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The fibonacci sequence and the principles of the golden section. "Golden Ratio" and Fibonacci Numbers Practical Fibonacci Numbers

is a comprehensive 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 acquainted with the golden rule, humanity no longer cheated on it.

Surely you have often wondered why Nature is able to create such amazing harmonious structures that delight and delight the eye. Why artists, poets, composers, architects create amazing works of art from century to century. What is the secret and what laws underlie these harmonious creatures? No one can unequivocally answer this question, but in our book we will try to open the veil and tell you about one of the secrets of the universe - the Golden Section or, as it is also called, the Golden or Divine Proportion. The Golden Section is called the PHI number (Phi) in honor of the great ancient Greek sculptor Phidias (Phidius), who used this number in his sculptures.

For more than one century, scientists have been using the unique mathematical properties of the PHI number, and these studies continue to this day. This number has found wide application in all areas of modern science, which we will also try to popularize on the pages. There are also a number of fibonacci sequence what is it You will find out more…

Definition 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 part refers to the whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of the parts of the whole will correlate as 62% by 38%. This ratio operates in the forms of space and time.

The ancients saw the golden section as 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 the idea of ​​​​golden proportions, they knew about them in Russia, but for the first time the monk Luca Pacioli explained the golden ratio scientifically in the book Divine Proportion, illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the Son, the large Father, and the whole the Holy Spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly connected with the golden section rule. 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 neighboring numbers of the Fibonacci series in the limit tends to the Golden Ratio. Kepler drew attention to the relationship of this sequence to the golden ratio: It is arranged in such a way that the two lower terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term. Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.

He also devoted a lot of time to studying the features of the golden section, most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in golden division.

With time rule the rule, depending on the stress and context, can mean the following: A rule is a requirement for the fulfillment of certain conditions (on behavior) by all participants in an action (game, the golden ratio turned into an academic routine, and only the philosopher Adolf Zeising in 1855 gave it 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 caused a lot of criticism.

Universal code of nature

Even without going into 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 fall under it, there is a golden section and in the shape of an egg, if a conditional line is drawn through its widest part.

The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with proportions of the golden section. In his opinion, one of the most interesting forms is spiraling.
Even Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted gravity nature the material world of the Universe, in essence - the main object of study of the natural sciences to spiral shapes, calling the spiral of the curve of life. Modern scientists have found that such manifestations of spiral forms in nature as the snail shell, the arrangement of sunflower seeds, web patterns, the movement of a 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 section. Man is universal the form can mean: The shape of the object - the relative position of the boundaries (contours) of the object, the object, as well as the relative position of the points of the line to test the laws of the golden section. 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 positions superimposed on each other. Based on the studies of the Roman architect Vitruvius, Leonardo similarly tried 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, exploring the proportionality of man, did a tremendous job. He measured about two thousand human bodies, as well as many ancient statues, and deduced that the golden ratio expresses the average law. AT man living intelligent social, subject of socio-historical activity and culture almost all parts of the body are subordinate to him, but the main indicator golden something made of gold section is a division body In mathematics: A body (algebra) is a set with two operations (addition and multiplication) that 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 golden section an ambiguous term meaning: Section in drawing - unlike a section, the image is only of a figure formed by cutting the body with a plane (planes) without depicting parts behind this than the proportions of the female body 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 to the 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 (Italian) da Vinci, the process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Dürer to define points can mean: A point is an abstract object in space that has no measurable characteristics other than coordinates the golden section used the proportional compass invented by him.

The art critic F. V. Kovalev, having studied in detail the painting by Nikolai Ge Alexander Sergeevich Pushkin in the village of Mikhailovsky, notes that every detail of the canvas, whether it be a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.

Researchers of the golden section tirelessly study and measure the masterpieces of architecture, claiming that they have become such because they were created according to the golden canons: their list includes 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 historians, they facilitate the perception of the work and form an aesthetic sensation in the viewer.

Word, sound and film

The forms of temporary art in their own way demonstrate to us the principle of golden division. Literary critics, for example, noticed that the most popular number of lines in the poems of the late period of Pushkin's work corresponds to the Fibonacci series 5, 8, 13, 21, 34.

The rule of the golden section also applies in individual works of the Russian classic. So the climax of the Queen of Spades is the dramatic scene of Herman and the Countess, ending with the death of the latter. There are 853 lines in the story, and the climax falls on the 535th line (853:535=1.6), this is the point of the golden section.

The Soviet musicologist E. K. Rozenov notes the amazing accuracy of the golden section ratios 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 golden ratio point usually accounts for the most striking or unexpected musical solution.
Film director Sergei Eisenstein deliberately coordinated the script for his film Battleship Potemkin with the golden section rule, dividing the film into five parts. In the first three sections, the action takes place on a 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 has gone 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-decimal number system and the positional principle of representing numbers, Euclid's trigonometry and geometry, incommensurable segments, the Golden Section and Platonic solids, the beginnings 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 the fundamental scientific ideas that permeate all stages of scientific and technological progress and have an impact on various areas 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." The deep faith of the outstanding physicist in the existence of universal laws of harmony of the universe is evidenced by another well-known statement by Einstein: "The religiosity 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 brilliant Russian philosopher Alexei Losev assesses the main achievements of the ancient Greeks in this area as follows:

“From the point of view of Plato, and indeed from the point of view of all ancient cosmology, the world is a kind of proportional whole, subject to the law of harmonic division - the Golden Section ... Their (ancient Greeks) system of cosmic proportions is often depicted in literature as a curious result of unbridled and wild fantasy. This kind of explanation shows the anti-scientific helplessness of those who claim it. However, this historical and aesthetic phenomenon can be understood only in connection with a holistic understanding of history, that is, using the dialectical-materialistic concept of culture and looking for an answer in the features of ancient social life.

