Fibonacci series golden ratio. Research work "mystery of fibonacci numbers". Golden ratio and fibonacci numbers in nature video

Ecology of life. Cognitively: Nature (including Man) develops according to the laws that are laid down in this numerical sequence...

Fibonacci numbers - numerical sequence, where each subsequent member of the series is equal to the sum of the two previous ones, that is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 6765 10946 17711 28657 46368 68021641812000,.. A variety of professional scientists and math lovers.

In 1997, several strange features of the series were described by the researcher Vladimir Mikhailov, who was convinced that Nature (including Man) develops according to the laws that are laid down in this numerical sequence.

A remarkable property of the Fibonacci number series is that as the numbers of the series increase, the ratio of two neighboring members of this series asymptotically approaches the exact proportion of the Golden Section (1: 1.618) - the basis of beauty and harmony in the nature around us, including in human relations.

Note that Fibonacci himself discovered his famous series, reflecting on the problem of the number of rabbits that should be born from one pair within one year. It turned out that in each subsequent month after the second, the number of pairs of rabbits exactly follows the digital series, which now bears his name. Therefore, it is no coincidence that man himself is arranged according to the Fibonacci series. Each organ is arranged according to internal or external duality.

Fibonacci numbers have attracted mathematicians because of their ability to appear in the most unexpected places. It has been noticed, for example, that the ratios of Fibonacci numbers, taken through one, correspond to the angle between adjacent leaves on the stem of plants, more precisely, they say what proportion of the turn this angle is: 1/2 - for elm and linden, 1/3 - for beech, 2/5 - for oak and apple, 3/8 - for poplar and rose, 5/13 - for willow and almond, etc. You will find the same numbers when counting seeds in sunflower spirals, in the number of rays reflected from two mirrors, in the number of options for crawling bees from one cell to another, in many mathematical games and tricks.

What is the difference between the Golden Ratio Spirals and the Fibonacci Spiral? The golden ratio spiral is perfect. It corresponds to the Primary source of harmony. This spiral has neither beginning nor end. She is endless. The Fibonacci spiral has a beginning, from which it starts “unwinding”. This is a very important property. It allows Nature, after the next closed cycle, to carry out the construction of a new spiral from “zero”.

It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helixes found all over the place. So, sunflower spirals always correlate with the Fibonacci series. Even in an ordinary pinecone, you can see this double Fibonacci spiral. The first spiral goes in one direction, the second - in the other. If we count the number of scales in a spiral rotating in one direction and the number of scales in the other spiral, we can see that these are always two consecutive numbers of the Fibonacci series. The number of these spirals is 8 and 13. There are pairs of spirals in sunflowers: 13 and 21, 21 and 34, 34 and 55, 55 and 89. And there are no deviations from these pairs!..

In Man, in the set of chromosomes of a somatic cell (there are 23 pairs of them), the source of hereditary diseases are 8, 13 and 21 pairs of chromosomes ...

But why does this series play a decisive role in Nature? The concept of triplicity, which determines the conditions for its self-preservation, can give an exhaustive answer to this question. If the "balance of interests" of the triad is violated by one of its "partners", the "opinions" of the other two "partners" must be corrected. The concept of triplicity manifests itself especially clearly in physics, where “almost” all elementary particles were built from quarks. If we recall that the ratios of fractional charges of quark particles make up a series, and these are the first members of the Fibonacci series, which are necessary for the formation of other elementary particles.

It is possible that the Fibonacci spiral can also play a decisive role in the formation of the pattern of limitedness and closedness of hierarchical spaces. Indeed, imagine that at some stage of evolution, the Fibonacci spiral has reached perfection (it has become indistinguishable from the golden section spiral) and for this reason the particle must be transformed into the next “category”.

These facts once again confirm that the law of duality gives not only qualitative but also quantitative results. They make us think that the Macrocosm and the Microcosm around us evolve according to the same laws - the laws of hierarchy, and that these laws are the same for living and inanimate matter.

All this indicates that a series of Fibonacci numbers is a kind of encrypted law of nature.

