Fibonacci numbers - a 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, .. 75025, .. 3478759200, 5628750625, .. 26093908980000, .. 422297015649625, .. 19581068021641812000, .. by the study of complex and amazing properties. a variety of professional scientists and amateurs of mathematics.

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 the fractional charges of quark particles make up a series, and these are the first members of the Fibonacci series, which are necessary to form 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 the 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 - is not a set of 24 digits a kind of digital code development of civilization?published

P.S. And remember, just by changing your consciousness - together we change the world! © econet

Some time ago, I promised to comment on Tolkachev's statement that St. Petersburg was built according to the principle of the Golden Section, and Moscow - according to the principle of symmetry, and that this is why the differences in the perception of these two cities are so tangible, and this is why a St. ”, And the Muscovite “gets sick with his head” when he comes to St. Petersburg. It takes some time to adjust to the city (as when flying to the states - you need to adjust over time).

The fact is that our eye looks - feeling the space with the help of certain eye movements - saccades (in translation - sail clap). The eye makes a “pop” and sends a signal to the brain “adhesion to the surface has occurred. Everything is fine. This is information." And during the life of the eye gets used to a certain rhythm of these saccades. And when this rhythm changes drastically (from the urban landscape to the forest, from the Golden Section to symmetry), then some brain work is required to reconfigure.

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 I took 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, at the heart of the ZS are the numbers of the Fibonacci sequence.
This video once again clearly demonstrates this connection between the AP and the Fibonacci numbers

Where else do the AP principle and the Fibonacci sequence numbers meet?

Plant leaves are described by the Fibonacci sequence. Sunflower seeds, pine cones, flower petals, pineapple cells are also arranged according to the Fibonacci sequence.

bird egg

The lengths of the phalanges of human fingers are approximately the same as the Fibonacci numbers. The golden ratio is seen in the proportions of the face.

Emil Rozenov studied the ZS in the music of the Baroque and Classicism eras using the works of Bach, Mozart, Beethoven as an example.

It is known that Sergei Eisenstein artificially built the film "Battleship Potemkin" according to the rules of the Legislative Assembly. He broke the tape into five parts. In the first three, the action develops on the ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city takes place exactly at the point of the golden ratio. Yes, and in each part there is a turning point, which occurs according to the law of the golden section. In the frame, scene, episode, there is a certain leap in the development of the theme: the plot, the mood. Eisenstein believed that, since such a transition is close to the golden section point, it is perceived as the most natural and natural.

Many decorative elements, as well as fonts, are created using GS. For example, the font of A. Dürer (the letter “A” in the figure)

It is believed that the term "Golden Ratio" was introduced by Leonardo Da Vinci, who said, "let no one, not being a mathematician, dare to read my works" and showed the proportions 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.”

The famous portrait of Mona Lisa or Gioconda (1503) was created on the principle of golden triangles.

Strictly speaking, the star itself or the pentacle is the construction of the AP.

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.

Now about the golden section in architecture

The Pyramid of Cheops represents the proportions of the GS. (I like the photo - with the Sphinx littered with sand).

According to Le Corbusier, in the relief from the temple of Pharaoh Seti I at Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the golden ratio. The facade of the ancient Greek temple of the Parthenon also has golden proportions.

Notredam de Paris Cathedral in Paris, France.

One of the outstanding buildings made according to the principle of the AP is the Smolny Cathedral in St. Petersburg. Two paths lead to the cathedral along the edges, and if you approach the cathedral along them, then it seems to rise in the air.

In Moscow, there are also buildings made using ZS. For example, St. Basil's Cathedral

However, buildings that use the principles of symmetry prevail.
For example, the Kremlin and the Spasskaya Tower.

The height of the Kremlin walls also nowhere reflects the AP principle regarding the height of the towers, for example. Or take the hotel Russia, or the hotel Cosmos.

At the same time, buildings built according to the AP principle represent a larger percentage in St. Petersburg, while these are street buildings. Liteiny Avenue.

Thus, the Golden Ratio uses a ratio of 1.68, and the symmetry is 50/50.
That is, symmetrical buildings are built on the principle of equality of sides.