“The law of golden division must be a dialectical necessity. This is the thought that, as far as I know, I am spending 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 Mean 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 that professionally study the Golden Section, 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 based on the Fibonacci numbers and the Golden Section have been made. The discovery of "quasi-crystals", made in 1982 by the Israeli scientist Dan Shechtman, based on the Golden Section and "pentagonal" symmetry, is of revolutionary importance for modern physics. A breakthrough in modern ideas about 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. The Belarusian philosopher Eduard Soroko formulated the "Law of structural harmony of systems", based on the Golden Section and playing an important role in the processes of self-organization. Thanks to the research of American scientists Elliott, Prechter and Fisher, Fibonacci numbers have actively entered the business sphere and have become the basis of optimal business and trading strategies. 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 life is based on the properties of the dodecahedron and icosahedron - two "Platonic solids" associated with the Golden Section. And finally, 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 the living cell - is built according to one principle - the dodecahedron and icosahedron infinitely inscribed into each other, which are in proportion to the Golden Section!

Ukrainian professor and doctor of science Stakhov A.P. was able to create some . The essence of this generalization is extremely simple. If we specify a non-negative integer p = 0, 1, 2, 3, ... and divide the segment “AB” by point C in such a proportion that it would be:

Then the universal formula of the golden section is the expression:

xp + 1 = xp + 1

The surrounding world, starting with the smallest invisible particles, and ending with distant galaxies of boundless space, is fraught with many unsolved mysteries. However, the veil of mystery has already been lifted over some of them thanks to the inquisitive minds of a number of scientists.

One such example is golden ratio and Fibonacci numbers that form its basis. This pattern has been displayed in mathematical form and is often found in the nature surrounding a person, once again excluding the possibility that it arose as a result of chance.

Fibonacci numbers and their sequence

Fibonacci number sequence called a series of numbers, each of which is the sum of the previous two:

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

A feature of this sequence is the numerical values ​​that are obtained by dividing the numbers of this series by each other.

A series of Fibonacci numbers has its own interesting patterns:

  • In the Fibonacci series, each number divided by the next will show a value tending towards 0,618 . The farther the numbers are from the beginning of the series, the more accurate the ratio will be. For example, the numbers taken at the beginning of the row 5 and 8 will show 0,625 (5/8=0,625 ). If we take the numbers 144 and 233 , then they will show the ratio 0.618 .
  • In turn, if in a series of Fibonacci numbers we divide the number by the previous one, then the result of the division will tend to 1,618 . For example, the same numbers were used as mentioned above: 8/5=1,6 and 233/144=1,618 .
  • The number divided by the next one after it will show a value approaching 0,382 . And the farther from the beginning of the series the numbers are taken, the more accurate the value of the ratio: 5/13=0,385 and 144/377=0,382 . Dividing the digits in reverse order will give the result 2,618 : 13/5=2,6 and 377/144=2,618 .

Using the above calculation methods and increasing the gaps between the numbers, you can display the following series of values: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236, which is widely used in Fibonacci tools in the forex market.

Golden Ratio or Divine Proportion

The “golden section” and Fibonacci numbers are very clearly represented by the analogy with a segment. If segment AB is divided by point C in such a ratio that the condition is met:

AC / BC \u003d BC / AB, then it will be the "golden section"

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Surprisingly, it is this ratio that can be traced in the series of Fibonacci numbers. Taking a few numbers from the series, you can check by calculation that this is so. For example, such a sequence of Fibonacci numbers ... 55, 89, 144 ... Let the number 144 be the whole segment AB, which was mentioned above. Since 144 is the sum of the two previous numbers, then 55+89=AC+BC=144.

Dividing the segments will show the following results:

AC/BC=55/89=0.618

BC/AB=89/144=0.618

If we take the segment AB as a whole, or as a unit, then AC \u003d 55 will be 0.382 of this whole, and BC \u003d 89 will be equal to 0.618.

Where are Fibonacci numbers found?

The regular sequence of Fibonacci numbers was known to the Greeks and Egyptians long before Leonardo Fibonacci himself. This number series acquired such a name after the famous mathematician ensured the wide distribution of this mathematical phenomenon in scientific ranks.

It is important to note that the golden Fibonacci numbers are not just science, but a mathematical representation of the world around them. Many natural phenomena, representatives of the flora and fauna have the "golden section" in their proportions. These are spiral curls of the shell, and the arrangement of sunflower seeds, cacti, pineapples.

The spiral, the proportions of the branches of which are subject to the laws of the "golden section", underlies the formation of a hurricane, the weaving of a web by a spider, the shape of many galaxies, the interweaving of DNA molecules and many other phenomena.

The length of the lizard's tail to its body has a ratio of 62 to 38. The chicory shoot, before releasing a leaf, makes a release. After the first sheet is released, a second ejection occurs before the release of the second sheet, equal in strength to 0.62 of the conditionally accepted unit of force of the first ejection. The third outlier is 0.38 and the fourth is 0.24.

It is also of great importance for a trader that the price movement in the Forex market is often subject to the patterns of golden Fibonacci numbers. Based on this sequence, a number of tools have been created that a trader can use in his arsenal.

Often used by traders, the instrument "" can accurately show the price movement targets, as well as the levels of its correction.

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Introduction

THE HIGHEST PURPOSE OF MATHEMATICS IS TO FIND THE HIDDEN ORDER IN THE CHAOS THAT SURROUNDS US.

Viner N.

A person strives for knowledge all his life, tries to study the world around him. And in the process of observation, he has questions that need to be answered. Answers are found, but new questions appear. In archaeological finds, in the traces of civilization, distant from each other in time and space, one and the same element is found - a pattern in the form of a spiral. Some consider it a symbol of the sun and associate it with the legendary Atlantis, but its true meaning is unknown. What do the shapes of a galaxy and an atmospheric cyclone, the arrangement of leaves on a stem and seeds in a sunflower have in common? These patterns come down to the so-called "golden" spiral, the amazing Fibonacci sequence, discovered by the great Italian mathematician of the 13th century.

