Italian mathematician Leonardo Fibonacci lived in the 13th century and one of the first in Europe began to use Arabic (Indian) figures. He came up with a somewhat artificial task of rabbits, which are grown on the farm, and all of them are considered females, the males are ignored. Rabbits begin to multiply after they are played for two months, and then every month give birth along the rabbit. Rabbits never die.

Need to determine how many rabbits will be on the farm through n. months, if only one newborn rabbit was at the initial moment of time.

Obviously, the farmer has one rabbit in the first month and one rabbit - in the second month. For the third month there will be two rabbits, on the fourth - three, etc. Denote the number of rabbits in n. Monthly like. In this way,
,
,
,
,
, …

You can build an algorithm that allows you to find with any n..

According to the condition of the problem, the total number of rabbits
in n.The month is folded into three components:

one-month rabbits that are not capable of reproduction in quantity

;

Thus, we get

. (8.1)

Formula (8.1) allows you to calculate a number of numbers: 0, 1, 1, 2, 3, 5, 55, 13, 21, 34, 55, 89, 144, 23, 55, 99, 144, 23, 55, 99, 144, 23, 55, 89, 144, 23

Numbers in this sequence are called fibonacci numbers .

If taken
and
, Using formula (8.1), you can define all other Fibonacci numbers. Formula (8.1) is called recurrent formula ( recurrence - "Return" in Latin).

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

If a n. \u003d 1, there is only one option to solve the problem. For n. \u003d 2 There are 2 options: two single steps or one double. For n. \u003d 3 There are 3 options: three single steps, or one single and one double, or one double and one one.

In the following case n. \u003d 4, we have 5 possibilities (1 + 1 + 1 + 1, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 2 + 2).

In order to answer a given question for arbitrary n., Denote the number of options as and try to determine
according to famous and
. If we start with a single step, we have combinations for the remaining n. Steps. If you start from a double step, we have
combinations for the remaining n.-1 steps. Total number of options for n.+1 Steps equals

. (8.2)

The resulting formula as a twin resembles formula (8.1). Nevertheless, it does not allow to identify the number of combinations with Fibonacci numbers . We see, for example, that
, but
. However, the following dependence is:

.

This is true for n. \u003d 1, 2, and also valid for each n.. Fibonacci numbers and number of combinations are calculated by the same formula, however the initial values
,
and
,
they differ from them.

Example 8.2.This example is practical for problematic coding problems. We find the number of all binary words of length n.not containing several zeros in a row. Denote by this number through . Obviously
, and the words of length 2, satisfying our limit, are: 10, 01, 11, i.e.
. Let be
- Such a word from n. Symbols. If symbol
T.
may be arbitrary (
) -Bust words that do not contain several zeros in a row. So, the number of words with unit at the end is equal
.

If symbol
, I must
, and first
symbol
may be arbitrary with regard to the limitations under consideration. Therefore, there is
length words n. with zero at the end. Thus, the total number of words of interest to us is equal

.

Considering that
and
The resulting sequence of numbers is the numbers of Fibonacci.

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

You can argue like that. Let be
number of zeros in the words under consideration. In any word there is
the gaps between the nearest zeros, in each of which there is one or several units. It is assumed that
. Otherwise, there is not a single word without nearby zeros.

If you remove exactly one unit from each gap, then we get the word length
Containing zeros. Any such word can be obtained indicated from some (and moreover only one) k.-Bended word containing zulos, no two of which are not nearby. So, the desired number coincides with the number of all the words of the length
containing smooth zeros, i.e. equally
.

Example 8.4.We prove that the amount
equal to Fibonacci numbers for any whole . Symbol
denotes the smallest integer, greater or equal . For example, if
T.
; what if
T.
ceil. ("ceiling"). Also occurs symbol
which means the greatest integer smaller or equal . In English this operation is called floor ("floor").

If a
T.
. If a
T.
. If a
T.
.

Thus, for the cases considered, the amount is really equal to Fibonacci numbers. Now we give proof for a general case. Since the numbers of fibonacci can be obtained using the recurrent equation (8.1), the equality should be performed:

.

And it really is done:

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

The amount of Fibonacci numbers

We define the amount of the first n. Fibonacci numbers.