The digital code for the development of civilization can be determined using various methods in numerology. For example, by converting complex numbers to single digits (for example, 15 is 1+5=6, etc.). Carrying out a similar addition procedure with all the complex numbers of the Fibonacci series, Mikhailov received the following series of these numbers: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 8, 1, 9, then everything repeats 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 4, 8, 8, .. and repeats again and again... This series also has the properties of the Fibonacci series, each infinitely subsequent term is equal to the sum of the previous ones. For example, the sum of the 13th and 14th terms is 15, i.e. 8 and 8=16, 16=1+6=7. It turns out that this series is periodic, with a period of 24 terms, after which the whole order of numbers is repeated. Having received this period, Mikhailov put forward an interesting assumption - Isn't a set of 24 digits a kind of digital code for the development of civilization? published

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Kanalieva Dana

In this paper, we have studied and analyzed the manifestation of the numbers of the Fibonacci sequence in the reality around us. We have discovered a surprising mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the numbers in the Fibonacci sequence. We also saw strict mathematics in the structure of man. The human DNA molecule, in which the entire program of the development of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical ratios.

We have seen that Nature has its own laws, expressed with the help of mathematics.

And mathematics is very important learning tool secrets of nature.

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MBOU "Pervomaiskaya secondary school"

Orenburgsky district of the Orenburg region

RESEARCH

"The riddle of numbers

Fibonacci"

Completed by: Kanalieva Dana

6th grade student

Scientific adviser:

Gazizova Valeria Valerievna

Mathematics teacher of the highest category

n. Experimental

2012

Explanatory note……………………………………………………………………........ 3.

Introduction. History of Fibonacci numbers.………………………………………………………..... 4.

Chapter 1. Fibonacci numbers in wildlife.......……. …………………………………... 5.

Chapter 2. Fibonacci Spiral............................................... ..........……………..... 9.

Chapter 3. Fibonacci numbers in human inventions .........…………………………….

Chapter 4. Our Research………………………………………………………………………………………………….

Chapter 5. Conclusion, conclusions……………………………………………………………….....

List of used literature and Internet sites……………………………………........21.

Object of study:

Man, mathematical abstractions created by man, inventions of man, the surrounding flora and fauna.

Subject of study:

the form and structure of the studied objects and phenomena.

Purpose of the study:

to study the manifestation of Fibonacci numbers and the law of the golden section associated with it in the structure of living and inanimate objects,

find examples of using Fibonacci numbers.

Work tasks:

Describe how to construct a Fibonacci series and a Fibonacci spiral.

To see mathematical patterns in the structure of man, the plant world and inanimate nature from the point of view of the phenomenon of the Golden Section.

Research novelty:

The discovery of Fibonacci numbers in the reality around us.

Practical significance:

Use of the acquired knowledge and research skills in the study of other school subjects.

Skills and abilities:

Organization and conduct of the experiment.

Use of specialized literature.

Acquisition of the ability to review the collected material (report, presentation)

Registration of work with drawings, diagrams, photographs.

Active participation in the discussion of their work.

Research methods:

empirical (observation, experiment, measurement).

theoretical (logical stage of knowledge).

Explanatory note.

“Numbers rule the world! Number is the power that reigns over gods and mortals!” - so said the ancient Pythagoreans. Is this basis of the Pythagorean teaching relevant today? Studying the science of numbers at school, we want to make sure that, indeed, the phenomena of the entire Universe are subject to certain numerical ratios, to find this invisible connection between mathematics and life!

Is it really in every flower,

Both in the molecule and in the galaxy,

Numerical patterns

This strict "dry" mathematics?

We turned to a modern source of information - the Internet and read about Fibonacci numbers, about magic numbers that are fraught with a great mystery. It turns out that these numbers can be found in sunflowers and pine cones, in dragonfly wings and starfish, in the rhythms of the human heart and in musical rhythms...

Why is this sequence of numbers so common in our world?

We wanted to learn about the secrets of Fibonacci numbers. This research work is the result of our work.

Hypothesis:

in the reality around us, everything is built according to surprisingly harmonious laws with mathematical precision.

Everything in the world is thought out and calculated by our most important designer - Nature!

Introduction. The history of the Fibonacci series.

Amazing numbers were discovered by the Italian mathematician of the Middle Ages, Leonardo of Pisa, better known as Fibonacci. Traveling in the East, he became acquainted with the achievements of Arabic mathematics and contributed to their transfer to the West. In one of his works, entitled "The Book of Calculations", he introduced Europe to one of the greatest discoveries of all times and peoples - the decimal number system.

Once, he puzzled over the solution of a mathematical problem. He was trying to create a formula describing the breeding sequence of rabbits.

The answer was a number series, each subsequent number of which is the sum of the two previous ones:

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

The numbers that form this sequence are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence.

"So what?" - you will say, - “Can we ourselves come up with similar numerical series, growing according to a given progression?” Indeed, when the Fibonacci series appeared, no one, including himself, suspected how close he managed to get closer to unraveling one of the greatest mysteries of the universe!