History of Fibonacci Numbers

For the first time about what Fibonacci numbers are, I heard from a mathematics teacher. But, besides, how the sequence of these numbers is formed, I did not know. This is what this sequence is actually famous for, how it affects a person, and I want to tell you. Little is known about Leonardo Fibonacci. There is not even an exact date of his birth. It is known that he was born in 1170 in the family of a merchant, in the city of Pisa in Italy. Fibonacci's father was often in Algiers on business, and Leonardo studied mathematics there with Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is the "Book of the abacus", which contains almost all the arithmetic and algebraic information of that time. 2

Fibonacci numbers are a sequence of numbers with a number of properties. Fibonacci discovered this numerical sequence by accident when he tried to solve a practical problem about rabbits in 1202. “Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, in order to find out how many pairs of rabbits will be born during the year, if the nature of rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth. When solving the problem, he took into account that each pair of rabbits gives birth to two more pairs during their life, and then dies. This is how the sequence of numbers appeared: 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence, each next number is equal to the sum of the two previous ones. It's called the Fibonacci sequence. Mathematical properties of a sequence

I wanted to explore this sequence, and I identified some of its properties. This rule is of great importance. The sequence slowly approaches some constant ratio of about 1.618, and the ratio of any number to the next is about 0.618.

One can notice a number of curious properties of Fibonacci numbers: two neighboring numbers are coprime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you choose any 10 neighboring numbers from the Fibonacci sequence and add them together, you will always get a number that is a multiple of 11. But that's not all. Each sum is equal to the number 11 multiplied by the seventh member of the given sequence. And here is another interesting feature. For any n, the sum of the first n members of the sequence will always be equal to the difference of the (n + 2) -th and first member of the sequence. This fact can be expressed by the formula: 1+1+2+3+5+…+an=a n+2 - 1. Now we have the following trick: to find the sum of all terms

sequence between two given members, it suffices to find the difference of the corresponding (n+2)-x members. For example, a 26 + ... + a 40 \u003d a 42 - a 27. Now let's look for a connection between Fibonacci, Pythagoras and the "golden section". The most famous evidence of the mathematical genius of mankind is the Pythagorean theorem: in any right triangle, the square of the hypotenuse is equal to the sum of the squares of its legs: c 2 \u003d b 2 + a 2. From a geometric point of view, we can consider all the sides of a right triangle as the sides of three squares built on them. The Pythagorean theorem says that the total area of ​​the squares built on the legs of a right triangle is equal to the area of ​​the square built on the hypotenuse. If the lengths of the sides of a right triangle are integers, then they form a group of three numbers called Pythagorean triples. Using the Fibonacci sequence, you can find such triples. Take any four consecutive numbers from the sequence, for example, 2, 3, 5 and 8, and construct three more numbers as follows: 1) the product of the two extreme numbers: 2*8=16; 2) the double product of the two numbers in the middle: 2* (3 * 5) \u003d 30; 3) the sum of the squares of two average numbers: 3 2 +5 2 \u003d 34; 34 2 =30 2 +16 2 . This method works for any four consecutive Fibonacci numbers. Predictably, any three consecutive numbers of the Fibonacci series behave in a predictable way. If you multiply the two extremes of them and compare the result with the square of the average number, then the result will always differ by one. For example, for numbers 5, 8 and 13 we get: 5*13=8 2 +1. If we consider this property from the point of view of geometry, we can notice something strange. Divide the square

size 8x8 (total 64 small squares) into four parts, the lengths of the sides of which are equal to the Fibonacci numbers. Now from these parts we will build a rectangle measuring 5x13. Its area is 65 small squares. Where does the extra square come from? The thing is that a perfect rectangle is not formed, but tiny gaps remain, which in total give this additional unit of area. Pascal's triangle also has a connection with the Fibonacci sequence. You just need to write the lines of Pascal's triangle one under the other, and then add the elements diagonally. Get the Fibonacci sequence.

Now consider a "golden" rectangle, one side of which is 1.618 times longer than the other. At first glance, it may seem like an ordinary rectangle to us. However, let's do a simple experiment with two ordinary bank cards. Let's put one of them horizontally and the other vertically so that their lower sides are on the same line. If we draw a diagonal line in a horizontal map and extend it, we will see that it will pass exactly through the upper right corner of the vertical map - a pleasant surprise. Maybe this is an accident, or maybe such rectangles and other geometric shapes using the "golden ratio" are especially pleasing to the eye. Did Leonardo da Vinci think about the golden ratio while working on his masterpiece? This seems unlikely. However, it can be argued that he attached great importance to the connection between aesthetics and mathematics.

Fibonacci numbers in nature

The connection of the golden section with beauty is not only a matter of human perception. It seems that nature itself has allocated a special role to F. If squares are sequentially inscribed in the "golden" rectangle, then an arc is drawn in each square, then an elegant curve is obtained, which is called a logarithmic spiral. It is not a mathematical curiosity at all. 5

On the contrary, this wonderful line is often found in the physical world: from the shell of a nautilus to the arms of galaxies, and in the elegant spiral of the petals of a full-blown rose. The connections between the golden ratio and Fibonacci numbers are numerous and unexpected. Consider a flower that looks very different from a rose - a sunflower with seeds. The first thing we see is that the seeds are arranged in two kinds of spirals: clockwise and counterclockwise. If we count the clockwise spirals, we get two seemingly ordinary numbers: 21 and 34. This is not the only example when you can find Fibonacci numbers in the structure of plants.

Nature gives us numerous examples of the arrangement of homogeneous objects described by Fibonacci numbers. In the various spiral arrangements of small plant parts, two families of spirals can usually be seen. In one of these families, the spirals curl clockwise, and in the other - counterclockwise. Spiral numbers of one type and another often turn out to be neighboring Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from bottom left to right up. On many cones, the seeds are arranged in three spirals, gently winding around the stem of the cone. They are arranged in five spirals, winding steeply in the opposite direction. In large cones, it is possible to observe 5 and 8, and even 8 and 13 spirals. The Fibonacci spirals are also clearly visible on the pineapple: there are usually 8 and 13 of them.

