The fibonacci sequence and the principles of the golden section. Fibonacci numbers: practical application

STATE EDUCATIONAL INSTITUTION

"KRIVLYANSKAYA SECONDARY SCHOOL"

ZHABINKO DISTRICT

FIBONACCI NUMBERS AND THE GOLDEN RATIO

Research

Work completed:

10th grade student

Gardener Valeria Alekseevna

Supervisor:

Lavrenyuk Larisa Nikolaevna,

computer science teacher and

mathematics 1 qualifying

Fibonacci numbers and nature

characteristic feature structure of plants and their development is helicity. Even Goethe, who was not only a great poet, but also a naturalist, considered helicity one of characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, tissues grow in a spiral in tree trunks, seeds in a sunflower are arranged in a spiral, spiral movements (nutations) are observed during the growth of roots and shoots.

At first glance, it may seem that the number of leaves, flowers can vary over a very wide range and take on any values. But such a conclusion turns out to be untenable. Studies have shown that the number of organs of the same name in plants is not arbitrary, there are values ​​that are often found and values ​​that are very rare.

In wildlife, forms based on pentagonal symmetry are widespread - starfish, sea ​​urchins, flowers.

Photo 13. Buttercup

A chamomile has 55 or 89 petals.

Photo 14. chamomile

Feverfew has 34 petals.

Phot. 15. Pyrethrum

Let's look at a pine cone. The scales on its surface are arranged in a strictly regular manner - along two spirals that intersect approximately at a right angle. The number of such spirals pine cones equals 8 and 13 or 13 and 21.

Photo 16. Cone

In sunflower baskets, seeds are also arranged in two spirals, their number is usually 34/55, 55/89.

Photo 17. Sunflower

Let's take a look at shells. If we count the number of "stiffening ribs" for the first shell taken at random - it turned out to be 21. Let's take the second, third, fifth, tenth shell - all will have 21 ribs on the surface. It can be seen that the mollusks were not only good engineers, they "knew" the Fibonacci numbers.

Photo 18. Shell

Here again we see a regular combination of Fibonacci numbers located side by side: 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89. Their ratio in the limit tends to the golden ratio, expressed by the number 0.61803 ...

Fibonacci numbers and animals

The number of rays in starfish corresponds to a series of Fibonacci numbers or very close to them and is equal to 5.8, 13.21.34.55.

Photo 19. Starfish

Modern arthropods are very diverse. The spiny lobster also has five pairs of legs, five feathers on the tail, the abdomen is divided into five segments, and each leg consists of five parts.

Phot. twenty. spiny lobster

In some insects, the abdomen consists of eight segments, there are three pairs of limbs, consisting of eight parts, and eight different antennae-like organs emerge from the mouth opening. Our well-known mosquito has three pairs of legs, the abdomen is divided into eight segments, and there are five antennae on the head. The mosquito larva is divided into 12 segments.

Phot. 21. Mosquito

In a cabbage fly, the abdomen is divided into five parts, there are three pairs of legs, and the larva is divided into eight segments. Each of the two wings is divided into eight parts by thin veins.

The caterpillars of many insects are divided into 13 segments, for example, in the skin-eater, flour-eater, Mauritanian booger. In most pest beetles, the caterpillar is divided into 13 segments. The structure of the legs of beetles is very characteristic. Each leg consists of three parts, as in higher animals - from the shoulder, forearm and paw. Thin, openwork paws of beetles are divided into five parts.

Openwork, transparent, weightless dragonfly wings are a masterpiece of "engineering" skill of nature. What proportions underlie the design of this tiny flying muscle car? The ratio of wingspan to body length in many dragonflies is 4/3. The body of a dragonfly is divided into two main parts: a massive body and a long thin tail. The body is divided into three parts: head, thorax, abdomen. The abdomen is divided into five segments, and the tail consists of eight parts. Here it is still necessary to add three pairs of legs with their division into three parts.

Phot. 22. Dragonfly

It is easy to see in this sequence of dividing the whole into parts the expansion of a series of Fibonacci numbers. The length of the tail, body and total length of the dragonfly are related by the golden ratio: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

It is not surprising that the dragonfly looks so perfect, because it is created according to the laws of the golden ratio.

The sight of a turtle against the background of a cracked takyr is an amazing phenomenon. In the center of the carapace there is a large oval field with large fused horny plates, and along the edges there is a border of smaller plates.

