The golden ratio - what is it? Are Fibonacci numbers? What do the DNA helix, shell, galaxy and Egyptian pyramids have in common? The Fibonacci spiral is an encrypted law of nature.

Leonardo Fibonacci is one of the greatest mathematicians of the Middle Ages. In one of his works "The Book of Computations" Fibonacci described the Indo-Arabic system of calculus and the advantages of using it over the Roman one.

Definition
Fibonacci numbers or Fibonacci Sequence is a numerical sequence that has a number of properties. For example, the sum of two adjacent numbers of the sequence gives the value of the next one (for example, 1 + 1 = 2; 2 + 3 = 5, etc.), which confirms the existence of the so-called Fibonacci ratios, i.e. constant ratios.

The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ...

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Complete definition of Fibonacci numbers

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Fibonacci sequence properties

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1. The ratio of each number to the next one more and more tends to 0.618 as the ordinal number increases. The ratio of each number to the previous one tends to 1.618 (inverse to 0.618). The number 0.618 is called (PI).

2. When dividing each number by the next one, after one, the number 0.382 is obtained; on the contrary - respectively 2.618.

3. By choosing the ratios in this way, we get the main set of Fibonacci coefficients:… 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

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The connection between the Fibonacci sequence and the "golden ratio"

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The Fibonacci sequence asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.

If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13: 8), the result will be a value that fluctuates around the irrational value 1.61803398875 ... and, once in a while, then it does not reach it. But even having touched Eternity on it, it is impossible to know the ratio exactly, up to the last decimal digit. For the sake of hardness, we will translate it in the form of 1.618. Special names for this ratio began to be given even before Luca Pacioli (a mid-century mathematician) called it the Divine Proportion. Among its modern names there are such as Golden Ratio, Golden Mean and the ratio of rotating squares. Keplep called this relationship one of the "treasures of geometry." In algebra, its designation is generally accepted by the Greek letter phi

Let's imagine the golden ratio using a line segment as an example.

Consider a segment with ends A and B. Let point C divide the segment AB so that,

AC / CB = CB / AB or

AB / CB = CB / AC.

You can think of it like this: A -– C --– B

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The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part as much as the larger part itself refers to the smaller one; or in other words, a smaller segment relates to a larger one as a larger one to everything.

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The segments of the golden ratio are expressed by the infinite irrational fraction 0.618 ... if AB is taken as one, AC = 0.382 .. As we already know the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.

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Fibonacci and Golden Ratios in nature and history

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It is important to note that Fibonacci, as it were, reminded his sequence to humanity. She was known even to the ancient Greeks and Egyptians. Indeed, since then in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other areas, patterns have been found described by the Fibonacci coefficients. It's amazing how many constants can be calculated using the Fibonacci sequence, and how its members appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a game with numbers, but the most important mathematical expression of natural phenomena ever discovered.

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The examples below show some interesting applications of this mathematical sequence.

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1. The shell is spirally wound. If you unfold it, you get a length slightly inferior to the length of the snake. A small 10-centimeter shell has a spiral 35 cm long. The shape of the spiral-wound shell attracted the attention of Archimedes. The point is that the ratio of the measurements of the shell curls is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn from this equation is named after him. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

2. Plants and animals. Even Goethe emphasized the tendency of nature to spiral. The helical and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore the law of the golden ratio manifests itself. The spider weaves the web in a spiral manner. A hurricane is spinning in a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral "the curve of life."

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at him. A process has formed from the main stem. The first sheet is located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but is shorter than the first, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and ejects again. If the first emission is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. The impulses of its growth gradually decreased in proportion to the golden section.

The lizard is viviparous. 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.

In both the plant and the animal world, the formative tendency of nature is persistently breaking through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In the parts, the repetition of the structure of the whole is manifested.

Pierre Curie at the beginning of this century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without considering the symmetry of the environment. The patterns of golden symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the genetic structures of living organisms. These patterns, as indicated above, are in the structure of individual organs of a person and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, with the help of this series (Fibonacci) found the regularity and order in the distances between the planets of the solar system

However, one case that seemingly contradicted the law: there was no planet between Mars and Jupiter. Concentrated observation of this region of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century.

The Fibonacci series is widely used: it is used to represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

4. Pyramids. Many have tried to unravel the secrets of the pyramid at Giza. Unlike other Egyptian pyramids, this is not a tomb, but rather an insoluble puzzle of number combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was preliterate, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the pyramid in Giza, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of ​​each of its faces was equal to the square of its height.