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

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

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

It is easy to notice that by adding to the right part of each equation, we again get the number of fibonacci. General formula for determining the amount of the first n. Fibonacci numbers has the form:

We prove this using the method of mathematical induction. To do this we write:

This amount must be equal
.

Reduced the left and right-hand part of the equation on -1, we obtain equation (6.1).

Formula for Fibonacci numbers

Theorem 8.1. Fibonacci numbers can be calculated by the formula

.

Evidence. Make sure of the justice of this formula for n. \u003d 0, 1, and then prove the validity of this formula for arbitrary n. By induction. Calculate the attitude of the two closest numbers of Fibonacci:

We see that the ratio of these numbers fluctuates near the value of 1.618 (if you ignore several first values). This property of Fibonacci is reminiscent of geometric progression. Institute
, (
). Then expression

transformed by B.

which after simplicity looks like

.

We received a square equation, the roots of which are equal:

Now we can write:

(Where c. is a constant). Both members and do not give numbers Fibonacci, for example
, while
. However, the difference
Satisfies a recurrent equation:

For n.\u003d 0 this difference gives , i.e:
. However n.\u003d 1 we have
. To obtain
, It is necessary to accept:
.

Now we have two sequences: and
which begin with the same two numbers and satisfy the same recurrent formula. They should be equal:
. Theorem is proved.

As an increase n. member becomes very big while
, and the role of a member the difference is reduced. Therefore, at large n. We can be recruited

.

We ignore 1/2 (since the numbers of Fibonacci increase to infinity with growth n. to infinity).

Attitude
called golden cross sectionIt is used outside of mathematics (for example, in sculpture and architecture). The golden cross section is the relationship between the diagonal and side the right pentagon (Fig. 8.1).

Fig. 8.1. The right pentagon and its diagonal

For the designation of the Golden section, it is customary to use the letter
in honor of the famous Athenian sculption Fidiya.

Simple numbers

All natural numbers, large units, disintegrate into two classes. The first includes numbers that have exactly two natural divisors, a unit and himself, to the second - all others. First class numbers are called simple, and the second - compound. Simple numbers within the first three tens: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

The properties of prime numbers and their connection with all natural numbers was studied by Euclide (3 century to our era). If you write simple numbers in a row, you can see that relative density decreases. For the first ten, they account for 4, i.e. 40%, on a hundred - 25, i.e. 25%, per thousand - 168, i.e. less than 17%, per million - 78498, i.e. less than 8%, etc .. However, their total number is infinite.

Among the simple numbers there are couples such, the difference between which is equal to two (the so-called simple twins) However, the limb or infinity of such steam has not been proven.

Euclid considered it obvious that with the help of multiplication of only prime numbers, all natural numbers can be obtained, and each natural number represents in the form of a product of prime numbers singularly (with an accuracy of the procedure for multipliers). Thus, simple numbers form a multiplicative basis of a natural row.

The study of the distribution of prime numbers led to the creation of an algorithm that allows you to receive tables of prime numbers. Such an algorithm is swelto Eratosthen (3 century BC). This method consists in selecting (for example, by overclocking) those integers of the specified sequence
who share at least one of the simple numbers smaller
.

Theorem 8 . 2 . (Euclide's theorem). The number of prime numbers is infinite.

Evidence. Euclide's theorem on the infinity of the number of proves the number of proves the method proposed by Leonard Euler (1707-1783). Euler reviewed the work on all simplicity p.:

for
. This product converges, and if it is revealed, then due to the uniqueness of the decomposition of natural numbers on ordinary factors it turns out that it equals the sum of the series From where the Euler identity follows:

.

Since when
the row on the right diverges (harmonic series), then the Euler's identity follows the Euclide's theorem.

Russian mathematician P.L. Chebyshev (1821-1894) brought the formula that determines the limits in which the number of prime numbers was concluded
not exceeding X.:

,

where
,
.

State Education Establishment

"Krivan Central School"

Zhabinkovsky District

Fibonacci numbers and golden section

Research

Work completed:

student 10 Class

Sadovnikchik Valery Alekseevna

Leader:

Lavrenyuk Larisa Nikolaevna,

teacher Informatics I.

mathematics 1 Qualification

Fibonacci and Nature

A characteristic feature of the structure of plants and their development is spirality. Another Goethe, who was not only a great poet, but also a naturalist, considered spirality with one of the characteristic signs of all organisms, manifestation of the most intimate essence of life. Plant mustache spirally, the spirals are tested in the trunks of the trees, the spirals are located in sunflower, spiral movements (nation) are observed with the growth of roots and shoots.