Fibonacci led a hermit life, spent a lot of time in nature, and walking in the forest, he noticed that these numbers literally began to haunt him. Everywhere in nature, he met these numbers again and again. For example, the petals and leaves of plants strictly fit into a given number series.

There is an interesting feature in the Fibonacci numbers: the quotient of dividing the next Fibonacci number by the previous one tends to 1.618 as the numbers themselves grow. It was this constant division number that was called the Divine Proportion in the Middle Ages, and is now referred to as the Golden Section or Golden Ratio.

In algebra, this number is denoted by the Greek letter phi (Ф)

So φ = 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

No matter how many times we divide one by the other, the number adjacent to it, we will always get 1.618. And if we do the opposite, that is, we divide the smaller number by the larger one, we get 0.618, this is the inverse of 1.618, also called the golden ratio.

The Fibonacci series could have remained only a mathematical incident if it were not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the golden division law.

Scientists, analyzing the further application of this number series to natural phenomena and processes, found that these numbers are contained in literally all objects of wildlife, in plants, in animals and in humans.

An amazing mathematical toy turned out to be a unique code embedded in all natural objects by the Creator of the Universe himself.

Consider examples where Fibonacci numbers are found in animate and inanimate nature.

Fibonacci numbers in wildlife.

If you look at the plants and trees around us, you can see how many leaves each of them has. From afar, it seems that the branches and leaves on the plants are arranged randomly, in an arbitrary order. However, in all plants it is miraculously, mathematically precisely planned which branch will grow from where, how branches and leaves will be located near the stem or trunk. From the first day of its appearance, the plant exactly follows these laws in its development, that is, not a single leaf, not a single flower appears by chance. Even before the appearance of the plant is already precisely programmed. How many branches will be on the future tree, where the branches will grow, how many leaves will be on each branch, and how, in what order the leaves will be arranged. 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 (phylotaxis), in the number of revolutions on the stem, in the number of leaves in the cycle, the Fibonacci series manifests itself, and therefore, the law of the golden section also manifests itself.

If you set out to find numerical patterns in wildlife, you will notice that these numbers are often found in various spiral forms, which the plant world is so rich in. For example, leaf cuttings adjoin the stem in a spiral that runs betweentwo adjacent leaves:full turn - at the hazel,- at the oak - at the poplar and pear,- at the willow.

The seeds of sunflower, Echinacea purpurea and many other plants are arranged in spirals, and the number of spirals in each direction is the Fibonacci number.

Sunflower, 21 and 34 spirals. Echinacea, 34 and 55 spirals.

A clear, symmetrical shape of flowers is also subject to a strict law.

Many flowers have the number of petals - exactly the numbers from the Fibonacci series. For example:

iris, 3 lep. buttercup, 5 lep. golden flower, 8 lep. delphinium,

13 lep.

chicory, 21 lep. aster, 34 lep. daisies, 55 lep.

The Fibonacci series characterizes the structural organization of many living systems.

We have already said that the ratio of neighboring numbers in the Fibonacci series is the number φ = 1.618. It turns out that the man himself is just a storehouse of the number phi.

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 to be ideally built. The principle of calculating the golden measure on the human body can be depicted in the form of a diagram.

M/m=1.618

The first example of the golden section in the structure of the human body:

If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.

Human hand

It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula 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 gives the golden ratio (with the exception of the thumb).

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

A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 ratio 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 section also exists in the structure of the human lungs.

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

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

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 section. 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.
There is another, more prosaic application of the proportions of the human body. For example, using these ratios, criminal analysts and archaeologists restore the appearance of the whole from fragments of parts of the human body.

Golden proportions in the structure of the DNA molecule.

All information about the physiological characteristics of living beings, be it a plant, an animal or a person, is stored in a microscopic DNA molecule, the structure of which also contains the law of the golden ratio. The DNA molecule consists of two vertically intertwined helices. Each of these spirals is 34 angstroms long and 21 angstroms wide. (1 angstrom is one hundred millionth of a centimeter).

So 21 and 34 are numbers following one after another in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic helix of the DNA molecule carries the formula of the golden section 1: 1.618.

Not only upright walkers, but also all those who swim, crawl, fly and jump, did not escape the fate of obeying the number phi. The human heart muscle contracts to 0.618 of its volume. The structure of the snail shell corresponds to the Fibonacci proportions. And there are plenty of such examples - there would be a desire to explore natural objects and processes. The world is so permeated with Fibonacci numbers that sometimes it seems that the Universe can be explained only by them.