Triangle area

356 x 440/2 = 78320

Square area

280 x 280 = 78400

The length of the edge of the base of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the base rib divided by the height leads to the ratio Ф = 1.618. A height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from the Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion Φ = 1.618. Some modern scholars are inclined to interpret that the ancient Egyptians built it with the sole purpose of transmitting knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.

Pyramids in Mexico. Not only are the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon is found in the Mexican pyramids. The idea arises that both Egyptian and Mexican pyramids were erected at approximately the same time by people of common descent.

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, constructing 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 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 as 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: 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 during 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 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 on proportions has been going on for centuries. In parallel with Leonardo da Vinci, the German artist Albrecht Durer also worked on the development of 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 raised 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.

based on the book by B. Biggs "the hedger came out of the fog"

About Fibonacci numbers and trading

As an introduction to the topic, let's briefly turn to technical analysis. In short, technical analysis aims to predict the future movement of an asset's price based on past historical data. The most famous wording of his supporters is that the price already includes all the necessary information. The implementation of technical analysis began with the development of stock speculation and is probably not completely finished so far, since it potentially promises unlimited earnings. The most famous techniques (terms) in technical analysis are support and resistance levels, Japanese candlesticks, patterns that foreshadow a price reversal, etc.

The paradox of the situation, in my opinion, is as follows - most of the described methods have become so widespread that, despite the lack of an evidence base for their effectiveness, they really got the opportunity to influence the behavior of the market. Therefore, even skeptics who use fundamental data should consider these concepts simply because they are taken into account by a very large number of other players ("techies"). Technical analysis can work well on history, but practically no one is able to consistently make money with it in practice - it is much easier to get rich by publishing a large circulation of the book "How to Become a Millionaire Using Technical Analysis" ...

In this sense, the Fibonacci theory stands apart, which is also used to predict prices for different periods. Its followers are usually referred to as “wave leaders”. It stands apart because it did not appear simultaneously with the market, but much earlier - as much as 800 years. Another feature of it is that the theory has found its reflection almost as a world concept for describing everything and everyone, and the market is only a special case for its application. The effectiveness of the theory and its duration provide it with both new supporters and new attempts to compose the least controversial and generally accepted description of the behavior of markets on its basis. But alas, the theory did not advance further than individual successful market predictions, which can be equated with luck.

The essence of Fibonacci theory

Fibonacci lived a long, especially for his time, life, which he devoted to solving a number of mathematical problems, formulating them in his voluminous work "The Book of Abacus" (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 before Fibonacci, for example, by the famous Omar Khayyam, a scientist and poet; however, Fibonacci formulated the problem of rabbit breeding, the conclusions from which brought him what allowed his name not to be lost in the 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 of rabbits gives birth to another pair every month, starting from the second month of its existence. The reproduction of rabbits in time will be described by the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. 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 equal to 1.618;

• the difference between the square of any number and the square of the 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

Of these conclusions, the second is the most interesting, as it uses the number 1.618, known as the Golden Ratio. This number was already known to the ancient Greeks, who used it in the construction of the Parthenon (by the way, according to some reports, the Central Bank served the Greeks). No less interesting is the fact that the number 1.618 can be found in nature both on a micro- and macroscale - from spiral turns on the shell of a snail to large spirals of cosmic galaxies. The pyramids in 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 sometimes used to create windows or doorways. Even a wave, as in the figure at the beginning of the article, can be represented 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. Nevertheless, there is an opinion that many statements that find Fibonacci numbers in natural and historical phenomena are incorrect - this is a common myth that often turns out to be an inaccurate fit to the desired result.

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 in vain in the sense that market descriptions using Fibonacci theory are often called "Elliot waves". The development of markets here was based on the model of human development from supercycles with three steps forward and two steps back. The fact that humanity develops nonlinearly is obvious to almost everyone - the knowledge of Ancient Egypt and the atomistic teaching of Democritus were completely lost in the Middle Ages, i.e. after about 2000 years; The 20th century gave rise to such horror and insignificance of human life, which was difficult to imagine even in the era of the Punic wars of the Greeks. 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. Prechter's Elliott Wave Theorist subscription grew to 20,000 that year.however, it declined in the early 1990s, as the predicted “doom and gloom” of the American market decided to postpone a little. 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. In the same way and with the same success, the theory is often tried to be applied to trading in the foreign exchange market.

The theory covers 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 is that 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.