At first glance, it may seem that the number of leaves, flowers can change in very wide limits and take any values. But this conclusion turns out to be insolvent. Studies have shown that the number of organs of the plants in plants is not arbitrary, there are values \u200b\u200bthat are often encountered and values \u200b\u200bthat are very rare.

In the wilderness, shapes based on pentagonal symmetry are widespread - starfish, marine hedgehogs, flowers.

Photo.13. Buttercup

In the chamomile number of petals 55 or 89.

Photo.14. Chamomile

Pyrethrum has 34 petals.

Fot. fifteen. Pyrethrum

Let's look at the pine bump. The scales on its surface are strictly natural - along two spirals that intersect approximately at right angles. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Photo.16. Cone

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

Photo.17. Sunflower

We look at the shells. If you recalculate the number of "ribs of stiffness" from the first, taken at rakushku rakoshai - it turned out 21. Take the second, third, fifth, the tenth sewer - everyone will have 21 edges on the surface. It can be seen, the mollusks were not only good engineers, they "knew" the numbers of Fibonacci.

Photo.18. Shell

Here again we see the regular combination of Fibonacci numbers nearby: 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89. Their attitude in the limit is striving for a gold proportion, a pronounced number of 0,61803 ...

Fibonacci and animal numbers

The number of rays from marine stars corresponds to a number of Fibonacci numbers or very close to them equal to 5.8, 13,21,34,55.

Photo.19. Starfish

Modern arthropods are very diverse. Langstum also has five feet pairs, on the tail five feathers, the abdomen is divided into five segments, and each foot consists of five parts.

Fot. twenty. Langust

In some insect, the abdomen consists of eight segments, there are three pairs of extremities consisting of eight parts, and eight different aggregate organs leave the mouth of the mouth. Our well-friendly mosquito - three pairs of legs, the abdomen is divided into eight segments, on the head five mustache - antennas. The mosquito larva is shaped on 12 segments.

Fot. 21. Mosquito

The flies of cabbage belly shacks five parts, there are three pairs of legs, and the larva is divided into eight segments. Each of the two wings is divided by subtle streaks to eight parts.

The caterpillars of many insects are members of the 13 segments, for example, at the hinewood, Mukeda, Kozhenki Moorish. In most pest beetles, the caterpillar is shaped into 13 segments. Very characteristic of the structure of the legs of the beetles. Each foot consists of three parts, like the highest animals, from the shoulder, forearm and paws. Thin, openwork paws of beetles are members of five parts.

Openwork, transparent, weightless dragonfly wings are a masterpiece of "engineering" mastery of nature. What proportions are based on the design of this tiny flying muscle? The ratio of the scope of wings to the length of the body in many dragonflies is 4/3. The body of dragonfly is divided into two main parts: a massive case and a long thin tail. Three parts are distinguished in the housing: head, chest, abdomen. The abdomen is broken by five segments, and the tail consists of eight parts. You still need to add three pairs of legs with their members to three parts.

Fot. 22. Dragonfly

It is easy to see in this sequence of membership of the whole part of the deployment of a number of Fibonacci numbers. The length of the tail, housing and the total length of the dragonflies are interconnected with a gold proportion: the ratio of the length of the tail and the housing is equal to the ratio of the total length to the tail length.

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

View of the turtle on the background of the tacty-covered tacty - the phenomenon is amazing. In the center of the shell, a large oval field with large controversy horny plates, and on the edges - a border of smaller plates.

Fot. 23. Turtle

Take any turtle - from closer to us to a giant sea, the soup turtle - and you will make sure that the drawing on the shell is similar to them: on the oval field there are 13 surrounding horny plates - 5 plates in the center and 8 - along the edges, and on the peripheral kimea About 21 plates (in the Chilean turtle on the periphery of the shell exactly 21 plates). On the paws in the skulls of 5 fingers, and the vertebral pole consists of 34 vertebrae. It is easy to notice that all the specified values \u200b\u200bcorrespond to Fibonacci numbers. Consequently, the development of the turtle, the formation of its body, the membership of the whole on the part was carried out under the law of a number of numbers of Fibonacci.

