elliot. His work was not wasted in the sense that market descriptions using the Fibonacci series are often called "Elliott waves". He based his search for market patterns on the model of human development from supercycles with three steps forward and two steps back. Below is an example of how you can try to use Fibonacci levels:
The fact that humanity develops nonlinearly is obvious to everyone - for example, the atomistic teaching of Democritus was completely lost until the end of the Middle Ages, i.e. forgotten for 2000 years. However, even if we accept the theory of steps and their number as true, the size of each step remains unclear, which makes Elliot waves comparable to the predictive power of heads and tails. The starting point and correct calculation of the number of waves were and will probably be the main weakness of the theory.
Nevertheless, the theory had local successes. Bob Pretcher, who can be considered a student of Elliot, correctly predicted the bull market of the early 80s, and 1987 - as a pivotal year. It actually happened, after which Bob obviously felt like a genius - at least in the eyes of others, he definitely became an investment guru. The world interest in Fibonacci levels has increased.
Subscriptions to Prechter's Elliott Wave Theorist surged to 20,000 that year, but declined in the early 1990s as the predicted “doom and gloom” of the American market took some time off. However, it worked for the Japanese market, and a number of supporters of the theory, who were "late" there by one wave, lost either their capital or the capital of their companies' clients.
Elliott Waves cover a variety of trading periods - from weekly, which makes it similar to standard technical analysis strategies, to calculating for decades, i.e. breaks into the territory of fundamental predictions. This is possible by varying the number of waves. The weaknesses of the theory, which were mentioned above, allow its adherents to talk not about the inconsistency of the waves, but about their own miscalculations, including the incorrect determination of the initial position.
It looks like a maze - even if you have the right map, you can only go through it if you understand exactly where you are. Otherwise, the card is useless. In the case of Elliott waves, there are all signs to doubt not only the correctness of its location, but also the correctness of the card as such.
conclusions
The wave development of mankind has a real basis - in the Middle Ages, waves of inflation and deflation alternated with each other, when wars replaced a relatively calm peaceful life. The observation of the Fibonacci sequence in nature, at least in some cases, is also beyond doubt. Therefore, everyone has the right to give his own answer to the question of who God is: a mathematician or a random number generator. My personal opinion: although the entire human history and markets can be represented in a wave concept, no one can predict the height and duration of each wave.
Is an all-encompassing manifestation of structural harmony. It is found in all spheres of the universe in nature, science, art in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity did not cheat on it anymore.
Surely you have often wondered why Nature is capable of creating such amazing harmonious structures that delight and delight the eye. Why artists, poets, composers, architects create delightful works of art from century to century. What is the secret and what laws are at the heart of these harmonious creatures? No one will be able to unequivocally answer this question, but in our book we will try to open the veil and tell you about one of the mysteries of the universe - the Golden Section or, as it is also called, the Golden or Divine Proportion. The Golden Ratio is called the PHI number (Phi) in honor of the great ancient Greek sculptor Phidius, who used this number in his sculptures.
For centuries, scientists have been using the unique mathematical properties of the PHI number, and this research continues to this day. This number has found wide application in all areas of modern science, which we will also try to popularly talk about on the pages. There are also a number of and what is fibonacci sequence You will learn more ...
Determination of the Golden Ratio
The simplest and most capacious definition of the golden ratio is that a small part refers to a larger one, as a large one refers to the whole whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of parts of a whole will relate as 62% to 38%. This relationship operates in the forms of space and time.
The ancients saw in the golden ratio a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as an asymmetric symmetry, calling it in a broad sense a universal rule that reflects the structure and order of our world order.
Fibonacci numbers in history
The ancient Egyptians had an idea of the golden proportions, they knew about them in Russia, but for the first time the golden ratio was explained scientifically by the monk Luca Pacioli in the book Divine Proportion, the illustrations of which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: a small segment personified the Son, the great Father, and the whole personified the Holy Spirit.
The name of the Italian mathematician Leonardo Fibonacci is directly related to the rule of the golden section. As a result of solving one of the problems, the scientist came up with a sequence of numbers, now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. The ratio of adjacent numbers in the Fibonacci series tends to the Golden Ratio in the limit. Kepler drew attention to the relation of this sequence to the golden ratio: It is arranged in such a way that the two lowest terms of this endless proportion add up to the third term, and any two last terms, if added, give the next term. Now the Fibonacci series is an arithmetic basis for calculating the proportions of the golden section in all its manifestations.
He also devoted a lot of time to the study of the features of the golden section, most likely, the term itself belongs to him. His drawings of a stereometric solid formed by regular pentagons prove that each of the rectangles obtained by cutting gives aspect ratios in gold division.
With time the rule a rule, depending on stress and context, can mean the following: Rule - a requirement for the fulfillment of certain conditions (behavior) by all participants in an action (game, golden ratio turned into an academic routine, and only the philosopher Adolf Zeising in 1855 gave him a second life. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his mathematical aestheticism drew much criticism.