The chicory shoot makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of lesser force, releases an even smaller leaf and ejection again. Its growth impulses gradually decrease in proportion to the "golden" section. To appreciate the huge role of Fibonacci numbers, just look at the beauty of the nature around us. Fibonacci numbers can be found in quantity

branches on the stem of each growing plant and in the number of petals.

Let's count the petals of some flowers - the iris with its 3 petals, the primrose with 5 petals, the ragweed with 13 petals, the daisy with 34 petals, the aster with 55 petals, and so on. Is this a coincidence, or is it the law of nature? Look at the stems and flowers of the yarrow. Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of the "Golden" numbers found in nature. These laws operate regardless of our consciousness and the desire to accept them or not. 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 gene structures of living organisms, in the structure of individual human organs and the body as a whole, and also manifest themselves in biorhythms and the functioning of the brain and visual perception.

Fibonacci numbers in architecture

The Golden Ratio also manifests itself in many remarkable architectural creations throughout the history of mankind. It turns out that even ancient Greek and Egyptian mathematicians knew these coefficients long before Fibonacci and called them the "golden section". The principle of the "golden section" was used by the Greeks in the construction of the Parthenon, the Egyptians - the Great Pyramid of Giza. Advances in building technology and the development of new materials opened up new possibilities for 20th-century architects. American Frank Lloyd Wright was one of the main proponents of organic architecture. Shortly before his death, he designed the Solomon Guggenheim Museum in New York, which is an inverted spiral, and the interior of the museum resembles a nautilus shell. Polish-Israeli architect Zvi Hecker also used spiral structures in the design of the Heinz Galinski School in Berlin, completed in 1995. Hecker started with the idea of ​​a sunflower with a central circle, from where

all architectural elements diverge. The building is a combination

orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and controlled chaos of nature. Its architecture mimics a plant that follows the movement of the sun, so the classrooms are lit up throughout the day.

In Quincy Park, located in Cambridge, Massachusetts (USA), the "golden" spiral can often be found. The park was designed in 1997 by artist David Phillips and is located near the Clay Mathematical Institute. This institution is a well-known center for mathematical research. In Quincy Park, you can walk among the "golden" spirals and metal curves, reliefs of two shells and a rock with a square root symbol. On the plate is written information about the "golden" proportion. Even bike parking uses the F symbol.

Fibonacci numbers in psychology

In psychology, there are turning points, crises, upheavals that mark the transformation of the structure and functions of the soul on a person's life path. If a person has successfully overcome these crises, then he becomes able to solve problems of a new class, which he had not even thought about before.

The presence of fundamental changes gives reason to consider the time of life as a decisive factor in the development of spiritual qualities. After all, nature measures time for us not generously, “no matter how much it will be, so much will be,” but just enough so that the development process materializes:

    in the structures of the body;

    in feelings, thinking and psychomotor - until they acquire harmony necessary for the emergence and launch of the mechanism

    creativity;

    in the structure of human energy potential.

The development of the body cannot be stopped: the child becomes an adult. With the mechanism of creativity, everything is not so simple. Its development can be stopped and its direction changed.

Is there a chance to catch up with time? Undoubtedly. But for this you need to do a lot of work on yourself. What develops freely, naturally, does not require special efforts: the child develops freely and does not notice this enormous work, because the process of free development is created without violence against oneself.

How is the meaning of the life path understood in everyday consciousness? The inhabitant sees it like this: at the foot - the birth, at the top - the prime of life, and then - everything goes downhill.

The wise man will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth ... Why is that? Few people are able to answer, although everyone is sure that these are closed, integral stages of life.

To find out how the mechanism of creativity develops, V.V. Klimenko used mathematics, namely the laws of Fibonacci numbers and the proportion of the "golden section" - the laws of nature and human life.

Fibonacci numbers divide our life into stages according to the number of years lived: 0 - the beginning of the countdown - the child was born. He still lacks not only psychomotor skills, thinking, feelings, imagination, but also operational energy potential. He is the beginning of a new life, a new harmony;

    1 - the child has mastered walking and masters the immediate environment;

    2 - understands speech and acts using verbal instructions;

    3 - acts through the word, asks questions;

    5 - "age of grace" - the harmony of psychomotor, memory, imagination and feelings, which already allow the child to embrace the world in all its integrity;

    8 - feelings come to the fore. They are served by imagination, and thinking, by the forces of its criticality, is aimed at supporting the internal and external harmony of life;

    13 - the mechanism of talent begins to work, aimed at transforming the material acquired in the process of inheritance, developing one's own talent;

    21 - the mechanism of creativity has approached a state of harmony and attempts are being made to perform talented work;

    34 - harmony of thinking, feelings, imagination and psychomotor skills: the ability to brilliant work is born;

    55 - at this age, subject to the preserved harmony of soul and body, a person is ready to become a creator. Etc…

What are Fibonacci serifs? They can be compared to dams on the path of life. These dams await each of us. First of all, it is necessary to overcome each of them, and then patiently raise your level of development, until one day it falls apart, opening the way to the next free flow.

Now that we understand the meaning of these nodal points of age development, let's try to decipher how it all happens.

At 1 year the child learns to walk. Before that, he knew the world with the front of his head. Now he knows the world with his hands - the exclusive privilege of man. The animal moves in space, and he, cognizing, masters the space and masters the territory on which he lives.

2 years understands the word and acts in accordance with it. It means that:

the child learns the minimum number of words - meanings and patterns of action;

    yet does not separate itself from the environment and is merged into integrity with the environment,

    Therefore, he acts on someone else's instructions. At this age, he is the most obedient and pleasant for parents. From a man of the senses, the child turns into a man of knowledge.

3 years- action with the help of one's own word. The separation of this person from the environment has already taken place - and he is learning to be an independently acting person. Hence he:

    consciously opposes the environment and parents, kindergarten teachers, etc.;

    is aware of its sovereignty and fights for independence;

    tries to subjugate close and well-known people to his will.