Another important characteristic of the GS is its dynamism and the desire to unfold, due to the sequence of Fibonacci numbers. Whereas symmetry, on the contrary, represents stability, stability and immobility.

In addition, the additional ZS introduces an abundance of water spaces into Peter's plan, spilling over the city and dictating the subordination of the city to their bends. And Peter's scheme itself resembles a spiral or an embryo at the same time.

The Pope, however, expressed a different version of why Muscovites and St. Petersburg residents have a “headache” when visiting the capitals. The Pope relates this to the energies of cities:
St. Petersburg - has a masculine gender and, accordingly, masculine energies,
Well, Moscow - respectively - female and has feminine energies.

So the residents of the capitals, who have tuned in to their certain balance of feminine and masculine in their bodies, find it difficult to rebuild when visiting a neighboring city, and someone may have some difficulties with the perception of one or another energy, and therefore the neighboring city may not at all be in love!

In support of this version, it also says that all Russian empresses it was in St. Petersburg that they ruled, while Moscow saw only male tsars!

Used resources.

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.

See mathematical patterns in the structure of man, flora and inanimate nature from the point of view of the Golden Section phenomenon.

Research novelty:

The discovery of Fibonacci numbers in the reality around us.

Practical significance:

Using acquired knowledge and skills research work when studying 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 presented to Europe one of greatest discoveries of all times and peoples - the decimal number system.

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

The solution was 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 reclusive life, spent a lot of time in nature, and while 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.

In Fibonacci numbers there is interesting feature: quotient from dividing the next Fibonacci number by the previous one, as the numbers themselves grow, tend to 1.618. 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 math toy turned out to be a unique code embedded in everything natural objects 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. Collaboration botanists and mathematicians 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 structural organization 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 ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.
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 physiological features living beings, be it a plant, an animal or a person, are 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 beings whom scientists world calls primitive life forms, to calculate that the spiral shape of the shell will 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 section.

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, bankruptcies, periods of prosperity, laws and waves of development in the stock and currency markets, cycles of family life, and so on , are organized on a time scale in the form of cycles, waves. These cycles and waves are also distributed according to numerical series Fibonacci!

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 cones

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 regularities of this number series, including the regularities of the "Golden" symmetry, are manifested in the energy transitions of elementary particles, in planetary and space 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 of space separate 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.:

He will tell about the concept of the Fibonacci series and how it is connected with the theory of waves, and also will refute the applicability of the series to natural processes.
, which the master developed in the 30s of the last century - this is one of the most exciting sections. By itself, it was isolated in new chapter the science that studies graphs. It is based on the developments of other specialists in the field of theory (I advise you to read - a book under the authorship).
So, for example, the great Italian mathematician Leonardo Fibonacci is considered a scientist (which I have already mentioned in articles -,), who created the basis for Eliot's theory.

Best Broker

The digital series of Fibonacci numbers - the golden ratio and coefficients or correction levels + video. Fibonacci numbers in nature.

The specialist lived in the XIII century. The scientist published a work called "The Book of Calculations". This book presented to Europe an important discovery for those times and not only the discovery - the decimal number system. This system introduced the usual numbers for us from zero to nine into circulation.

This system was the first important achievements Europe since the fall of Rome. Fibonacci saved numerical science for the Middle Ages. He also laid deep foundations for the development of other sciences, such as higher mathematics, physics, astronomy, and mechanical engineering.

Watch the video

How did numbers and their derivatives appear?

Solving an applied problem, Leonardo stumbled upon curious series of Fibonacci numbers, at the beginning of which there are two units.

Each subsequent term is the sum of the previous two. The most curious thing is that the Fibonacci number series is a remarkable sequence in that if you divide any term by the previous one, you get a number that is close to 0.618. This number was named golden ratio».

It turned out that this number has been known to mankind for a very long time. For example, in ancient egypt built pyramids using it, and the ancient Greeks built their temples on it. Leonardo da Vinci showed how the structure of the human body obeys this number.