At the same time, 200 years of observing the American market and more than 100 years after the rest make it clear that the stock market is growing, going through various periods of growth and stagnation. This fact is quite enough for long-term earnings in the stock market, without resorting to controversial theories and entrusting them with more capital than should be within reasonable risks.

Let's find out what is common between the ancient Egyptian pyramids, the painting by Leonardo da Vinci "Mona Lisa", a sunflower, a snail, a pine cone and human fingers?

The answer to this question is hidden in amazing numbers that have been discovered the Italian mathematician of the Middle Ages Leonardo of Pisa, better known by the name of Fibonacci (born c. 1170 - died after 1228), italian mathematician ... Traveling in the East, I got acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called by the name of the famous mathematician. The amazing essence of the Fibonacci sequence is that each number in this sequence is obtained from the sum of the two previous numbers.

So, the numbers forming the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence.

There is one very interesting feature in Fibonacci numbers. When dividing any number from the sequence by the number in front of it in the row, the result will always be a value that fluctuates around the irrational value 1.61803398875 ... and through times it either rises or does not reach it. (Note: an irrational number, i.e. a number whose decimal representation is infinite and not periodic)

Moreover, after the 13th in the sequence, this division result becomes constant indefinitely ... It is this constant number of divisions in the Middle Ages that was called the Divine proportion, and nowadays it is called the golden ratio, golden mean or golden proportion ... In algebra, this number is denoted by the Greek letter phi (Ф)

So, the Golden Ratio = 1: 1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, also created according to the principle of the golden ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.

The most important book of all modern architects, E. Neufert's reference book "Building Design" contains the basic calculations of the parameters of the human body, containing the golden proportion.

The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered perfectly folded. The principle of calculating the golden measure on the human body can be depicted as a diagram:

M / m = 1.618

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.

In addition, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1: 1.618;

* distance from shoulder level to crown of head and head size is 1: 1.618;

* the distance from the navel point to the crown of the head and from the shoulder level to the crown of the head is 1: 1.618;

* the distance of the navel point to the knees and from the knees to the feet is 1: 1.618;

* the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618:

The golden ratio in human facial features as a criterion for perfect beauty.

In the structure of human facial features, there are also many examples that approach the value of the golden ratio formula. However, do not rush immediately after the ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and people of art, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden ratio in a person's face is the ideal of beauty for the human eye.

For example, if we add up the width of the two front upper teeth and divide this amount by the height of the teeth, then, having received the Golden Ratio number, it can be argued that the structure of these teeth is ideal.

On the human face, there are other incarnations of the rule of the golden ratio. Here are some of these relationships:

* Face height / face width;

* The center point of the junction of the lips to the base of the nose / length of the nose;

* Face height / distance from the tip of the chin to the center point of the junction of the lips;

* Width of the mouth / width of the nose;

* Width of nose / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand

It is enough just to bring your palm closer to you now and carefully look at the index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.

* The sum of the first two phalanges of the finger in relation to the entire length of the finger and gives the number of the golden ratio (excluding the thumb);

* In addition, the ratio between middle finger and little finger is also equal to the golden ratio;

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, a total of 10, but with the exception of two biphalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:

The golden proportion in the structure of the human lungs

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, found that the golden ratio also exists in the structure of the human lungs.

The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi are made up of two main airways, one of which (left) is longer and the other (right) is shorter.

* It was found that this asymmetry continues in the branches of the bronchi, in all the smaller airways. Moreover, the ratio of the length of the short and long bronchi also makes up the golden ratio and is equal to 1: 1.618.

The structure of the golden orthogonal quadrilateral and spiral

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part in the same way as the larger part itself refers to the smaller one; or in other words, a smaller segment relates to a larger one as a larger one to everything.

In geometry, a rectangle with this aspect ratio has come to be called the golden rectangle. Its long sides compare to short sides in a 1.168: 1 ratio.

The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. Cutting off a square from the gold rectangle, the side of which is equal to the smaller side of the rectangle, we again get a smaller gold rectangle. This process can be continued indefinitely. As we continue to cut out the squares, we will get smaller and smaller golden rectangles. Moreover, they will be located along a logarithmic spiral, which is important in mathematical models of natural objects (for example, snail shells).

The spiral pole lies at the intersection of the diagonals of the initial rectangle and the first vertical cut to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also a golden triangle.

The English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for millennia, explaining it this way:

"We like the look of the spiral, because visually we can easily see it."