Now for a child, a word is an action. This is where the acting person begins.

5 years- Age of Grace. He is the personification of harmony. Games, dances, dexterous movements - everything is saturated with harmony, which a person tries to master with his own strength. Harmonious psychomotor contributes to bringing to a new state. Therefore, the child is directed to psychomotor activity and strives for the most active actions.

Materialization of the products of the work of sensitivity is carried out through:

    the ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all sense organs work for this process);

    ability to design the outside world, including yourself

    (creation of a second nature, hypotheses - to do both tomorrow, build a new machine, solve a problem), by the forces of critical thinking, feelings and imagination;

    the ability to create a second, man-made nature, products of activity (implementation of the plan, specific mental or psychomotor actions with specific objects and processes).

After 5 years, the imagination mechanism comes forward and begins to dominate the rest. The child does a gigantic job, creating fantastic images, and lives in the world of fairy tales and myths. The hypertrophy of the child's imagination causes surprise in adults, because the imagination does not correspond to reality in any way.

8 years- feelings come to the fore and their own measurements of feelings (cognitive, moral, aesthetic) arise when the child unmistakably:

    evaluates the known and the unknown;

    distinguishes the moral from the immoral, the moral from the immoral;

    beauty from what threatens life, harmony from chaos.

13 years old- the mechanism of creativity begins to work. But that doesn't mean it's working at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If even in this age period of development harmony is preserved, which almost all the time rebuilds its structure, then the child will painlessly get to the next dam, overcome it imperceptibly and will live at the age of a revolutionary. At the age of a revolutionary, the youth must take a new step forward: to separate from the nearest society and live in it a harmonious life and activity. Not everyone can solve this problem that arises before each of us.

21 years old If a revolutionary has successfully overcome the first harmonious peak of life, then his mechanism of talent is capable of fulfilling a talented

work. Feelings (cognitive, moral, or aesthetic) sometimes overshadow thinking, but in general, all elements work in harmony: feelings are open to the world, and logical thinking is able to name and find measures of things from this peak.

The mechanism of creativity, developing normally, reaches a state that allows it to receive certain fruits. He starts to work. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by feelings and thinking, antagonism arises between them. Feelings win. This ability is gradually gaining power, and the boy begins to use it.

34 years- balance and harmony, productive effectiveness of talent. Harmony of thinking, feelings and imagination, psychomotor skills, which is replenished with optimal energy potential, and the mechanism as a whole - an opportunity is born to perform brilliant work.

55 years- a person can become a creator. The third harmonious peak of life: thinking subdues the power of feelings.

Fibonacci numbers name the stages of human development. Whether a person goes through this path without stopping depends on parents and teachers, the educational system, and then on himself and on how a person will learn and overcome himself.

On the path of life, a person discovers 7 objects of relationships:

    From birthday to 2 years - the discovery of the physical and objective world of the immediate environment.

    From 2 to 3 years - the discovery of oneself: "I am Myself."

    From 3 to 5 years - speech, the effective world of words, harmony and the "I - You" system.

    From 5 to 8 years old - the discovery of the world of other people's thoughts, feelings and images - the "I - We" system.

    From 8 to 13 years old - the discovery of the world of tasks and problems solved by the geniuses and talents of mankind - the system "I - Spirituality".

    From 13 to 21 years old - the discovery of the ability to independently solve well-known tasks, when thoughts, feelings and imagination begin to work actively, the "I - Noosphere" system arises.

    From 21 to 34 years old - the discovery of the ability to create a new world or its fragments - the realization of the self-concept "I am the Creator".

The life path has a space-time structure. It consists of age and individual phases, determined by many parameters of life. A person masters to a certain extent the circumstances of his life, becomes the creator of his history and the creator of the history of society. A truly creative attitude to life, however, does not appear immediately and not even in every person. There are genetic links between the phases of the life path, and this determines its natural character. It follows that, in principle, it is possible to predict future development on the basis of knowledge of its early phases.

Fibonacci numbers in astronomy

It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found regularity and order in the distances between the planets of the solar system. But one case seemed to be against the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the XIX century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.

Conclusion

In the process of research, I found out that Fibonacci numbers are widely used in the technical analysis of stock prices. One of the simplest ways to use Fibonacci numbers in practice is to determine the length of time after which an event will occur, for example, a price change. The analyst counts a certain number of Fibonacci days or weeks (13,21,34,55, etc.) from the previous similar event and makes a forecast. But this is too hard for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are the statue in front of the Leaning Tower of Pisa and two streets that bear his name, one in Pisa and the other in Florence. And yet, in connection with everything I have seen and read, quite natural questions arise. Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? What will be next? Finding the answer to one question, you get the next. If you solve it, you get two new ones. Deal with them, three more will appear. Having solved them, you will acquire five unresolved ones. Then eight, thirteen, and so on. Do not forget that there are five fingers on two hands, two of which consist of two phalanges, and eight of which consist of three.

Literature:

    Voloshinov A.V. "Mathematics and Art", M., Enlightenment, 1992

    Vorobyov N.N. "Fibonacci numbers", M., Nauka, 1984

    Stakhov A.P. "The Da Vinci Code and the Fibonacci Series", Peter Format, 2006

    F. Corvalan “The Golden Ratio. Mathematical language of beauty”, M., De Agostini, 2014

    Maksimenko S.D. "Sensitive periods of life and their codes".

    "Fibonacci numbers". Wikipedia

Have you ever heard that mathematics is called "the queen of all sciences"? Do you agree with this statement? As long as mathematics remains a boring textbook puzzle for you, you can hardly feel the beauty, versatility and even humor of this science.

But there are topics in mathematics that help to make curious observations on things and phenomena that are common to us. And even try to penetrate the veil of the mystery of the creation of our universe. There are curious patterns in the world that can be described with the help of mathematics.

Introducing Fibonacci Numbers

Fibonacci numbers name the elements of a sequence. In it, each next number in the series is obtained by summing the two previous numbers.