Nature's universal code
Without even going into the calculations, the Golden Ratio and Fibonacci numbers can be easily found in nature. So, the ratio of the tail and body of the lizard, the distance between the leaves on the branch, there is a golden ratio and in the shape of an egg, if a conditional line is drawn through its widest part, fall under it.
The Belarusian scientist Eduard Soroko, who studied the forms of gold divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting.
Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted gravitation nature the material world of the Universe, in essence, is the main object of study of natural sciences to spiral forms, calling the spiral the curve of life. Modern scientists have found that such manifestations of spiral forms in nature such as the snail shell, the arrangement of sunflower seeds, the patterns of the cobweb, the movement of the hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.
Golden ratio formula
Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is universal the form can mean: The shape of the object - the relative position of the boundaries (contours) of the object, object, as well as the relative position of the points of the line to test the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In the diary of Leonardo da Vinci there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo tried in a similar way to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo's Vitruvian Man, created his own scale of harmonic proportions, which influenced the aesthetics of 20th century architecture.
Adolf Zeising, investigating the proportionality of man, did a tremendous job. He measured about two thousand human bodies, as well as many antique statues, and deduced that the golden ratio expresses the average law. IN man living rational social, subject of socio-historical activity and culture almost all parts of the body are subordinate to him, but the main indicator gold something made of gold section is division body In mathematics: Body (algebra) - a set with two operations (addition and multiplication), which has certain properties navel point.
As a result of measurements, the researcher found that the proportions of the male body 13: 8 are closer to gold cross-section a polysemantic term meaning: Section in drawing - unlike a section, the image only of the figure formed by the dissection of the body by the plane (planes) without depicting the parts behind this than the proportion of the female body is 8: 5.
The art of spatial forms
The artist Vasily Surikov said that there is an immutable law in the composition, when nothing can be removed or added in a picture, even an extra point cannot be put, this is real mathematics. For a long time, artists followed this law intuitively, but after Leonardo di ser Piero da Vinci (ital da Vinci, the process of creating a painting can no longer do without solving geometric problems. For example, Albrecht Durer to define points can mean: A point is an abstract object in space that has no measurable characteristics other than coordinates the golden ratio used the proportional compass invented by him.
The art critic FV Kovalev, having examined in detail the painting of Nikolai Ge, Alexander Sergeevich Pushkin in the village of Mikhailovskoye, notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.
Researchers of the Golden Ratio tirelessly study and measure the masterpieces of architecture, claiming that they became such because they were created according to the golden canons: in their list are the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling for the viewer.
Word, sound and filmstrip
Forms of temporary art in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin's work corresponds to the Fibonacci series 5, 8, 13, 21, 34.
The rule of the golden ratio also applies in individual works of the Russian classic. So the climax of the Queen of Spades is the dramatic scene of Hermann and the Countess, ending with the death of the latter. There are 853 lines in the story, and the culmination is on line 535 (853: 535 = 1.6), this is the point of the golden section.
The Soviet musicologist E.K. Rosenov notes the amazing accuracy of the golden ratio in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical decision usually falls on the golden section.
Film director Sergei Eisenstein deliberately coordinated the script of his film Battleship Potemkin with the rule of the golden section, dividing the tape into five parts. In the first three sections, the action takes place on the ship, and in the last two in Odessa. The transition to the scenes in the city is the golden mean of the film.
Harmony of the Golden Ratio
Scientific and technological progress has a long history and went through several stages in its historical development (Babylonian and ancient Egyptian culture, the culture of Ancient China and Ancient India, ancient Greek culture, the Middle Ages, the Renaissance, the industrial revolution of the 18th century, the great scientific discoveries of the 19th century, scientific and technological revolution of the 20th century) and entered the 21st century, which opens a new era in the history of mankind - the era of Harmony. It was during the ancient period that a number of outstanding mathematical discoveries were made that had a decisive influence on the development of material and spiritual culture, including the Babylonian 60-ary number system and the positional principle of representing numbers, trigonometry and Euclidean geometry, incommensurable segments, the Golden Section and Platonic Solids, the beginning number theory and measurement theory. And, although each of these stages has its own specifics, at the same time it necessarily includes the content of the previous stages. This is the continuity in the development of science. Succession can take many forms. One of the essential forms of its expression is fundamental scientific ideas that permeate all stages of scientific and technological progress and influence various fields of science, art, philosophy and technology.
The idea of Harmony associated with the Golden Section belongs to the category of such fundamental ideas. According to B.G. Kuznetsov, a researcher of the work of Albert Einstein, the great physicist firmly believed that science, physics in particular, has always had its eternal fundamental goal “To find objective harmony in the labyrinth of observed facts”. Another well-known statement of Einstein testifies to the deep belief of the outstanding physicist in the existence of universal laws of the harmony of the universe: "The religiousness of a scientist consists in an enthusiastic admiration for the laws of harmony."