Nature uses the Fibonacci numbers in her most intimate and advanced areas. From atomic structures and other small forms, like DNA molecules and brain microcapillaries, to huge ones, like planetary orbits and galaxy structures. The number of examples is so great that it should be argued that in nature there really is a certain basic law of proportions.

Therefore, it is not surprising that the Fibonacci series and the golden ratio made their way to stock charts. And not just one number 0.618, but also its derivatives.

If you raise the number of the golden section to the first, second, third and fourth powers and subtract the result from one, then you get new row which is called " Fibonacci retracement ratios". It remains only to add a mark of five tenths - this is fifty percent.

However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then we get 1.618, if we square it, then we get 2.618, if we 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 programs technical analysis, to determine the amount of correction and expansion. In addition, these same rows play important role in the Elliot 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, 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. As long as we can guess 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 opened, 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 the most different 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 wave theory, may be incomprehensible, over time it will become more accessible and understandable. 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 french revolution. It has nothing to do with the golden section of Fibonacci.

There are infinitely many spirals, the most popular are: natural logarithm spiral, Archimedes spiral, hyperbolic spiral.

Together with the publishing house "" we publish an excerpt from the book of Professor applied mathematics Edward Scheinerman "Guide for those in love with mathematics", dedicated to non-standard issues exciting mathematics, puzzles, universe of numbers and shapes. Translation from English by Alexey Ognev.

This chapter talks about the famous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, etc. This series was named after Leonardo of Pisa, better known as Fibonacci. Leonardo of Pisa (1170–1250) – one of the first major mathematicians medieval Europe. The nickname Fibonacci means "son of Bonacci". Author of the Book of the Abacus, which outlines the decimal number system.

Squares and dominoes

Let's start by laying squares and dominoes. Imagine a long horizontal 1×10 frame. We want to completely fill it with 1×1 squares and 1×2 dominoes, leaving no gaps. Here is the picture:

Question: How many ways can this be done?

For convenience, we denote the number of options by F10. Go through them all and then recalculate - hard labour fraught with errors. Much better to simplify the task. Let's not look for F10 right off the bat, let's start with F1. It's easier than ever! We need to fill a 1 × 1 frame with 1 × 1 squares and 1 × 2 dominoes. Dominoes will not fit, the only solution remains: take one square. In other words, F1 = 1.

Now let's deal with F2. The size of the frame is 1 × 2. You can fill it with two squares or one domino. So there are two options, and F2 = 2.

Next: In how many ways can a 1 × 3 frame be filled? The first option: three squares. Two other options: one domino (two will not fit) and a square on the left or right. So, F3 = 3. One more step: take a 1 × 4 frame. The figure shows all the filling options:

We have found five possibilities, but what is the guarantee that we have not missed anything? There is a way to test yourself. At the left end of the frame there can be either a square or a domino. In the top row in the picture - options when the square is on the left, in the bottom row - when dominoes are on the left.

Let's say it's a square on the left. The rest must be filled with squares and dominoes. In other words, you need to fill the box 1 × 3. This gives 3 options, since F3 = 3. If there are dominoes on the left, the size of the remaining part is 1 × 2, and there are two options to fill it, since F2 = 2.

So we have 3 + 2 = 5 options and we made sure F4 = 5.

Now you. Think for a couple of minutes and find all the infill options for the 1×5 frame. There are not many of them. The solution is at the end of the chapter. You can relax and think.

Let's get back to our squares. I would like to believe that you have found 8 options, since there are 5 ways of laying, where the square is on the left, and 3 more ways, where the dominoes are on the left. So F5 = 8.

Let's summarize. We have labeled FN the number of ways to fill a 1 × n frame with squares and dominoes. We need to find F10. Here's what we already know:

We move on. What is F6 equal to? You can draw all the options, but it's boring. Let's break the question into two parts. In how many ways can a 1 × 6 frame be filled in if on the left is (a) a square and (b) a domino? The good news is we already know the answer! In the first case, we are left with five squares, and we know that F5 = 8. In the second case, we need to fill in four squares; we know that F4 = 5. So F5 + F4 = 13.

What is F7 equal to? Based on the same considerations, F7 = F6 + F5 = 13 + 8 = 21. What about F8? Obviously F8 = F7 + F6 = 21 + 13 = 34. And so on. We found the following relationship: Fn = Fn-1 + Fn-2.