Fibonacci spiral.

There is no other form in mathematics that has the same unique properties as a spiral, because
The structure of the spiral is based on the rule of the Golden Section!

To understand the mathematical construction of the spiral, let's repeat what the Golden Ratio is.

The Golden Ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part in the same way as the larger part itself is related to the smaller one, or, in other words, the smaller segment is related to the larger one as the larger one is to everything.

That is, (a + b) / a = a / b

A rectangle with exactly this ratio of sides was called the golden rectangle. Its long sides are related to the short sides in a ratio of 1.168:1.
The golden rectangle has many unusual properties. Cutting off from the golden rectangle a square whose side is equal to the smaller side of the rectangle,

we again get a smaller golden rectangle.

This process can be continued ad infinitum. As we keep cutting off the squares, we'll get smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects.

For example, a spiral shape can also be seen in the arrangement of sunflower seeds, in pineapples, cacti, the structure of rose petals, and so on.

We are surprised and delighted by the spiral structure of shells.

In most snails that have shells, the shell grows in a spiral shape. However, there is no doubt that these unreasonable beings not only have no idea about the spiral, but do not even have the simplest mathematical knowledge to create a spiral shell for themselves.
But then how could these unintelligent 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 scientific world calls primitive life forms, have calculated that the spiral shape of the shell would be ideal for their existence?

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

Spirals are also in man. With the help of spirals we hear:

Also, in the human inner ear there is an organ Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bone-like structure is filled with liquid and created in the form of a snail with golden proportions.

Spirals are on our palms and fingers:

In the animal kingdom, we can also find many examples of spirals.

The horns and tusks of animals develop in a spiral form, the claws of lions and the beaks of parrots are logarithmic forms and resemble the shape of an axis that tends to turn into a spiral.

It is interesting that a hurricane, cyclone clouds are spiraling, and this is clearly visible from space:

In ocean and sea waves, the spiral can be mathematically plotted with points 1,1,2,3,5,8,13,21,34 and 55.

Everyone will also recognize such a “everyday” and “prosaic” spiral.

After all, water runs away from the bathroom in a spiral:

Yes, and we live in a spiral, because the galaxy is a spiral that corresponds to the formula of the Golden Section!

So, we found out that if we take the Golden Rectangle and break it into smaller rectanglesin the exact Fibonacci sequence, and then divide each of them in such proportions again and again, you get a system called the Fibonacci spiral.

We found this spiral in the most unexpected objects and phenomena. Now it’s clear why the spiral is also called the “curve of life”.
The spiral has become a symbol of evolution, because everything develops in a spiral.

Fibonacci numbers in human inventions.

Having peeped from nature the law expressed by the sequence of Fibonacci numbers, scientists and people of art try to imitate it, to embody this law in their creations.

The proportion of phi allows you to create masterpieces of painting, competently fit architectural structures into space.

Not only scientists, but also architects, designers and artists are amazed at this flawless spiral at the nautilus shell,

occupying the smallest space and providing the least heat loss. Inspired by the “camera nautilus” example of putting the maximum in the minimum of space, American and Thai architects are busy developing designs to match.

Since time immemorial, the proportion of the Golden Ratio has been considered the highest proportion of perfection, harmony, and even divinity. The golden ratio can be found in sculptures, and even in music. An example is the musical works of Mozart. Even stock prices and the Hebrew alphabet contain a golden ratio.

But we want to dwell on a unique example of creating an efficient solar installation. American schoolboy from New York Aidan Dwyer brought together his knowledge of trees and discovered that the efficiency of solar power plants can be increased by using mathematics. While on a winter walk, Dwyer wondered why the trees needed such a “pattern” of branches and leaves. He knew that the branches on the trees are arranged according to the Fibonacci sequence, and the leaves carry out photosynthesis.

At some point, a smart little boy decided to check if this position of the branches helps to collect more sunlight. Aidan built a pilot plant in his backyard with small solar panels instead of leaves and tested it in action. It turned out that in comparison with a conventional flat solar panel, his “tree” collects 20% more energy and works effectively for 2.5 hours longer.

Dwyer's solar tree model and student plots.

“It also takes up less space than a flat panel, collects 50% more sun in winter even where it does not face south, and it does not accumulate snow as much. In addition, the design in the form of a tree is much more suitable for the urban landscape," notes the young inventor.

Aidan recognized one of the best young natural scientists of 2011. The 2011 Young Naturalist competition was hosted by the New York Museum of Natural History. Aidan filed a provisional patent application for his invention.