Sample sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

You can write it like this:

F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2

You can start a series of Fibonacci numbers with negative values n. Moreover, the sequence in this case is two-sided (that is, it covers negative and positive numbers) and tends to infinity in both directions.

An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

The formula in this case looks like this:

F n = F n+1 - F n+2 or otherwise you can do it like this: F-n = (-1) n+1 Fn.

What we now know as "Fibonacci numbers" was known to ancient Indian mathematicians long before they were used in Europe. And with this name, in general, one continuous historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only several centuries after his death. But let's talk about everything in order.

Leonardo of Pisa aka Fibonacci

The son of a merchant who became a mathematician, and subsequently received the recognition of his descendants as the first major mathematician of Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which then, we recall, were not yet called that). Which he described at the beginning of the 13th century in his work “Liber abaci” (“The Book of the Abacus”, 1202).

Traveling with his father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were one of the best specialists in this matter, and in many other sciences). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.

Having properly comprehended everything he read and connected his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the “Book of the Abacus” already mentioned above. In addition to her, he created:

  • "Practica geometriae" ("Practice of Geometry", 1220);
  • "Flos" ("Flower", 1225 - a study on cubic equations);
  • "Liber quadratorum" ("The Book of Squares", 1225 - problems on indefinite quadratic equations).

He was a great lover of mathematical tournaments, so in his treatises he paid much attention to the analysis of various mathematical problems.

Very little biographical information remains about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was fixed to him only in the 19th century.

Fibonacci and his problems

After Fibonacci, a large number of problems remained, which were very popular among mathematicians in the following centuries. We will consider the problem of rabbits, in the solution of which the Fibonacci numbers are used.

Rabbits are not only valuable fur

Fibonacci set the following conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, male and female.

These conditional rabbits are placed in a closed space and breed enthusiastically. It is also stipulated that no rabbit dies from some mysterious rabbit disease.

We need to calculate how many rabbits we will get in a year.

  • At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
  • The second month - we already have 2 pairs of rabbits (a pair has parents + 1 pair - their offspring).
  • Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
  • Fourth month: The first couple gives birth to a new couple, the second couple does not lose time and also gives birth to a new couple, the third couple is just mating. Total - 5 pairs of rabbits.

Number of rabbits in n-th month = number of pairs of rabbits from the previous month + number of newborn pairs (there are the same number of pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: F n \u003d F n-1 + F n-2.

Thus, we obtain a recurrent (explanation of recursion- below) numerical sequence. In which each next number is equal to the sum of the previous two:

  1. 1 + 1 = 2
  2. 2 + 1 = 3
  3. 3 + 2 = 5
  4. 5 + 3 = 8
  5. 8 + 5 = 13
  6. 13 + 8 = 21
  7. 21 + 13 = 34
  8. 34 + 21 = 55
  9. 55 + 34 = 89
  10. 89 + 55 = 144
  11. 144 + 89 = 233
  12. 233+ 144 = 377 <…>

You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we have set a specific period - a year, we are interested in the result obtained on the 12th "move". Those. 13th member of the sequence: 377.

The answer is in the problem: 377 rabbits will be obtained if all the stated conditions are met.

One of the properties of the Fibonacci sequence is very curious. If you take two consecutive pairs from a row and divide the larger number by the smaller one, the result will gradually approach golden ratio(You can read more about it later in the article).

In the language of mathematics, "relationship limit a n+1 to a n equal to the golden ratio.

More problems in number theory

  1. Find a number that can be divided by 7. Also, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
  2. Find a square number. It is known about him that if you add 5 to it or subtract 5, you again get a square number.

We invite you to find answers to these questions on your own. You can leave us your options in the comments to this article. And then we will tell you if your calculations were correct.

An explanation about recursion

recursion- definition, description, image of an object or process, which contains this object or process itself. That is, in fact, an object or process is a part of itself.

Recursion finds wide application in mathematics and computer science, and even in art and popular culture.

Fibonacci numbers are defined using a recursive relation. For number n>2 n- e number is (n - 1) + (n - 2).

Explanation of the golden ratio

golden ratio- the division of a whole (for example, a segment) into such parts that are related according to the following principle: a large part relates to a smaller one in the same way as the entire value (for example, the sum of two segments) to a larger part.

The first mention of the golden ratio can be found in Euclid's treatise "Beginnings" (about 300 BC). In the context of building a regular rectangle.

The term familiar to us in 1835 was introduced by the German mathematician Martin Ohm.

If you describe the golden ratio approximately, it is a proportional division into two unequal parts: approximately 62% and 38%. Numerically, the golden ratio is the number 1,6180339887 .

The golden ratio finds practical application in the visual arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (S. Ezenstein's Battleship Potemkin) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This view is still popular today. Although, according to the results of research, visually, most people do not perceive such a proportion as the most successful option and consider it too elongated (disproportionate).

  • Cut length with = 1, a = 0,618, b = 0,382.
  • Attitude with to a = 1, 618.
  • Attitude with to b = 2,618

Now back to the Fibonacci numbers. Take two successive terms from its sequence. Divide the larger number by the smaller and get approximately 1.618. And now let's use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.

Here is an example: 144, 233, 377.

233/144 = 1.618 and 233/377 = 0.618

By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is almost not respected for the beginning of the sequence. But on the other hand, as you move along the row and the numbers increase, it works fine.

And in order to calculate the entire series of Fibonacci numbers, it is enough to know three members of the sequence, following each other. You can see for yourself!

Golden Rectangle and Fibonacci Spiral

Another curious parallel between the Fibonacci numbers and the golden ratio allows us to draw the so-called "golden rectangle": its sides are related in the proportion of 1.618 to 1. But we already know what the number 1.618 is, right?

For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and build a rectangle with the following parameters: width = 8, length = 13.

And then we break the large rectangle into smaller ones. Mandatory condition: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. the side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.

The way it is done in this figure (for convenience, the figures are signed in Latin letters).