The highest type of animals on the planet is mammals. The number of ribs in many animal species is equal to or close to thirteen. In completely different mammals - whale, camel, deer, tour - the number of edges is 13 ± 1. The number of vertebrae changes very much, especially due to the tailings, which can be different lengths even in the same type of animal. But many of them have the number of verteons equally or close to 34 and 55. So, 34 vertebra at the giant deer, 55 - in China.

The skeleton of domestic limbs consists of three identical bone links: shoulder (pelvic) bones, bones of the forearm (tibia) and paw bones (foot). Stop, in turn, consists of three bone links.

The number of teeth from many pets to Fibonacci's numbers: a rabbit has 14 pairs, dogs, pigs, horses - 21 ± 1 pair of teeth. In wild animals, the number of teeth changes more widely: one short predator is equal to 54, the hyena is 34, one of the types of dolphins reaches 233. The total number of bones in the skeleton of pets (taking into account the teeth) at the same group close to 230, and In the other - to 300. It should be noted that small hearing bones and non-permanent bones are not included in the number of skeleton bones. With their account, the total number of skeleton bones in many animals will become close to 233, and others will exceed 300. As we see, the body membership, 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 very Close to them, forming a number 3, 5, 8, 134, 21, 34, 55, 89, 144, 233. The ratio of the size of most chicken eggs is 4: 3 (in some 3/2), pumpkin seeds - 3: 2 , watermelon seeds - 3/2. The ratio of the length of pine cones to their diameter turned out to be 2: 1. The dimensions of the birch leaves are very close to, and the acorns - 5: 2.

It is believed that if you need to split the flower lawn (grass and flowers) into two parts, then you should not make these bands equal in width, it will be more beautiful if you take them in terms of 5: 8 or 8: 13, i.e. Take advantage of such a proportion called "golden cross section."

Fibonacci numbers and photo

With regard to photographic art, the Rule of the Golden section divides the frame with two horizontal and two vertical lines on 9 unequal rectangles. To facilitate the task of shooting balanced images, photographers have slightly simplified the task and began to share the frame on 9 equal rectangles in accordance with Fibonacci numbers. So the golden cross section rule was transformed into a rule of the third, which refers to one of the principles of constructing the composition.

Fot. 24. Frame and Golden section

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

Photo.25. Digital camera and focus points

Photo.26.

Photo.27. Photography and focus points

The usur of the third is applicable to all plot compositions: you take off the landscape or portrait, still life or report. While your feeling of harmony has become acquired and unconscious, adherence to the religious rule of the third will allow you to take pictures expressive, harmonious, balanced.

Photo.28. Photography and the attitude of the sky and earth 1 to 2.

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

Photo.29. Flower photography spirals

Fibonacci and space

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

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

Fot.30. Sizes of land and moon

The figure shows the relative dimensions of the Earth and the Moon on the scale.

Draw the land radius. We carry out a segment from the central point of the Earth to the central point of the moon, the length of which will be equal to). Draw a segment to connect two segments data to form a triangle. We get a golden triangle.

Saturn shows a gold proportion in several of its measurements

Photo.31. Saturn and his rings

The diameter of Saturn is very close in relation to the gold proportion with the diameter of the rings, as shown by green lines. Radius B.the nutrine part of the rings is in relation to, 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 proportion.

Photo.32. Distance Planets from the Sun

Golden section in everyday life

The golden section is also used to give style and attractive in the field of marketing and design of everyday consumer goods. There are many examples, but we will illustrate only some.

Photo.33. EmblemToyota.

Photo.34. Golden section and clothing

Photo.34. Golden section and automotive design

Photo.35. EmblemApple.

Photo.36. EmblemGoogle

Practical research

Now we apply the knowledge gained in practice. We first take measurements among students of grade 8.

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

Conducting measurements and constructions in CorelDRAW X4 and using the Russian News Channel Frame 24, you can detect the following:

a) The length ratio 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, we turn to the official microblogging of the newspaper "Izvestia", in other words, to the twitter page. For the monitor screen with the sides of 4: 3vidim, that the "cap" page is 3/8 from the entire height of the page.

Carefully looking at the fee of the military, you can find the following:

a) The forage of the Minister of Defense of the Russian Federation has the ratio of the specified parts of 21.73 K 15.52, equal to 1.4.

b) The forage of the border guard RB has the dimensions of the specified parts of 44.42 to 21.33, which is 2.1.

c) Puffing the time of the USSR has the dimensions of the specified parts of 49.67 to 31.04, which is 1.6.