In ancient Greek philosophy, Harmony opposed Chaos and meant the organization of the Universe, the Cosmos. The genius Russian philosopher Alexei Losev assesses the main achievements of the ancient Greeks in this area:
“From the point of view of Plato, and in general from the point of view of all ancient cosmology, the world is a kind of proportional whole, obeying the law of harmonic division - the Golden Section ... Their (ancient Greeks) system of cosmic proportions is often portrayed in literature as a curious result of unrestrained and wild imagination. This kind of explanation reveals the antiscientific helplessness of those who claim it. However, this historical and aesthetic phenomenon can only be understood in connection with a holistic understanding of history, that is, using the dialectical-materialistic idea of culture and looking for an answer in the peculiarities of ancient social life. "
“The law of the golden division must be a dialectical necessity. This is the thought that, as far as I know, I am conducting for the first time ", - Losev spoke with conviction more than half a century ago in connection with the analysis of the cultural heritage of the ancient Greeks.
And here is another statement regarding the Golden Section. It was made in the 17th century and belongs to the brilliant astronomer Johannes Kepler, the author of the three famous Kepler's Laws. Kepler expressed his admiration for the Golden Ratio in the following words:
“In geometry, there are two treasures - and the division of a segment in the extreme and average ratio. The first can be compared with the value of gold, the second can be called a precious stone. "
Recall that the old problem of dividing a segment in the extreme and average ratio, which is mentioned in this statement, is the Golden Section!
Fibonacci numbers in science
In modern science, there are many scientific groups professionally studying the Golden Ratio, Fibonacci numbers and their numerous applications in mathematics, physics, philosophy, botany, biology, medicine, and computer science. Many artists, poets, musicians use the "Principle of the Golden Section" in their work. In modern science, a number of outstanding discoveries have been made based on the Fibonacci numbers and the Golden Section. The discovery of "quasi-crystals", made in 1982 by the Israeli scientist Dan Shechtman, based on the Golden Section and "pentagonal" symmetry, has revolutionary implications for modern physics. A breakthrough in modern concepts of the nature of the formation of biological objects was made in the early 90s by the Ukrainian scientist Oleg Bodnar, who created a new geometric theory of phyllotaxis. Belarusian philosopher Eduard Soroko formulated the “Law of Structural Harmony of Systems” based on the Golden Section and playing an important role in self-organization processes. Thanks to the research of American scientists Elliott, Prechter and Fisher, Fibonacci numbers actively entered the business sphere and became the basis for optimal strategies in business and trade. These discoveries confirm the hypothesis of the American scientist D. Winter, head of the "Planetary Heartbeats" group, according to which not only the energy frame of the Earth, but also the structure of all living things are based on the properties of the dodecahedron and icosahedron - two "Platonic solids" associated with the Golden Section. And finally, and perhaps most importantly, the DNA structure of the genetic code of life is a four-dimensional sweep (along the time axis) of a rotating dodecahedron! Thus, it turns out that the entire Universe - from the Metagalaxy to a living cell - is built according to the same principle - dodecahedron and icosahedron, infinitely inscribing into each other, which are in the proportion of the Golden Section!
Ukrainian professor and doctor of sciences Stakhov A.P. I was able to create something. The essence of this generalization is extremely simple. If you set a non-negative integer p = 0, 1, 2, 3, ... and divide the segment “AB” by point C in such proportion as to be:
Then the universal formula for the golden ratio is the expression:
x p + 1 = x p + 1
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 form the basis for solving the world around, 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 size 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 down 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 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 by going from a large rectangle to small ones so that the lengths of the sides are 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: 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 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, which Leonardo da Vinci used to illustrate 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 of the 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 of the head 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.
Research on the Golden Ratio in the 16th and 19th centuries
Using the golden ratio and Fibonacci numbers, research on proportions has been going on for centuries. 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 relationship 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 all natural phenomena. He conducted studies of a huge number of people, or rather their bodily proportions (about 2 thousand), based on which conclusions were drawn about statistically 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 structures), musical tones, dimensions when writing poems were also studied - 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 leaves or branches of any plants, as well as distances, are related to a 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 are the spiraling snail shells, the patterns on the cobwebs, the movement of the wind inside the hurricane, the 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 exploring in detail various architectural objects and paintings. Famous sculptural works, whose creators adhered to 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", united together in a regular pentagon-star. All of 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 architectural masterpieces created according to the rules of the "golden section": the Egyptian pyramids, the Pantheon, the 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 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.
The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N.I. Pirogov), was one of the architects who used laws in the design and construction about the golden ratio.
Applying proportions in design
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 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 the elements of the park are in such proportions that with the help of the geometric structure, mutual arrangement, illumination and light, make an impression on a person 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 that make up chemical compounds, in space systems, in the genetic 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 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 patterns 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.
Since the 1970s, The Fibonacci Quarterly magazine has been published in the United States, 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 fields of knowledge. For example, for coding information, chemical research, biological, etc.
All this confirms the conclusions of ancient and modern scientists that the golden ratio 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.
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 the golden ratio and can be traced throughout the nature of our planet.
An amazing property of these numbers is that the sum of all the 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 world. 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 caught - 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. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the 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. This ancient structure with its harmonious proportions 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 whether 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.