Scientists continue to actively develop the theory of Fibonacci numbers and the golden ratio.

Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers.

There are elegant methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section.

In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

So, we see that the scope of the Fibonacci sequence is very multifaceted:

Observing the phenomena occurring in nature, scientists have made amazing conclusions that the whole sequence of events occurring in life, revolutions, collapses, bankruptcy, periods of prosperity, laws and waves of development in the stock and currency markets, cycles of family life, and so on, are organized on the timeline in the form of cycles, waves. These cycles and waves are also distributed according to the Fibonacci number series!

Based on this knowledge, a person will learn to predict various events in the future and manage them.

4. Our research.

We continued our observations and studied the structure

Pine cone

yarrow

mosquito

human

And we made sure that in these objects, so different at first glance, the very numbers of the Fibonacci sequence are invisibly present.

So step 1.

Let's take a pine cone:

Let's take a closer look at it:

We notice two series of Fibonacci spirals: one - clockwise, the other - against, their number 8 and 13.

Step 2

Let's take a yarrow:

Let's take a closer look at the structure of stems and flowers:

Note that each new branch of the yarrow grows from the sinus, and new branches grow from the new branch. Adding old and new branches, we found the Fibonacci number in each horizontal plane.

Step 3

Do Fibonacci numbers show up in the morphology of various organisms? Consider the well-known mosquito:

We see: 3 pair of legs, head 5 antennae - antennae, the abdomen is divided into 8 segments.

Conclusion:

In our research, we saw that in the plants around us, living organisms, and even in the human structure, numbers from the Fibonacci sequence manifest themselves, which reflects the harmony of their structure.

Pine cone, yarrow, mosquito, man are arranged with mathematical precision.

We were looking for an answer to the question: how does the Fibonacci series manifest itself in the reality around us? But, answering it, received new and new questions.

Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? Is the coil twisting or untwisting?

How amazingly man knows this world!!!

Having found the answer to one question, he receives the next one. Solve it, get two new ones. Deal with them, three more will appear. Having solved them, he will acquire five unresolved ones. Then eight, then thirteen, 21, 34, 55...

Do you recognize?

Conclusion.

By the creator himself in all objects

A unique code has been assigned

And the one who is friendly with mathematics,

He will know and understand!

We have studied and analyzed the manifestation of the numbers of the Fibonacci sequence in the reality around us. We also learned that the patterns of this number series, including the patterns of the "Golden" symmetry, are manifested in the energy transitions of elementary particles, in planetary and cosmic systems, in the gene structures of living organisms.

We have discovered a surprising mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the numbers in the Fibonacci sequence. We have seen how the morphology of various organisms also obeys this mysterious law. We also saw strict mathematics in the structure of man. The human DNA molecule, in which the whole program of the development of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical ratios.

We have learned that pine cones, snail shells, ocean waves, animal horns, cyclone clouds, and galaxies all form logarithmic spirals. Even the human finger, which is made up of three phalanges in relation to each other in the Golden ratio, takes on a spiral shape when compressed.

eternity of time and light years space divide a pinecone and a spiral galaxy, but the structure remains the same: the coefficient 1,618 ! Perhaps this is the supreme law that governs natural phenomena.

Thus, our hypothesis about the existence of special numerical patterns that are responsible for harmony is confirmed.

Indeed, everything in the world is thought out and calculated by our most important designer - Nature!

We are convinced that Nature has its own laws, expressed with the help of mathematics. And math is a very important tool

to discover the mysteries of nature.

List of literature and Internet sites:

1. Vorobyov N. N. Fibonacci numbers. - M., Nauka, 1984.
2. Gika M. Aesthetics of proportions in nature and art. - M., 1936.

3. Dmitriev A. Chaos, fractals and information. // Science and Life, No. 5, 2001.
4. Kashnitsky S. E. Harmony woven from paradoxes // Culture and

Life. - 1982.- No. 10.
5. Malay G. Harmony - the identity of paradoxes // MN. - 1982.- No. 19.
6. Sokolov A. Secrets of the golden section // Technique of youth. - 1978.- No. 5.
7. Stakhov A. P. Codes of the golden ratio. - M., 1984.
8. Urmantsev Yu. A. Symmetry of nature and the nature of symmetry. - M., 1974.
9. Urmantsev Yu. A. Golden section // Priroda. - 1968.- No. 11.

10. Shevelev I.Sh., Marutaev M.A., Shmelev I.P. Golden Ratio/Three

A look at the nature of harmony.-M., 1990.