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

If you "draw" a dress to the "perfect" length, then we get this picture.

All dimensions have some error, as they were held in photography, which does not prevent you from seeing a tendency - everything that perfectly contains a golden cross section to one degree or another.

Conclusion

The world of wildlife appears before us completely different - movable, volatile and amazingly diverse. Life demonstrates us a fantastic carnival of diversity and uniqueness of creative combinations! The world of inanimate nature is primarily the world of symmetry, who has sustainability and beauty. The world of nature is primarily the world of harmony in which the "Golden section" is valid.

“Golden cross section appears to be the moment of truth, without the execution of which it is not possible, in general, something that is. Whatever we have taken an element of research, the "golden cross section" will be everywhere; If there is no visible observance of it, it necessarily takes place on the energy, molecular or cellular levels.

Truly, nature turns out to be monotonous (and therefore one!) In the manifestation of its fundamental patterns. They found by it "Most successful" solutions apply to a variety of objects, on a wide variety of forms of organization. Continuity and discreteness of the Organization proceeds from the dialos of matter - its corpuscular and wave nature, penetrates the chemistry, where the laws of integer stoichiometry, chemical compounds of continuous and variable composition, gives. In the botanist, continuity and discreteness find their specific expression in philloaxis, discreteness quanta, growth quanta, unity of discreteness and continuity of the spatial-temporal organization. And already in the numerical relationships of plant organs, the "principle of multiple relations" appears, introduced by A. Gursky, is the full repetition of the main law of chemistry.

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

There is a suggestion that a number of fibonacci is an attempt to adapt to a more fundamental and perfect gold logarithmic sequence, which is practically the same, just starts from nowhere and goes nowhere. The nature must necessarily need some kind of principle, from which you can push off, it cannot create something from nothing. The relations of the first members of Fibonacci sequence are far from the golden section. But the farther we are moving on it, the more these deviations are smoothed. To determine any row, it is enough to know three of its member, coming together. But not for the gold sequence, it is enough for it, it is geometric and arithmetic progress at the same time. You might think it seems to be the basis for all other sequences.

Each member of the golden logarithmic sequence is the degree of gold proportion (). Part of the row looks like this:... ; ; ; ; ; ; ; ; ; ; ... If we round the value of the golden proportion to three characters, then 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 member can be obtained not only by multiplying the previous on1,618 , but also the addition of the two previous ones. Thus, exponential growth is provided by simply addition of two adjacent elements. This is a number without start and end, and it is at him that it is trying to be a similar Fibonacci sequence. Having a well-defined start, she strives for the ideal, never reaching it. That is life.

Nevertheless, in connection with all 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 all the way he wanted? And if so, why did it come down? Mutations? Free choice? What will be next? Spiral twists or spinned?

Finding an answer to one question, you will get the next. I solve it, you will get two new ones. You'll figure them out, three more will appear. Deciding and to have them, get five unresolved. Then eight, then thirteen, 21, 34, 55 ...

List of sources used

Vasyutinsky, N. Golden proportion / Vasyutinsky N, Moscow, Young Guard, 1990, - 238 p. - (Eureka).

Vorobev, 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.

according to the materials of the book B. Biggs "Hedger came out of fog"

About Fibonacci and Trading

As an entry to the topic, we turn to the technical analysis for a while. If we talk briefly, the technical analysis puts the task of predicting the future movement of the price of the asset, based on past historical data. The most famous wording of his supporters is the price already includes all the necessary information. The implementation of technical analysis began with the development of exchange specs and is probably completely not completed so far, because there is potentially unlimited earnings. The most famous techniques (terms) in tectalize are the levels of support and resistance, Japanese candles, figures that foreshadow price and others.

The paradoxicity of the situation in my opinion is the following - most of the methods described got so great distribution that, despite the lack of an evidence base for their effectiveness, they really got the opportunity to influence the behavior of the market. Therefore, even skeptics that enjoy fundamental data should take into account these concepts simply because they take into account a very large number of other players ("tech"). Technical analysis can work well on history, but it is not possible to make it possible to earn anyone in practice almost anyone - it's much easier to get rich, making a big edition book "How to become a millionaire using a technical analysis" ...