Phot. 23. Turtle

Take any turtle - from the swamp turtle close to us to the giant sea turtle, soup turtle - and you will see that the pattern on their shell is similar: on the oval field there are 13 fused horn plates - 5 plates in the center and 8 at the edges, and on the peripheral border about 21 plates (the Chilean tortoise has exactly 21 plates along the periphery of the shell). Turtles have 5 fingers on their paws, and the vertebral column consists of 34 vertebrae. It is easy to see that all of these quantities correspond to the Fibonacci numbers. Consequently, the development of the turtle, the formation of its body, the division of the whole into parts was carried out according to the law of a series of Fibonacci numbers.

Supreme type animals on the planet are mammals. The number of ribs in many animal species is equal to or close to thirteen. In completely different mammals - a whale, a camel, a deer, a tour - the number of ribs is 13 ± 1. The number of vertebrae varies greatly, especially due to the tails, which can be of different lengths even in the same animal species. But in many of them the number of vertebrae is equal to or close to 34 and 55. So, 34 vertebrae in a giant deer, 55 in a whale.

The skeleton of the limbs of domestic animals consists of three identical bone links: the humerus (pelvic) bone, the bone of the forearm (shin) and the bone of the paw (foot). The foot, in turn, consists of three bone links.

The number of teeth in many domestic animals tends to Fibonacci numbers: a rabbit has 14 pairs, a dog, a pig, a horse has 21 ± 1 pairs of teeth. In wild animals, the number of teeth varies more widely: in one marsupial predator it is 54, in a hyena - 34, in one of the species of dolphins it reaches 233. The total number of bones in the skeleton of domestic animals (including teeth) in one group is close to 230, and the other - to 300. It should be noted that the small auditory ossicles and non-permanent ossicles are not included in the number of bones of the skeleton. Taking them into account total number bones of the skeleton in many animals will become close to 233, and in others it will exceed 300. As you can see, the division of the body, accompanied by the development of the skeleton, is characterized by a discrete change in the number of bones in various organs of animals, and these numbers correspond to Fibonacci numbers or are very close to them, forming row 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. The ratio of sizes in most chicken eggs equals 4:3 (some 3/2), pumpkin seeds - 3:2, watermelon seeds - 3/2. The ratio of the length of pine cones to their diameter was found to be 2:1. The size of birch leaves on average is very close to, and acorns - 5:2.

It is believed that if it is necessary to divide a flower lawn into two parts (grass and flowers), then these stripes should not be made equal in width, it will be more beautiful if you take them in a ratio of 5: 8 or 8: 13, i.e. use a ratio called the golden ratio.

Fibonacci numbers and photography

As applied to photographic art, the golden section rule divides the frame with two horizontal and two vertical lines into 9 unequal rectangles. To make it easier for themselves to shoot balanced images, photographers have simplified the task a little and began to divide the frame into 9 equal rectangles according to the Fibonacci numbers. So the rule of the golden section was transformed into the rule of thirds, which refers to one of the principles of composition.

Phot. 24. Frame and golden ratio

In the viewfinders of modern digital cameras, focus points are located at positions 2/8 or on imaginary lines dividing the frame according to the golden section rule.

Photo 25. Digital camera and focus points

Photo 26.

Photo 27. Photography and focus points

The rule of thirds applies to all subject compositions: you are shooting a landscape or a portrait, a still life or a reportage. Until your sense of harmony has become acquired and unconscious, the observance simple rule the third will allow you to take pictures expressive, harmonious, balanced.

Photo 28. Photography and the ratio of heaven and earth 1 to 2.

The most successful example for demonstration is the landscape. The principle of composition is that the sky and land (or water surface) should have a ratio of 1:2. One third of the frame should be taken under the sky, and two thirds under the land, or vice versa.

Photo 29. Photo of a flower spiraling

Fibonacci and space

The ratio of water and land on planet Earth is 62% and 38%.

The dimensions of the Earth and the Moon are in the golden ratio.

Photo 30. Dimensions of the Earth and Moon

The figure shows the relative sizes of the Earth and the Moon to scale.

Let's draw the radius of the Earth. Let's draw a segment from the central point of the Earth to the central point of the Moon, the length of which will be equal). Let's draw a line to connect these two lines to form a triangle. We get the golden triangle.

Saturn shows the golden ratio in several of its dimensions

Photo 31. Saturn and its rings

The diameter of Saturn is very close in relation to the golden ratio with the diameter of the rings, as shown by the green lines.Radius inthe inside of the rings is in a ratio very close to the outer diameter of the rings, as shown by the blue line.