11. Shubnikov A. V., Koptsik V. A. Symmetry in science and art. -M.:

However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then we get 1.618, if we square it, then we get 2.618, if we raise it into a cube, we get the number 4.236. These are the Fibonacci expansion coefficients. The only thing missing here is the number 3.236, which was proposed by John Murphy.

What do experts think about sequence?

Some will say that these numbers are already familiar because they are used in technical analysis programs to determine the amount of correction and expansion. In addition, these same series play an important role in the Eliot wave theory. They are its numerical basis.

Our expert Nikolay Proven portfolio manager of Vostok investment company.

• — Nikolai, what do you think, is the appearance of Fibonacci numbers and its derivatives on the charts of various instruments by chance? And is it possible to say: "The Fibonacci series practical use" occurs?
• - I have a bad attitude towards mysticism. And even more so on the stock exchange charts. Everything has its reasons. in the book "Fibonacci Levels" he beautifully told where the golden ratio appears, that he was not surprised that it appeared on the stock exchange charts. But in vain! Pi often appears in many of the examples he gave. But for some reason it is not in the price ratio.
• - So you do not believe in the effectiveness of the Elliot wave principle?
• “No, no, that’s not the point. The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
• What do you think are the reasons for the appearance of the golden section on stock charts?
• - The correct answer to this question may be able to deserve Nobel Prize on economics. While we can guess the true reasons. They are clearly out of harmony with nature. There are many models of exchange pricing. They do not explain the indicated phenomenon. But not understanding the nature of the phenomenon should not deny the phenomenon as such.
• - And if this law is ever open, will it be able to destroy the exchange process?
• - As the same theory of waves shows, the law of change in stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and will not be able to destroy the stock exchange.

The material is provided by webmaster Maxim's blog.

The coincidence of the foundations of the principles of mathematics in a variety of theories seems incredible. Maybe it's fantasy or an adjustment to the end result. Wait and see. Much of what was previously considered unusual or impossible: space exploration, for example, has become commonplace and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will become more accessible and understandable with time. What was previously unnecessary, in the hands of an experienced analyst, will become a powerful tool for predicting future behavior.

Fibonacci numbers in nature.

Look

And now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.

Let's take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next member of the sequence is equal to the sum of the two previous ones. For example, let's take two numbers: 6 and 51. Now we will build a sequence that we will complete with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.

At the same time, the numbers that were obtained by dividing other pairs decreased from the first to the last, which allows us to assert that if this series is continued indefinitely, then we will get a number equal to the golden ratio.

Thus, the Fibonacci numbers themselves are not distinguished by anything. There are other sequences of numbers, of which there are an infinite number, which result in the golden number phi as a result of the same operations.

Fibonacci was not an esotericist. He didn't want to put any mysticism into the numbers, he was just solving an ordinary rabbit problem. And he wrote a sequence of numbers that followed from his task, in the first, second and other months, how many rabbits there would be after breeding. Within a year, he received that same sequence. And didn't make a relationship. There was no golden ratio, no Divine relation. All this was invented after him in the Renaissance.

Before mathematics, Fibonacci's virtues are enormous. He adopted the number system from the Arabs and proved its validity. It was a hard and long struggle. From the Roman number system: heavy and inconvenient for counting. She disappeared after the French Revolution. It has nothing to do with the golden section of Fibonacci.

Fibonacci sequence in mathematics and in nature

Fibonacci sequence, known to everyone from the film "The Da Vinci Code" - a series of numbers described as a riddle by the Italian mathematician Leonardo of Pisa, better known by the nickname Fibonacci, in the 13th century. Briefly, the essence of the riddle:

Someone placed a pair of rabbits in a certain closed space to find out how many pairs of rabbits would be born during the year, if the nature of the rabbits is such that every month a pair of rabbits produces another pair, and the ability to produce offspring appears on reaching two months old.

The result is the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 , where the number of pairs of rabbits in each of the twelve months is shown, separated by commas.

This sequence can be continued indefinitely. Its essence is that each next number is the sum of the previous two.

This sequence has a number of mathematical features that must be touched upon. This 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 cannot be expressed exactly.

So the ratio of any member of the sequence to the one preceding it fluctuates around the number 1,618 , sometimes surpassing it, sometimes not reaching it. The ratio to the following similarly approaches the number 0,618 , which is inversely proportional 1,618 . If we divide the elements of the sequence through one, then we get the numbers 2,618 And 0,382 , which are also inversely proportional. These are the so-called Fibonacci ratios.