A few more steps - and we will find the desired number F10. The correct answer is at the end of the chapter.

Fibonacci numbers

Fibonacci numbers are the sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

It is built according to the following rules:

— first two numbers 1 and 1;

— each next number is obtained by adding the two previous ones.

We will denote the n-th element of the sequence Fn, starting from zero: F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5, ... We calculate the next element by the formula: Fn = Fn-1 + Fn-2 .

As we can see, the problem of stacking squares and dominoes led us to the sequence of Fibonacci numbers [ 1 ]In the squares and dominoes problem, we found out: F1 = 1, and F2 = 2. But the Fibonacci numbers start with F0 = 1. How does this agree with the conditions of the problem? How many ways are there to fill the 0 × 1 frame under the same conditions? The length of the square and the length of the domino are both greater than zero, so there is a temptation to say that the answer is zero, but it is not. The 0 × 1 rectangle is already filled, there are no gaps; we don't need a square or a domino. Thus, there is only one course of action: do not take a square or dominoes. Do you understand? In that case, congratulations. You have the soul of a mathematician!

Sum of Fibonacci Numbers

Let's try adding the first few Fibonacci numbers. What can we say about the sum F0 + F1 +… + Fn for any n? Let's do some calculations and see what happens. Notice the addition results below. Do you see a pattern? Wait a bit before moving on: it's better if you find the answer yourself, rather than reading a ready-made solution.

I would like to believe that you saw that the results of the summation, if one is added to them, also line up in a sequence of Fibonacci numbers. For example, adding the numbers F0 to F5 gives: F0 + F1 + F2 + F3 + F4 + F5 = 1 + 1 + 2 + 3 + 5 + 8 = 20 = F7 - 1. Adding the numbers F0 to F6 gives 33, which is one less than F8 = 34. We can write the formula for non-negative integers n: F0 +F1 +F2 +…+Fn =Fn+2 –1. (*)

It will probably be enough for you personally to see that the formula [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1.. works in a dozen cases to make you believe it's true, but mathematicians are hungry for proof. We are happy to present you with two possible proofs that it is true for all non-negative integers n.

The first is called proof by induction, the second is called combinatorial proof.

Proof by induction

Formula [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. is an infinite number of formulas in folded form. Prove that [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. true for a particular value of n, say for n = 6, is a simple arithmetic problem. It will be enough to write down the numbers from F0 to F6 and add them: F0 + F2 + ... + F6 = 1 + 1 + 2 + 3 + 5 + 8 + 13 = 33.

It's easy to see that F8 = 34, so the formula works. Let's move on to F7. Let's not waste time and add up all the numbers: we already know the sum up to F6. Thus, (F0 +F1 +…+F6)+F7 =33+21=54. As before, everything converges: F9 = 55.

If now we start checking whether the formula for n = 8 works, our strength will finally run out. But still, let's see what we already know and what we want to find out:

F0 +F1 +…+F7 =F9.

F0 +F1 +…+F7 +F7 =?

Let's use the previous result: (F0 +F1 +…+F7)+F8 =(F9-1)+F8.

We can of course calculate (F9-1) + F8 arithmetically. But this will make us even more tired. At the same time, we know that F8 + F9 = F10. Thus, we do not need to calculate anything or look into the table of Fibonacci numbers:

(F0 + F1 +… + F7) + F8 = (F9-1) + F8 = (F8 + F9-1) = F10-1.

We verified that the formula works for n = 8 based on what we knew about n = 7.

In the case of n = 9, we rely on the result for n = 8 in the same way (see for yourself). Of course, proving the correctness of [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. for n, we can be sure that [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. true for n + 1 as well.

We are ready to give a full proof. As already mentioned, [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. represents an infinite number of formulas for all values ​​of n from zero to infinity. Let's see how the proof works.

First we prove [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. in the simplest case, for n = 0. We simply check that F0 = F0+2 - 1. Since F0 = 1 and F2 = 2, obviously 1 = 2 - 1 and F0 = F2-1.