The distance of the planets from the Sun also obeys the golden ratio.

Photo 32. Distance of the planets from the Sun

golden ratio at home

The golden ratio is also used to add style and appeal to the marketing and design of everyday consumer products. There are many examples, but we will illustrate only a few.

Photo 33. EmblemToyota

Photo 34. The golden ratio and clothes

Photo 34. The Golden Ratio and Automotive Design

Photo 35. EmblemApple

Photo 36. EmblemGoogle

Practical research

Now we will apply the knowledge gained in practice. Let's first take measurements among students in grade 8.

The experiment involved 7 students of the 8th grade, 5 girls and 2 boys. Height and distance from the navel to the floor were measured. The results are reflected in the tables. One student of ideal physique, for her the ratio of height to the distance from the navel to the floor is 1.6185. Another student is very close to the golden ratio, . As a result of the measurements, 29% of the participants have ideal parameters. These percentage results are also close to the golden ratio of 68% and 32%. For the first subject, we see that 3 ratios out of 5 are close to the golden ratio, in percentage terms it is 60% to 40%. And for the second - 4 out of 5, that is, 80% to 20%.

If you look closely at the television picture, then its dimensions will be 16 to 9 or 16 to 10, which is also close to the golden ratio.

Carrying out measurements and constructions in CorelDRAW X4 and using a frame from the news channel Russia 24, you can find the following:

a) the ratio of the length to the width of the frame is 1.7.

b) the person in the frame is located exactly at the focus points located at a distance of 3/8.

Next, let's turn to the official microblog of the Izvestia newspaper, in other words, to the Twitter page. For a monitor screen with 4:3 sides, we see that the “header” of the page is 3/8 of the entire height of the page.

Looking closely at the caps of the military, you can find the following:

a) the cap of the Minister of Defense of the Russian Federation has the ratio of the indicated parts 21.73 to 15.52, equal to 1.4.

b) the cap of the border guard of the Republic of Belarus has the dimensions of the indicated parts 44.42 to 21.33, which is equal to 2.1.

c) the cap of the times of the USSR has the dimensions of the indicated parts 49.67 to 31.04, which is equal to 1.6.

For this model, the length of the dress is 113.13 mm.

If you “finish” the dress to the “ideal” length, we get this picture.

All measurements have some error, since they were taken from a photograph, which does not prevent us from seeing a trend - everything that is ideal contains the golden ratio to one degree or another.

Conclusion

The world of wildlife appears to us in a completely different way - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and originality of creative combinations! Peace inanimate nature is, first of all, a world of symmetry, which gives stability and beauty to his creations. The world of nature is, first of all, a world of harmony, in which the "law of the golden section" operates.

“The Golden Ratio” appears to be that moment of truth, without which, in general, anything that exists is not possible. Whatever we take as an element of research, the "golden section" will be everywhere; even if there is no visible observance of it, then it necessarily takes place at the energy, molecular or cellular levels.

Indeed, nature turns out to be monotonous (and therefore uniform!) in the manifestation of its fundamental laws. The “most successful” solutions found by it apply to a wide variety of objects, to the most various forms organizations. The continuity and discreteness of the organization comes from the dual unity of matter - its corpuscular and wave nature, penetrates into chemistry, where it gives the laws of integer stoichiometry, chemical compounds constant and variable composition. In botany, continuity and discreteness find their specific expression in phyllotaxis, discreteness quanta, growth quanta, unity of discreteness and continuity of space-time organization. And now, in the numerical ratios of plant organs, the “principle of multiple ratios” introduced by A. Gursky appears - a complete repetition of the basic law of chemistry.

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

There is speculation that the Fibonacci series is an attempt by nature to adapt to a more fundamental and perfect golden section logarithmic sequence, which is almost 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 series, it is enough to know three of its members, going one after another. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. You might think that it is the basis for all other sequences.

Each member of the golden logarithmic sequence is a degree of the Golden Ratio (). Part of the row looks something like this:... ; ; ; ; ; ; ; ; ; ; ... If we round the value of the Golden Ratio to three decimal places, we get=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 by1,618 , but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two neighboring 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...

List of sources used

Vasyutinskiy, N. golden ratio/ Vasyutinskiy N, Moscow, Young Guard, 1990, - 238 p. - (Eureka).

Vorobyov, N.N. Fibonacci numbers,

Access mode: . Access date: 17.11.2015.

Access mode: . Access date: 16/11/2015.

Access mode: . Access date: 13. 11. 2015.