In this sense, the theory of Fibonacci is worth a mansion, also used to predict prices for different dates. Her followers are usually called "wavewings." It is precisely a mansion because it appeared simultaneously with the market, but much earlier - as well as 800 years. Another feature is that the theory was reflected almost as a world concept for the description of everything and everything, and the market is only a special case for its application. The effectiveness of the theory and its existence provides it with both new supporters and new attempts to draw up the least controversial and generally accepted description of the behavior of the markets on its basis. But alas - further some successful market predictions that can be equated with luck, the theory still has not advanced.

The essence of Fibonacci theory

Fibonacci lived for a long time, especially for his time, the life that was devoted to solving a number of mathematical tasks, formulating them in their voluminous labor "Book of Accounts" (beginning of the 13th century). He was always interested in the Mystic numbers - probably he was not less genious than Archimedes or Euclide. The challenges associated with the square equations were made and partially solved before Fibonacci, for example, by the famous Omar Khayiam - scientists and a poet; However, Fibonacci formulated the task of reproduction of rabbits, the conclusions from which they brought him what allowed his behalf not to get lost in the centuries.

In short, the task is as follows. In place, fenced from all sides by the wall, placed a couple of rabbits, and any pair of rabbits takes on the light of another couple every month, starting from the second month of its existence. The reproduction of rabbits in time will be described by the sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. From a mathematical point of view, the sequence was simply unique, because possessed a number of outstanding properties:

• the sum of two any consecutive numbers is the following number of sequences;

• the ratio of each number of sequences, starting with the fifth, to the previous one, is 1.618;

• the difference between the square of any number and the square of the number into two positions to the left will be the number of fibonacci;

• the sum of squares standing next to numbers will be the number of fibonacci, which stands through two positions after more elevated in the square of the numbers

Of these conclusions, the second is most interesting, since it uses the number 1.618, known as the Golden Section. This number was known to the ancient Greeks that used it during the construction of Parfenon (by the way, according to some data served by Grekam Central Bank). No less interesting and the fact that the number of 1.618 can be found in nature both in micro and macroscale - from the turns of the spiral on the snail shell to large spirals of cosmic galaxies. Pyramids in Giza, created by the ancient Egyptians, also contained several parameters of the Fibonacci row. The rectangle, one side of which is 1.618 times, looks most nice for the eye - this ratio used Leonardo da Vinci for his paintings, and at a more everybody plan, they sometimes used when creating windows or doorways. Even a wave, as in the picture at the beginning of the article, can be represented as a spiral of Fibonacci.

In the wilderness, Fibonacci sequence manifests itself no less often - it can be found in claws, teeth, sunflower, web and even reproduction of bacteria. If desired, the sequence is found in almost everything, including the human face and body. Nevertheless, there is an opinion that many allegations that are Fibonacci's numbers in natural and historical phenomena are incorrect - this is a common myth, which is often inaccurate under the desired result.

Fibonacci numbers in financial markets

One of the first, who most densely engaged in the appointment of Fibonacci numbers to the financial market, was R. Elliot. His works did not disappear in the sense that market descriptions with the use of Fibonacci theory are often referred to as "Elliot's waves". The basis of the development of markets here was a model for the development of humanity from supercickers with three steps ahead and two backwards. The fact that humanity is developing non-linearly obviously almost to everyone - the knowledge of the ancient Egypt and the atomistic teaching of the democritus was completely lost in the Middle Ages, i.e. approximately 2000 years; The 20th century gave rise to such horror and the insignificance of human life, which was difficult to imagine even in the era of the Punic Wars of the Greeks. However, even if we take the theory of steps and their number for the truth, it remains unclear the size of each step, which makes Elliot's waves comparable to the predictive force of the eagle and the rush. The starting point and the correct calculation of the number of waves were and apparently will be the main weakness of the theory.

Nevertheless, local progress in theory was. Bob Postecher, who can be considered a student of Elliot, correctly predicted the bullish market of early 80s, and 1987 - as a swivel. It really happened, after which Bob obviously felt like a genius - at least in the eyes of others, he exactly became an investment guru. Subscribe to Elliott Wave TheORIST Poster grew to 20,000,however, it decreased in the early 1990s, since the "death and darkness" predicted by the American market decided to wait a little. However, for the Japanese market it worked, and a number of supporters of the theory, "late" there for one wave, lost either their capital, or capital customers of their companies. Equally, with the same successes, the theory is often trying to apply to trade in the foreign exchange market.