By the way, you can build rectangles in the reverse order. Those. start building from squares with a side of 1. To which, guided by the principle voiced above, figures with sides equal to the Fibonacci numbers are completed. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.

If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Rather, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.

Such a spiral is often found in nature. Mollusk shells are one of the most striking examples. Moreover, some galaxies that can be seen from Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when shooting them from satellites.

It is curious that the DNA helix also obeys the golden section rule - the corresponding pattern can be seen in the intervals of its bends.

Such amazing “coincidences” cannot but excite the minds and give rise to talk about a certain single algorithm that all phenomena in the life of the Universe obey. Now do you understand why this article is called that way? And the doors to what amazing worlds can mathematics open for you?

Fibonacci numbers in nature

The connection between Fibonacci numbers and the golden ratio suggests curious patterns. So curious that it is tempting to try to find sequences like Fibonacci numbers in nature and even in the course of historical events. And nature indeed gives rise to such assumptions. But can everything in our life be explained and described with the help of mathematics?

Examples of wildlife that can be described using the Fibonacci sequence:

  • the order of arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);

  • the location of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row is clockwise, the other is counterclockwise);

  • location of scales of pine cones;
  • flower petals;
  • pineapple cells;
  • the ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.

Problems in combinatorics

Fibonacci numbers are widely used in solving problems in combinatorics.

Combinatorics- this is a branch of mathematics that deals with the study of a selection of a given number of elements from a designated set, enumeration, etc.

Let's look at examples of combinatorics tasks designed for the high school level (source - http://www.problems.ru/).

Task #1:

Lesha climbs a ladder of 10 steps. He jumps up either one step or two steps at a time. In how many ways can Lesha climb the stairs?

The number of ways that Lesha can climb the stairs from n steps, denote and n. Hence it follows that a 1 = 1, a 2= 2 (after all, Lesha jumps either one or two steps).

It is also agreed that Lesha jumps up the stairs from n > 2 steps. Suppose he jumped two steps the first time. So, according to the condition of the problem, he needs to jump another n - 2 steps. Then the number of ways to complete the climb is described as a n-2. And if we assume that for the first time Lesha jumped only one step, then we will describe the number of ways to finish the climb as a n-1.

From here we get the following equality: a n = a n–1 + a n–2(looks familiar, doesn't it?).

Since we know a 1 and a 2 and remember that there are 10 steps according to the condition of the problem, calculate in order all a n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.

Answer: 89 ways.

Task #2:

It is required to find the number of words with a length of 10 letters, which consist only of the letters "a" and "b" and should not contain two letters "b" in a row.

Denote by a n number of words long n letters that consist only of the letters "a" and "b" and do not contain two letters "b" in a row. Means, a 1= 2, a 2= 3.

In sequence a 1, a 2, <…>, a n we will express each next term in terms of the previous ones. Therefore, the number of words of length n letters that also do not contain a doubled letter "b" and begin with the letter "a", this a n-1. And if the word is long n letters begins with the letter "b", it is logical that the next letter in such a word is "a" (after all, there cannot be two "b" according to the condition of the problem). Therefore, the number of words of length n letters in this case, denoted as a n-2. In both the first and second cases, any word (of length n - 1 and n - 2 letters respectively) without doubled "b".

We were able to explain why a n = a n–1 + a n–2.

Let's calculate now a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 144. And we get the familiar Fibonacci sequence.

Answer: 144.

Task #3:

Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first cell of the tape. On whichever of the cells of the tape he is, he can only move to the right: either one cell, or two. How many ways are there for a grasshopper to jump from the beginning of the ribbon to n th cell?

Let us denote the number of ways the grasshopper moves along the tape up to n th cell as a n. In this case a 1 = a 2= 1. Also in n + 1-th cell the grasshopper can get either from n th cell, or by jumping over it. From here n + 1 = a n – 1 + a n. Where a n = F n – 1.

Answer: F n – 1.

You can create similar problems yourself and try to solve them in math lessons with your classmates.

Fibonacci numbers in popular culture

Of course, such an unusual phenomenon as the Fibonacci numbers cannot but attract attention. There is still something attractive and even mysterious in this strictly verified pattern. It is not surprising that the Fibonacci sequence somehow “lit up” in many works of modern mass culture of various genres.

We will tell you about some of them. And you try to look for yourself more. If you find it, share it with us in the comments - we are also curious!

  • Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code by which the main characters of the book open the safe.
  • In the 2009 American film Mr. Nobody, in one of the episodes, the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode, the main character must call the phone number, which is essentially the same, but slightly distorted (an extra number after the number 5) sequence: 123-581-1321.
  • In the 2012 TV series The Connection, the main character, an autistic boy, is able to discern patterns in the events taking place in the world. Including through the Fibonacci numbers. And manage these events also through numbers.
  • The developers of the java-game for Doom RPG mobile phones placed a secret door on one of the levels. The code that opens it is the Fibonacci sequence.
  • In 2012, the Russian rock band Splin released a concept album called Illusion. The eighth track is called "Fibonacci". In the verses of the leader of the group Alexander Vasiliev, the sequence of Fibonacci numbers is beaten. For each of the nine consecutive members, there is a corresponding number of rows (0, 1, 1, 2, 3, 5, 8, 13, 21):

0 Set off on the road

1 Clicked one joint

1 One sleeve trembled

2 Everything, get the staff

Everything, get the staff

3 Request for boiling water

The train goes to the river

The train goes to the taiga<…>.

  • limerick (a short poem of a certain form - usually five lines, with a certain rhyming scheme, comic in content, in which the first and last lines are repeated or partially duplicate each other) by James Lyndon also uses a reference to the Fibonacci sequence as a humorous motif:

Dense food of the Fibonacci wives

It was only for their benefit, not otherwise.

The wives weighed, according to rumor,

Each is like the previous two.

Summing up

We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now look for the Fibonacci spiral in the nature around you. Suddenly, it is you who will be able to unravel the "secret of life, the universe and in general."