Further, it is enough for us to show that the correctness of the formula for one value of n (say, n = k) automatically means the correctness for n + 1 (in our example, n = k + 1). We just need to demonstrate how it works “automatically”. What do we need to do?

Let's take some number k. Suppose we already know that F0+F1+…+Fk =Fk+2–1. We are looking for the value F0 + F1 +… + Fk + Fk+1.

We already know the sum of the Fibonacci numbers up to Fk, so we get:

(F0+F1+…+Fk)+Fk+1 =(Fk+2–1)+Fk+1.

The right side is equal to Fk+2 - 1 + Fk+1, and we know what the sum of consecutive Fibonacci numbers is equal to:

Fk+2–1 + Fk+1 = (Fk+2 + Fk+1) - 1 = Fk+3– 1

Substitute into our equation:

(F0+F1+…+Fk)+Fk+1 =Fk+3–1

Now I will explain what we have done. If we know that [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. is true when we sum the numbers up to Fk, then [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. should be true if we add Fk+1.

To summarize:

Formula [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. true for n = 0.

If the formula [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. is true for n, it is also true for n + 1.

We can confidently say that [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. is true for any values ​​of n. Is it true [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. for n=4987? This is true if the expression is true for n = 4986, which is based on the expression being true for n = 4985, and so on up to n = 0. Therefore, the formula [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. is true for all possible values. This method of proof is known as mathematical induction (or proof by induction). We check the base case and give a template by which each next case can be proven based on the previous one.

Combinatorial proof

And here is a completely different proof of the identity [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1.. The main approach here is to take advantage of the fact that the number Fn is the number of ways to cover a 1 × n rectangle with squares and dominoes.

Let me remind you that we need to prove:

F0 + F1 + F2 +… + Fn = Fn+2- 1. (*)

The idea is to treat both sides of the equation as a solution to the cladding problem. If we prove that the left and right parts are the solution for the same rectangle, they will coincide with each other. This technique is called combinatorial proof[ 2 ]The word "combinatorial" is derived from the noun "combinatorial" - the name of the branch of mathematics, the subject of which is the calculation of options in problems similar to covering a rectangle. The word "combinatorics", in turn, is derived from the word "combinations"..

What question in combinatorics is the equation [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. gives two correct answers? This puzzle is similar to those found in the show Jeopardy! [ 3 ]A popular TV show in the USA. Similar to Jeopardy! come out in different countries; in Russia it is "Own game". - Approx. ed., where participants must formulate a question, knowing the correct answer in advance.

The right side looks simpler, so let's start with it. Answer: Fn+2– 1. What is the question? If the answer were simply Fn+2, we could easily formulate the question: in how many ways can a 1 × (n + 2) rectangle be tiled with squares and dominoes? This is almost exactly what you need, but the answer is less than one. Let's try to gently change the question and reduce the answer. Let's remove one version of the lining and recalculate the rest. The difficulty is to find one option that is radically different from the rest. Is there one?

Each cladding method involves the use of squares or dominoes. Only the squares are involved in the only option, in the others there is at least one domino. Let's take this as the basis of a new question.

Question: How many options are there for covering a 1 × (n + 2) rectangular frame with squares and dominoes, including at least one domino?

Now we will find two answers to this question. Since both will be true, between the numbers we can confidently put an equal sign.

We have already discussed one of the answers. There are Fn+2 stacking options. Only one of them involves the use of squares exclusively, without dominoes. Thus, answer #1 to our question is: Fn+2– 1.

The second answer should be - I hope - the left side of the equation [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1.. Let's see how it works.

It is necessary to recalculate the options for filling the frame, including at least one domino. Let's think about where the very first bone will be located. There are n + 2 positions, and the first tile can be in positions 1 to n + 1.

Consider the case n = 4. We are looking for ways to fill a 1 × 6 frame that involve at least one domino. We know the answer: F6 - 1 = 13 - 1 = 12, but we need to get it in a different way.

The first domino can take the following positions:

The first column shows the case when the knuckle is in the first position, the second - when the knuckle is in the second position, and so on.

How many options are in each column?