In nature

* The rule of the golden ratio underlying the structure of the spiral is found in nature very often in creations that are incomparable in beauty. The most vivid examples - a spiral shape can be seen in the arrangement of sunflower seeds, and in pine cones, in pineapples, cacti, the structure of rose petals, etc .;

* Botanists have established that in the arrangement of leaves on a branch, sunflower seeds or pine cones, the Fibonacci series is clearly manifested, and therefore, the law of the golden section is manifested;

The Supreme Lord established a special measure and proportionality for each of His creation, which is confirmed by examples found in nature. A great many examples can be cited when the growth process of living organisms occurs in strict accordance with the shape of a logarithmic spiral.

All the springs in the coil have the same shape. Mathematicians have found that even with an increase in the size of the springs, the shape of the spiral remains unchanged. There is no other form in mathematics that has the same unique properties as a spiral.

The structure of sea shells

Scientists who have studied the internal and external structure of the shells of soft-bodied mollusks living at the bottom of the seas stated:

“The inner surface of the shells is impeccably smooth, while the outer surface is covered with roughness and irregularities. The clam was in the shell, and for this the inner surface of the shell had to be perfectly smooth. External corners-bends of the shell increase its strength, hardness and thus increase its strength. The perfection and amazing intelligence of the structure of the shell (snail) is amazing. The spiral idea of ​​shells is a perfect geometric shape and is astonishing in its polished beauty. "

In most snails that have shells, the shell grows in a logarithmic spiral. However, there is no doubt that these unreasonable creatures have no idea not only of a logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral shell for themselves.

But then how could these unreasonable beings determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientists of the world call primitive forms of life, calculate that the logarithmic form of a shell would be ideal for their existence?

Of course not, because such a plan cannot be realized without the presence of reason and knowledge. But neither primitive mollusks, nor unconscious nature, which, however, some scientists call the creator of life on earth (?!)

Trying to explain the origin of such even the most primitive form of life by a random coincidence of certain natural circumstances is at least absurd. It is clear that this project is a conscious creation.

Biologist Sir D'arkey Thompson calls this type of growth of sea shells "The growth form of the gnomes."

Sir Thompson makes the following comment:

“There is no simpler system than the growth of seashells, which grow and expand proportionately, keeping the same shape. The shell, most surprisingly, grows, but never changes shape. "

The nautilus, a few centimeters in diameter, is the most striking example of the gnome type of growth. S. Morrison describes this process of growth of the nautilus in the following way, which is rather difficult to plan even with the human mind:

“Inside the nautilus shell there are many compartments-rooms with mother-of-pearl partitions, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in the front part of the shell, but already larger than the previous one, and the partitions of the room left behind are covered with a layer of mother-of-pearl. Thus, the spiral is expanding proportionally all the time. "

Here are just some types of spiral shells with a logarithmic growth in accordance with their scientific names:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa, Solarium Trochleare.

All discovered fossil remains of shells also had a developed spiral shape.

However, the logarithmic form of growth is found in the animal kingdom not only in molluscs. The horns of antelopes, wild goats, rams and other similar animals also develop in the form of a spiral according to the laws of the golden ratio.

The golden ratio in the human ear

In the inner ear of a person there is an organ called Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also created in the form of a snail, containing a stable logarithmic spiral shape = 73º 43 ’.

Horns and tusks of animals developing in a spiral shape

The tusks of elephants and extinct mammoths, the claws of lions and the beaks of parrots are logarithmic and resemble the shape of an axis that tends to turn into a spiral. Spiders always spin their webs in a logarithmic spiral. The structure of microorganisms such as plankton (species globigerinae, planorbis, vortex, terebra, turitellae and trochida) are also spiral-shaped.

The golden ratio in the structure of microworlds

Geometric shapes are not limited to just triangles, squares, pentagons, or hexagons. If we connect these figures in different ways with each other, then we get new three-dimensional geometric shapes. Examples of this are shapes such as a cube or a pyramid. However, in addition to them, there are also other three-dimensional figures that we did not have to meet in everyday life, and whose names we hear, perhaps for the first time. These three-dimensional figures include a tetrahedron (a regular four-sided figure), an octahedron, a dodecahedron, an icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula for the logarithmic spiral of the golden ratio.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. ... For example, many viruses have a three-dimensional geometric shape of the icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein coat of the adeno virus is formed from 252 units of protein cells arranged in a specific sequence. In each corner of the icosahedron there are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.