Why all this? So we are approaching one of the most mysterious phenomena of nature. Fibonacci did not actually discover anything new, he just reminded the world of such a phenomenon as Golden Section, which is not inferior in importance to the Pythagorean theorem

We distinguish all the objects around us, including in form. We like some more, some less, some completely repulse the eye. Sometimes interest can be dictated by a life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape contributes to the best visual perception and evokes a sense of beauty and harmony. A holistic image always consists of parts of different sizes, which are in a certain relationship with each other and the whole.

golden ratio- the highest manifestation of the perfection of the whole and its parts in science, art and nature.

If on a simple example, then the Golden Section is the division of a segment into two parts in such a ratio in which the larger part relates to the smaller one, as their sum (the entire segment) to the larger one.

If we take the entire segment c behind 1 , then the segment a will be equal to 0,618 , line segment b - 0,382 , only in this way the condition of the Golden Section will be observed (0.618 / 0.382 = 1,618 ; 1/0,618=1,618 ). Attitude c To a equals 1,618 , A With To b2,618. These are all the same, already familiar to us, Fibonacci coefficients.

Of course, there is a golden rectangle, a golden triangle, and even a golden cuboid. The proportions of the human body in many respects are close to the Golden Section.

Image: marcus-frings.de

But the most interesting begins when we combine the knowledge gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. From above we add a square of the second size. We paint next to a square with a side equal to the sum of the sides of the previous two, the third size. By analogy, a square of the fifth size appears. And so on until you get bored, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the two previous ones. We see a series of rectangles whose side lengths are Fibonacci numbers, and oddly enough they are called Fibonacci rectangles.

If we draw a smooth line through the corners of our squares, we get nothing more than an Archimedes spiral, the increase in the pitch of which is always uniform.

Doesn't it remind you of anything?

Photo: ethanhein on Flickr

And not only in the shell of a mollusk you can find the spirals of Archimedes, but in many flowers and plants, they are just not so obvious.

Aloe multileaf:

Photo: brewbooks on Flickr

Photo: beart.org.uk

Photo: esdrascalderan on Flickr

Photo: manj98 on Flickr

And then it's time to remember the Golden Section! Are any of the most beautiful and harmonious creations of nature depicted in these photographs? And that's not all. Looking closely, you can find similar patterns in many forms.

Of course, the statement that all these phenomena are built on the Fibonacci sequence sounds too loud, but the trend is on the face. And besides, the sequence itself is far from perfect, like everything else in this world.

There is speculation that the Fibonacci sequence is nature's attempt to adapt to a more fundamental and perfect golden section logarithmic sequence, which is practically the same, just starts from nowhere and goes nowhere. Nature, on the other hand, definitely needs some kind of whole beginning, from which you can push off, it cannot create something out of nothing. The ratios of the first members of the Fibonacci sequence are far from the Golden Section. But the further we move along it, the more these deviations are smoothed out. To determine any sequence, it is enough to know its three terms, going one after another. But not for the golden sequence, two are enough for it, it is geometric and arithmetic progression simultaneously. You might think that it is the basis for all other sequences.

Each member of the golden logarithmic sequence is a power of the Golden Ratio ( z). Part of the row looks something like this: ... z -5 ; z-4; z-3; z-2; z -1 ; z0; z1; z2; z3; z4; z 5 ... If we round the value of the Golden Ratio to three decimal places, we get z=1.618, then the row looks like this: ... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next term can be obtained not only by multiplying the previous one by 1,618 , but also by adding the two previous ones. Thus, exponential growth in the sequence is provided by simply adding two adjacent elements. This is a series without beginning and end, and it is precisely this that the Fibonacci sequence tries to be like. Having a well-defined beginning, it strives for the ideal, never reaching it. That is life.

And yet, in connection with everything 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? Was it ever the way he wanted it to be? And if so, why did it fail? Mutations? Free choice? What will be next? Is the coil twisting or untwisting?

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, then thirteen, 21, 34, 55...

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Fibonacci numbers and the golden ratio form the basis for unraveling the surrounding world, building its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.

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 founded as a result of research by ancient scientists on the nature of numbers.

Evidence of the use of the golden ratio by ancient thinkers is given in the book of Euclid's "Beginnings", written back in the 3rd century. BC, who used 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 the Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time gives a pattern of numbers, in a series of which each number is the sum of the 2 previous digits. The sequence of Fibonacci numbers is as follows:

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

The scientist also cited a number of patterns:

Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will be more and more accurate.

If you divide the number from the series by the previous one, then the result will tend to 1.618.