The first column contains five options. If we discard the dominoes on the left, we get exactly F4 = 5 options for a 1 × 4 rectangle. In the second column, there are three options. Let's drop the dominoes and the square on the left. We get F3 = 3 options for a 1×3 rectangle. Similarly for the other columns. Here's what we found:

Thus, the number of ways to tile squares and dominoes (at least one bone) on a 1 × 6 rectangular frame is F4 + F3 + F2 + F1 + F0 = 12.

Output: F0+F1+F2+F3+F4=12=F6–1.

Let's consider the general case. We are given a frame of length n + 2. How many ways are there to fill it, in which the first domino is at some position k? In this case, the first k - 1 positions are occupied by squares. Thus, k + 1 positions are occupied in total [ 4 ]The number k can take values ​​from 1 to n + 1, but no more, because otherwise the last domino will stick out of the frame.. The remaining (n + 2) - (k + 1) = n - k + 1 can be filled in by any means. This gives Fn-k+1 options. Let's build a diagram:

If k changes from 1 to n + 1, the value of n - k + 1 changes from 0 to n. Thus, the number of options for filling our frame with at least one domino is Fn + Fn-1 +… + F1 + F0.

If we put the terms in reverse order, we get the left side of the expression (*). Thus, we have found the second answer to the question posed: F0 +F1 +…+Fn.

So we have two answers to the question. The values ​​obtained using the two formulas we have derived coincide, and the identity [ * ]F0 +F1 +F2 +…+Fn =Fn+2 –1. proven.

Fibonacci ratio and golden ratio

Adding two successive Fibonacci numbers gives the next Fibonacci number. In this section, we will touch on a more interesting question: what happens if we divide the Fibonacci number by the one preceding it in the series? Let's calculate the ratio Fk1. For increasing values ​​of k.

In the table you can see the ratios from F1/F0 to F20/19.

The larger the Fibonacci numbers become, the closer the Fk+1/Fk ratio is to a constant approximately equal to 1.61803. This number is - you'll be surprised - well known, and if you enter it into a search engine, a lot of pages about the golden ratio will fall out. What it is? The ratio of neighboring Fibonacci numbers is not the same. However, it is almost the same if the numbers are large enough. Let's find a formula for the number 1.61803 and for this we will assume for a while that all ratios are the same. We introduce the notation x:

x=Fk+1/ Fk=/ Fk+2/ Fk+1= Fk+3/ Fk+2=…

This means that Fk+1 = xFk, Fk+2 = xFk+1, etc. We can reformulate:

Fk+2=xFk+1=x2>Fk.

But we know that Fk+2= Fk+1 + Fk. Thus, x2>FkFk = xFk + Fk.

If we divide both sides by Fk and rearrange the terms, we get quadratic equation: x2-x-1=0. It has two solutions:

The ratio must be positive. And so we got the number we know. Usually, the Greek letter φ (phi) is used to denote the golden ratio:

We have already noted that the ratio of neighboring Fibonacci numbers approaches (tends) to φ. This is amazing. This gives us another way to calculate approximate Fibonacci numbers. The sequence of Fibonacci numbers is the series F0 F1, F2, F3, F4, F5… If all ratios Fk+1/Fk are the same, we get the formula:

Here With is another constant. Let's compare the rounded values ​​of Fn and φn for different n:

For large values ​​of n, the ratio Fn/ φn≈0.723607. This number is exactly φ/root5. In other words,

Note that if we round up to the nearest whole number, we get exactly Fn.

If you don't want to bother with rounding to an integer, then the formula named after Jacques Binet [ 5 ]Jacques Binet (1786–1856) – French mathematician, mechanician and astronomer The formula for Fibonacci numbers is named after Binet, although Abraham de Moivre (1667–1754) had derived it almost a hundred years earlier. - Approx. per., will give you the exact value:

Fill frame 1×5

Our frame can be filled with squares and dominoes in the following ways:

There are F4 = 5 options when there is a square at the beginning, and F3 = 3 options when there is a domino at the beginning. In total, this gives F5 = F4 + F3 = 8 options.

F10 value(answer to next question concerning stacking) is equal to 89.