For the first time, the golden ratio in the structure of viruses was discovered in the 1950s. scientists from London Birkbeck College A. Klug and D. Kaspar. 13 Polyo virus was the first to appear in the logarithmic form. The form of this virus was found to be similar to that of the Rhino 14 virus.

The question arises, how do viruses form such complex three-dimensional forms, the structure of which contains the golden ratio, which even our human mind is quite difficult to construct? The discoverer of these forms of viruses, virologist A. Klug, gives the following comment:

“Dr. Kaspar and I have shown that for the spherical envelope of the virus, the most optimal shape is symmetry, such as the shape of the icosahedron. This arrangement minimizes the number of connecting elements ... Most of the Buckminster Fuller geodesic hemispherical cubes are built on a similar geometric principle. 14 The installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units. "

The Italian mathematician Leonardo Fibonacci lived in the 13th century and was one of the first in Europe to use Arabic (Indian) numbers. He came up with a somewhat artificial problem about rabbits that are raised on a farm, all of which are considered females, males are ignored. Rabbits start breeding after they are two months old, and then give birth to a rabbit every month. Rabbits never die.

It is necessary to determine how many rabbits will be on the farm in n months, if at the initial time there was only one newborn rabbit.

Obviously, the farmer has one rabbit in the first month and one rabbit in the second month. In the third month there will be two rabbits, in the fourth - three, etc. Let us denote the number of rabbits in n month like. Thus,
,
,
,
,
, …

An algorithm can be constructed to find for any n.

According to the condition of the problem, the total number of rabbits
v n+1 month is decomposed into three components:

one-month-old rabbits not capable of breeding, in quantity

;

Thus, we get

. (8.1)

Formula (8.1) allows you to calculate a series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...

The numbers in this sequence are called Fibonacci numbers .

If you accept
and
, then using formula (8.1) all other Fibonacci numbers can be determined. Formula (8.1) is called recurrent by the formula ( recurrence - "return" in Latin).

Example 8.1. Suppose there is a staircase in n steps. We can climb it with a step of one step, or - with a step of two steps. How many combinations of different lifting methods are there?

If n= 1, there is only one solution to the problem. For n= 2 there are 2 options: two single steps or one double. For n= 3 there are 3 options: three unit steps, or one unit and one double, or one double and one unit.

In the next case n= 4, we have 5 possibilities (1 + 1 + 1 + 1, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 2 + 2).

In order to answer the question asked with an arbitrary n, we denote the number of options as , and try to define
according to well-known and
... If we start with a single step, then we have combinations for the remaining n steps. If we start with a double step, then we have
combinations for the remaining n–1 steps. The total number of options for n+1 rung equals

. (8.2)

The resulting formula resembles formula (8.1) as a twin. However, this does not allow identifying the number of combinations with Fibonacci numbers ... We see, for example, that
, but
... However, the following relationship takes place:

.

This is true for n= 1, 2, and it is also true for each n... Fibonacci numbers and the number of combinations are calculated using the same formula, but the initial values
,
and
,
they differ.

Example 8.2. This example is of practical importance for error-correcting coding problems. Find the number of all binary words of length n that do not contain several zeros in a row. We denote this number by ... Obviously,
, and words of length 2 that satisfy our constraint are: 10, 01, 11, i.e.
... Let be
- such a word from n characters. If the symbol
, then
can be arbitrary (
) -literal word that does not contain several zeros in a row. Hence, the number of words with a unit at the end is
.

If the symbol
, then surely
and the first
symbol
can be arbitrary, taking into account the considered restrictions. Therefore, there is
word length n with a zero at the end. Thus, the total number of words of interest to us is equal to

.

Given that
and
, the resulting sequence of numbers is Fibonacci numbers.

Example 8.3. In Example 7.6, we found that the number of binary words of constant weight t(and length k) equals ... Now we find the number of binary words of constant weight t that do not contain several zeros in a row.

You can reason like this. Let be
the number of zeros in the words in question. Any word has
gaps between the nearest zeros, each of which contains one or more ones. It is assumed that
... Otherwise, there is not a single word without adjacent zeros.

If we remove exactly one unit from each interval, then we get a word of length
containing zeros. Any such word can be obtained in the indicated way from some (and, moreover, only one) k-literal word containing zeros, no two of which are next to each other. Hence, the required number coincides with the number of all words of length
containing exactly zeros, i.e. equals
.

Example 8.4. Let us prove that the sum
is equal to the Fibonacci numbers for any integer ... Symbol
denotes smallest integer greater than or equal to ... For example, if
, then
; what if
, then
ceil("ceiling"). Also occurs the symbol
which stands for largest integer less than or equal to ... In English, this operation is called floor ("floor").