One number divided by the next one will show a value tending to 0.382.

The application of the connection and patterns of the golden section, 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.

For practical purposes, they are limited to an approximate value of Φ = 1.618 or Φ = 1.62. In a rounded percentage, the golden ratio is the division of any value in relation to 62% and 38%.

Historically, the division of segment AB by point C into two parts (a smaller segment AC and a larger segment BC) was originally called the golden section, so that AC / BC = BC / AB was true for the lengths of the segments. talking in simple terms, the segment is divided by the golden section into two unequal parts so that the smaller part is related to the larger one, as the larger one is to the entire segment. Later this concept was extended to arbitrary quantities.

The number Φ is also called golden number.

The golden ratio has many wonderful properties, but in addition, many fictional properties are attributed to it.

Now the details:

The definition of ZS is the division of a segment into two parts in such a ratio that the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.

That is, if we take the entire segment c as 1, then segment a will be equal to 0.618, segment b - 0.382. Thus, if we take a building, for example, a temple built according to the principle of GS, then with its height, say, 10 meters, the height of the drum with the dome will be 3.82 cm, and the height of the base of the building will be 6.18 cm. (It is clear that the numbers taken equal for clarity)

And what is the relationship between GL and Fibonacci numbers?

The Fibonacci sequence numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597…

The pattern of numbers is that each subsequent number is equal to the sum of the two previous numbers.
0 + 1 = 1;
1 + 1 = 2;
2 + 3 = 5;
3 + 5 = 8;
5 + 8 = 13;
8 + 13 = 21 etc.

and the ratio of adjacent numbers approaches the ratio of 3S.
So, 21:34 = 0.617, and 34:55 = 0.618.

That is, the basis of the ZS are the numbers of the Fibonacci sequence.

It is believed that the term "Golden Ratio" was introduced by Leonardo Da Vinci, who said, "let no one who is not a mathematician dare to read my works" and showed the proportions of the human body in his famous drawing "Vitruvian Man". “If we tie a human figure – the most perfect creation of the Universe – with a belt and then measure the distance from the belt to the feet, then this value will refer to the distance from the same belt to the top of the head, as the entire height of a person to the length from the belt to the feet.”

A series of Fibonacci numbers is visually modeled (materialized) in the form of a spiral.

And in nature, the 3S spiral looks like this:

At the same time, the spiral is observed everywhere (in nature and not only):

Seeds in most plants are arranged in a spiral
- A spider weaves a web in a spiral
- A hurricane spirals
- A frightened herd of reindeer scatters in a spiral.
- The DNA molecule is twisted in a double helix. The DNA molecule consists of two vertically intertwined helices 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.
- The embryo develops in the form of a spiral
- Spiral "cochlea in the inner ear"
- Water goes down the drain in a spiral
- Spiral dynamics shows the development of a person's personality and his values ​​in a spiral.
- And of course, the Galaxy itself has the shape of a spiral

Thus, it can be argued that nature itself is built on the principle of the Golden Section, which is why this proportion is more harmoniously perceived by the human eye. It does not require "fixing" or supplementing the resulting picture of the world.

Movie. God number. Irrefutable Proof of God; The number of God. The incontrovertible proof of God.

Golden proportions in the structure of the DNA molecule

All information about the physiological characteristics of living beings is stored in a microscopic DNA molecule, the structure of which also contains the law of the golden ratio. The DNA molecule consists of two vertically intertwined helices. Each of these spirals is 34 angstroms long and 21 angstroms wide. (1 angstrom is one hundred millionth of a centimeter).

21 and 34 are numbers following one after another in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic helix of the DNA molecule carries the golden section formula 1: 1.618

The golden section in the structure of microworlds

Geometric shapes are not limited to just a triangle, square, five- or hexagon. If we combine these figures in various ways with each other, then we will get new three-dimensional geometric shapes. Examples of this are figures such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we have not encountered in everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures one can name 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 easy to transform, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden section.

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 an icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a certain sequence. In each corner of the icosahedron are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.

The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from London's Birkbeck College A.Klug and D.Kaspar. 13 The Polyo virus was the first to show a 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 section, which is quite difficult to construct even with our human mind? The discoverer of these forms of viruses, virologist A. Klug makes the following comment:

“Dr. Kaspar and I have shown that for a spherical shell of a virus, the most optimal shape is symmetry like the shape of an icosahedron. This order minimizes the number of connecting elements ... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. 14 Installation of such cubes requires an extremely precise and detailed explanation scheme. Whereas unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units.