The theory covers a variety of trading periods - from weekly, which relates it to standard strategies to thekanalysis, up to the calculation for decades, i.e. Closes on the territory of fundamental predictions. This is possible due to the variation of the number of waves. The weaknesses of the theory mentioned above allow its adepts to speak not about the insolvency of the waves, but about their own miscalculations among them and incorrect definition of the initial position. It looks like a labyrinth - even if you have a faithful card, then you can go on it only if you understand where you are. Otherwise, there is no benefit from the map. In the case of Elliot's waves, there are all signs to doubt not only in the correctness of their location, but also in loyalty of the card as such.

conclusions

The wave development of humanity has a real basis - in the Middle Ages, the waves of inflation and deflation alternated with each other, when the war replaced a relatively peaceful peaceful life. The observation of Fibonacci sequence in nature at least in some cases of doubt also does not cause. Therefore, each to the question of who is God: a mathematician or a random number generator - has the right to give his own answer. Personally, my opinion is such that although all human history and markets can be represented in the wave concept, the height and duration of each wave is not given to predict anyone.

At the same time, 200 years of observations over the American market and more than 100 years are allowed to clearly say that the stock market is growing, passing through various periods of growth and stagnation. This fact is enough for long-term earnings in the stock market, without resorting to controversial theories and trusting them more capital than it follows as part of reasonable risks.

Khanaliyeva Dana

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

We were convinced that nature has its own laws expressed by mathematics.

And mathematics so An important tool of knowledge secrets of nature.

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MBOU "Pervomaisk Secondary School"

Orenburg district of the Orenburg region

RESEARCH

"Mystery of numbers

Fibonacci "

Performed: Canaliyeva Dana

grade 6 student

Scientific adviser:

Gazizova Valery Valerievna

Mathematics teacher of the highest category

p. Experimental

2012

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

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

Chapter 1. The numbers of Fibonacci in the wildlife ....... ....... …………………………………... five.

Chapter 2. Spiral Fibonacci ................................................. ..........……………..... nine.

Chapter 3. The numbers of Fibonacci in the inventions of a person ......... ................................. .. 13

Chapter 4. Our research ........................................................................ .... 16.

Chapter 5. Conclusion, conclusions .......................................................................... 19.

List of used literature and websites of the Internet ....................................... ........ 21.

Object of study:

Man, mathematical abstractions created by man, inventions of a person surrounding the plant and animal world.

Subject of study:

the form and structure of the studied objects and phenomena.

Purpose of the study:

explore the manifestation of Fibonacci numbers and the law of the golden section in the structure of living and non-living objects associated with it

find examples of using Fibonacci numbers.

Tasks of work:

Describe the method of building a row of Fibonacci and Spiral Fibonacci.

See mathematical patterns, in the structure of man, plant peace and inanimate nature from the point of view of the phenomenon of the golden cross section.

Novelty Studies:

Opening of Fibonacci numbers in the surrounding reality.

Practical significance:

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

Skills and abilities:

Organization and conduct of the experiment.

Using special literature.

Acquisition of the ability to make a review of the assembled material (report, presentation)

Design work with drawings, diagrams, photographs.

Active participation in the discussion of your work.

Research methods:

empirical (observation, experiment, measurement).

theoretical (logical level of knowledge).

Explanatory note.

"Numbers manage the world! The number is the power reigning over the gods and mortals! " - So they said more ancient Pythagoreans. Is this the basis of the teachings of Pythagora today? Studying in school science numbers, we want to make sure that indeed, the phenomena of the whole universe are subordinated to certain numerical relations, find this invisible connection between mathematics and life!

Is it really in every flower

And in the molecule and in the galaxy,

Numerical patterns

This strict "dry" mathematics?

We turned to a modern source of information - to the Internet and read about Fibonacci numbers, about magical numbers that make up a great riddle. It turns out that these numbers can be found in sunflowers and pine cones, in the wings of dragonfly 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. The result of our activity and was this research work.

Hypothesis:

in the surrounding reality, everything is built in surprisingly harmonious laws with mathematical accuracy.

Everything in the world is thoughtful and calculated the most importantly our designer - Nature!