If
, then
... If
, then
... If
, then
.

Thus, for the cases considered, the sum is really equal to the Fibonacci numbers. Now we give a proof for the general case. Since the Fibonacci numbers can be obtained using the recurrent equation (8.1), the following equality must hold:

.

And it actually does:

Here we used the formula (4.4) obtained earlier:
.

Sum of Fibonacci numbers

Let us determine the sum of the first n Fibonacci numbers.

0+1+1+2+3+5 = 12,

0+1+1+2+3+5+8 = 20,

0+1+1+2+3+5+8+13 = 33.

It is easy to see that by adding unity to the right side of each equation, we again get the Fibonacci number. General formula for determining the sum of the first n Fibonacci numbers is:

Let us prove this using the method of mathematical induction. To do this, write down:

This amount should be equal to
.

Reducing the left and right sides of the equation by –1, we obtain equation (6.1).

Formula for Fibonacci numbers

Theorem 8.1. Fibonacci numbers can be calculated using the formula

.

Proof... Let us verify the validity of this formula for n= 0, 1, and then prove the validity of this formula for an arbitrary n by induction. Let's calculate the ratio of the two nearest Fibonacci numbers:

We see that the ratio of these numbers fluctuates around 1.618 (if we ignore the first few values). By this property, the Fibonacci numbers resemble the members of a geometric progression. We will accept
, (
). Then the expression

converted to

which after simplifications looks like this

.

We got a quadratic equation, the roots of which are equal:

Now we can write:

(where c is constant). Both members and do not give Fibonacci numbers like
, while
... However, the difference
satisfies the recurrent equation:

For n= 0 this difference gives , that is:
... However, with n= 1 we have
... To obtain
, it is necessary to accept:
.

We now have two sequences: and
that start with the same two numbers and satisfy the same recurrence formula. They must be equal:
... The theorem is proved.

Ascending n member becomes very large, while
, and the role of the member in the difference is reduced. Therefore, for large n we can roughly write

.

We ignore 1/2 (since Fibonacci numbers go up to infinity with n to infinity).

Attitude
called golden ratio, it is used outside of mathematics (for example, in sculpture and architecture). The golden ratio is the ratio between the diagonal and the side regular pentagon(fig. 8.1).

Rice. 8.1. Regular pentagon and its diagonals

To denote the golden ratio, it is customary to use the letter
in honor of the famous Athenian sculptor Phidias.

Prime numbers

All natural numbers, large units, fall into two classes. The first includes numbers that have exactly two natural divisors, one and oneself, to the second - all the others. First class numbers are called simple, and the second - constituent... Prime numbers within the first three tens: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

The properties of prime numbers and their relationship with all natural numbers were studied by Euclid (3rd century BC). If you write out the prime numbers in a row, you will notice that their relative density decreases. There are 4 of them in the first ten, i.e. 40%, in a hundred - 25, i.e. 25%, per thousand - 168, i.e. less than 17%, per million - 78498, i.e. less than 8%, etc. However, their total number is infinite.

Among the primes, there are pairs of such, the difference between which is equal to two (the so-called simple twins), but the finiteness or infinity of such pairs has not been proved.

Euclid considered it obvious that by multiplying only prime numbers one can obtain all natural numbers, and each natural number can be represented as a product of prime numbers uniquely (up to the order of factors). Thus, the primes form a multiplicative basis for the natural series.

The study of the distribution of prime numbers led to the creation of an algorithm that allows obtaining tables of prime numbers. This algorithm is sieve of Eratosthenes(3rd century BC). This method consists in weeding out (for example, by strikethrough) those integers of a given sequence
that are divisible by at least one of the prime numbers less than
.

Theorem 8 . 2 . (Euclid's theorem). The number of primes is infinite.

Proof... Let us prove Euclid's theorem on the infinity of the number of primes by the method proposed by Leonard Euler (1707–1783). Euler considered the product over all primes p:

at
... This product converges, and if we expand it, then, due to the uniqueness of the decomposition of natural numbers into prime factors, it turns out that it is equal to the sum of the series , whence Euler's identity follows:

.

Since at
the series on the right diverges (harmonic series), then Euclid's theorem follows from Euler's identity.

Russian mathematician P.L. Chebyshev (1821-1894) deduced a formula that determines the limits in which the number of primes is enclosed
not exceeding X:

,

where
,
.