Use the formula for Fibonacci numbers when solving problems in combinatorics. You can build on the examples described in this article.

site, with full or partial copying of the material, a link to the source is required.

Recently, working in individual and group processes with people, I returned to the idea of ​​unifying all processes (karmic, mental, physiological, spiritual, transformational, etc.) into one.

Friends behind the veil more and more revealed the image of the multidimensional Man and the interconnection of everything in everything.

An inner impulse prompted me to return to the old studies with numbers and once again look through Drunvalo Melchizedek's book "The Ancient Secret of the Flower of Life".

At this time, the film "The Da Vinci Code" was shown in cinemas. I do not intend to discuss the quality, value and truth of this film. But the moment with the code, when the numbers began to scroll rapidly, became one of the key moments in this film for me.

Intuition told me that it is worth paying attention to the Fibonacci number sequence and the Golden Section. If you look on the Internet to find something about Fibonacci, you will be bombarded with information. You will find out that this sequence was known at all times. It is represented in nature and space, in technology and science, in architecture and painting, in music and proportions in the human body, in DNA and RNA. Many researchers of this sequence have come to the conclusion that key events in the life of a person, state, civilization are also subject to the law of the golden section.

It seems that the Human has been given a fundamental clue.

Then the thought arises that a Person can consciously apply the principle of the Golden Section to restore health and correct fate, i.e. streamlining the ongoing processes in one's own universe, expanding the Consciousness, returning to the Welfare.

Let's remember the Fibonacci sequence together:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025…

Each subsequent number is formed by adding the previous two:

1+1=2, 1+2=3, 2+3=5 etc.

Now I propose to bring each number of the series to one digit: 1, 1, 2, 3, 5, 8,

13=1+3(4), 21=2+1(3), 34=3+4(7), 55=5+5(1), 89= 8+9(8), 144=1+4+4(9)…

Here's what we got:

1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9…1, 1, 2…

a sequence of 24 numbers that repeats again from the 25th:

75025=7+5+0+2+5=19=1+0=1, 121393=1+2+1+3+9+3=19=1+0=1…

Doesn't it seem strange or natural to you that

  • in a day - 24 hours,
  • space houses - 24,
  • strands of DNA - 24,
  • 24 elders from the God Star Sirius,
  • repeating sequence in the Fibonacci series - 24 digits.

If the resulting sequence is written as follows,

1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9

8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9

9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,

then we will see that the 1st and 13th number of the sequence, the 2nd and 14th, the 3rd and 15th, the 4th and 16th ... the 12th and 24th add up to 9 .

3 3 6 9 6 6 3 9

When testing these numerical series, we got:

  • Child Principle;
  • Father Principle;
  • Mother Principle;
  • the principle of unity.

Matrix of the Golden Section

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

2 2 4 6 1 7 8 6 5 2 7 9 7 7 5 3 8 2 1 3 4 7 2 9

4 4 8 3 2 5 7 3 1 4 5 9 5 5 1 6 7 4 2 6 8 5 4 9

3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9

7 7 5 3 8 2 1 3 4 7 2 9 2 2 4 6 1 7 8 6 5 2 7 9

4 4 8 3 2 5 7 3 1 4 5 9 5 5 1 6 7 4 2 6 8 5 4 9

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

5 5 1 6 7 4 2 6 8 5 4 9 4 4 8 3 2 5 7 3 1 4 5 9

6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9

2 2 4 6 1 7 8 6 5 2 7 9 7 7 5 3 8 2 1 3 4 7 2 9

8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8 9

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

Practical application of the Fibonacci series

A friend of mine expressed his intention to work individually with him on the development of his abilities and abilities.

Suddenly, at the very beginning, Sai Baba came into the process and invited me to follow him.

We began to rise up inside the Divine Monad of a friend and, having left it through the Causal Body, we found ourselves in another reality at the level of the Cosmic House.

Those who have studied the works of Mark and Elizabeth Clair Prophetov know the teaching about the Cosmic Clock, which was passed on to them by Mother Mary.

At the level of the Space House, Yuri saw a circle with an inner center with 12 arrows.

The elder, who met us at this level, said that before us is the Divine Clock and 12 hands represent 12 (24) Manifestations of the Divine Aspects… (perhaps the Creators).

As for the Cosmic Clock, they were located under the Divine ones according to the principle of the energy eight.

- In what mode are the Divine Clocks in relation to you?

- The hands of the Clock are standing, there is no movement.Thoughts come to me now that many eons ago I abandoned the Divine Consciousness and went a different path, the path of a Magician. All my magical artifacts and amulets that have accumulated in me and in me over many incarnations look like baby rattles at this level. On the subtle plane, they represent an image of magical energy clothes.

- Completed.However, I bless my magical experience.Living this experience sincerely prompted me to return to the original source, to wholeness.I am offered to take off my magical artifacts and stand in the center of the Clock.

— What needs to be done to activate the Divine Clock?

- Sai Baba appeared again and offered to express the intention to connect the Silver String with the Clock. He also says that you have some kind of number series. He is the key to activation. The image of Leonard da Vinci's Man appears before the inner eye.

- 12 times.

“I ask you to God-center the whole process and direct the action of the energy of the number series to the activation of the Divine Clock.

Read aloud 12 times

1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9…

In the process of reading, the hands on the clock went.

An energy went through the silver string, which connected all the levels of the Yurina Monad, as well as the earthly and heavenly energies…

The most unexpected thing in this process was that four Essences appeared on the Clock, which are some parts of the One Whole with Yura.

During the communication, it turned out that once there was a division of the Central Soul, and each part chose its own area in the universe for realization.

A decision was made to integrate, which happened in the center of the Divine Clock.

The result of this process was the creation of the Common Crystal at this level.

After that, I remembered that Sai Baba once spoke about a certain Plan, which involves first combining two Essences into one, then four, and so on according to the binary principle.

Of course, this number series is not a panacea. This is just a tool that allows you to quickly perform the necessary work with a person, to tune him vertically with different levels of Existence.