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

One such example is golden ratio and Fibonacci numbers that form its basis. This pattern has been displayed in mathematical form and is often found in human environment nature, once again excluding the possibility that it arose by chance.

Fibonacci numbers and their sequence

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

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

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

A series of Fibonacci numbers has its own interesting patterns:

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

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

Golden Ratio or Divine Proportion

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

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

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

Dividing the segments will show the following results:

AC/BC=55/89=0.618

BC/AB=89/144=0.618

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

Where are Fibonacci numbers found?

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

It is important to note that the golden Fibonacci numbers are not just science, but a mathematical representation of the world around us. Lots of natural phenomena, representatives of the plant and animal world has the "golden section" in its proportions. These are spiral curls of the shell, and the arrangement of sunflower seeds, cacti, pineapples.

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

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

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

Often used by traders, the tool "" can high precision show the price movement targets, as well as the levels of its correction.

IN Lately, working in individual and group processes with people, I returned to thoughts about the unification of all processes (karmic, mental, physiological, spiritual, transformational, etc.) into one.

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

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

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

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

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

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

Let's remember the Fibonacci sequence together:

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

Each subsequent number is formed by adding the previous two:

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

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

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

Here's what we got:

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

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

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

Doesn't it seem strange or natural to you that

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

If the resulting sequence is written as follows,

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

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

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

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

3 3 6 9 6 6 3 9

When testing these numerical series, we got:

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

Matrix of the Golden Section

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Practical application of the Fibonacci series

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

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

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

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

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

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

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

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

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

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

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

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

- 12 times.

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

Read aloud 12 times

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

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

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

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

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

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

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

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

Of course, this number series is not a panacea. It's just a tool that allows you to quickly produce necessary work with a person, align him vertically with different levels Genesis.

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

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

Introducing Fibonacci Numbers

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

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

You can write it like this:

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

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

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

The formula in this case looks like this:

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

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

Leonardo of Pisa aka Fibonacci

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

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

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

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

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

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

Fibonacci and his tasks

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

Rabbits are not only valuable fur

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

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

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

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

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

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

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

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

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

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

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

More problems in number theory

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

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

An explanation about recursion

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

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

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

Explanation of the golden ratio

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

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

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

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

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

• Cut length from = 1, but = 0,618, b = 0,382.
• Attitude from to but = 1, 618.
• Attitude from to b = 2,618

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

Here is an example: 144, 233, 377.

233/144 = 1.618 and 233/377 = 0.618

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

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

Golden Rectangle and Fibonacci Spiral

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

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

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

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

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

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

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

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

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

Fibonacci numbers in nature

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

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

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

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

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

Problems in combinatorics

Fibonacci numbers are widely used in solving problems in combinatorics.

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

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

Task #1:

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

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

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

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

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

Answer: 89 ways.

Task #2:

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

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

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

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

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

Answer: 144.

Task #3:

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

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

Answer: F n – 1.

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

Fibonacci numbers in popular culture

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

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

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

0 Set off on the road

1 Clicked one joint

1 One sleeve trembled

2 Everything, get the staff

Everything, get the staff

3 Request for boiling water

The train goes to the river

The train goes to the taiga<…>.

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

Dense food of the Fibonacci wives

It was only for their benefit, not otherwise.

The wives weighed, according to rumor,

Each is like the previous two.

Summing up

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

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

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

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

The result is a series of numbers: 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. It can be continued indefinitely. Its essence is that each next number is the sum of the previous two.

This series has several mathematical features that must be touched upon. It 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 series 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 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 nature. The savvy Leonardo, in fact, did not discover anything new, he simply 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 life situation, and sometimes 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 different sizes, which are in a certain relationship with each other and the whole. golden ratio - supreme manifestation perfection of the whole and its parts in science, art and nature.

If on simple example, then the Golden Section is the division of the 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 , section b - 0,382 , only in this way the condition of the Golden Section will be met (0,618/0,382=1,618 ; 1/0,618=1,618 ) . Attitude c to a equals 1,618 , but from to b 2,618 . 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. Proportions human body in many respects 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?

A 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:

A photo: brewbooks on Flickr

A photo: beart.org.uk
A photo: esdrascalderan on Flickr
A 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, she herself is far from perfect, like everything else in this world.

There is speculation that the Fibonacci series is an attempt by nature to adapt to a more fundamental and perfect golden section logarithmic sequence, which is almost 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 series, it is enough to know three of its members, going one after another. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. 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 is achieved by simply adding two neighboring 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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