Introduction The story of a number of Fibonacci.

Amazing numbers were opened by the Italian mathematician Middle Ages Leonardo Pisansky, more famous under the name Fibonacci. Traveling in the east, he met the achievements of Arab Mathematics, contributed to the transfer of them to the West. In one of his works, under the name "Computing Book", he presented to Europe one of the greatest discoveries of all times and peoples - a decimal number system.

Once, he broke his head over the solution of one mathematical task. He tried to create a formula describing the sequence of breeding rabbits.

The rallying was a numeric number, each subsequent number of which is the sum of the two previous two:

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

The numbers forming this sequence are called "Fibonacci Numbers", and the sequence itself is a fibonacci sequence.

"So what?" - You will tell you, "Did we ourselves come up with such numeric rows that grow up for a given progression?" Indeed, when a number of Fibonacci appeared, no one, including he himself, did not suspect how closely he managed to get closer to one of the greatest secrets of the universe!

Fibonacci was led by a lesible lifestyle, spent a lot of time in nature, and, walking in the forest, he noticed that these numbers were literally pursuing him. Everywhere in nature, he again met these numbers. For example, petals and leaves of plants strictly laid in this numerical series.

In Fibonacci numbers, there is an interesting feature: private from dividing the subsequent number of Fibonacci to the previous one, as the numbers themselves grow, strive for 1.618. It is this constant number of division in the Middle Ages that was called the divine proportion, and now it is called a gold cross section or a golden proportion.

In the algebree, this number is indicated by the Gpeech letter FI (f)

So, φ \u003d 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

How many times we did not divide one thing to another, the number neighbors with him, we always get 1, 618. And if we do the other hand, that is, we divide a smaller number to more, then we get 0, 618, this is the number inverse to 1, 618, Also called gold proportion.

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

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

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

Consider the examples where Fibonacci numbers live and inanimate nature are found.

Fibonacci numbers in wildlife.

If you look at the plants and trees around us, it can be seen how many leaves on each of them. From afar, it seems that branches and leaves on plants are randomly located in any order. However, in all plants it is miraculously, mathematically accurately planned which twig from where it will grow like branches and leaves will be located near the stem or trunk. From the first day of appearance, the plant should be exactly in its development by these laws, that is, no sheet, no flower appears by chance. Even before the appearance, the plant is already accogrammed. 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 leaves will be located. The joint work of nerds and mathematicians shed light on these amazing phenomena of nature. It turned out that in the location of the leaves on the branch (phyotaxis), among the revolutions on the stem, among the leaves in the cycle, a number of fibonacci is manifested in the cycle, and therefore, the law of the golden section is manifested.

If you specify the goal of finding numerical patterns in wildlife, then notice that these numbers are often found in various spiral forms that the world of plants are so rich. For example, the cuttings of the leaves are adjacent to the stalk of the spiral, which passes betweentwo adjacent leaves: Full turnover - Oshnik, - Oak, - Poplar and pear, - Willow.

Sunflower seeds, echinacea of \u200b\u200bpurple and many other plants are located spirals, and the number of spirals of each direction - the number of fibonacci.

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

A clear, symmetric form of colors is also subordinated to a strict law..

Many colors have the number of petals - exactly the numbers from the Fibonacci range. For example:

iris, 3let. Buttercup, 5 lep. Zlatocevet, 8 lep. delphinium,

13 lep.

chicory, 21let. Astra, 34 Lep. Daisy, 55p.

A number of Fibonacci characterizes the structural organization of many living systems.

We have already said that the relations of neighboring numbers in a row of Fibonacci have the number φ \u003d 1.618. It turns out that both the person himself is just a storehouse of Fi.

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

M / M \u003d 1,618

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

If you take the center of the Pupa's human body, and the distance between the feet of a person and the PUP point per unit of measurement, then the human height is equivalent to the number 1.618.

Human hand

It is enough just to bring your palm now to yourself and carefully look at the index finger, and you immediately find in it the formula of the Golden Section. Each finger of our hand consists of three phalanges.
The sum of the two first phalanches of the finger in the ratio from the entire finger length and gives the number of golden section (except for thumb).

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

A person has 2 hands, fingers on each hand consist of 3 phalanges (except for thumb). On each hand there are 5 fingers, that is, only 10, but with the exception of two two-phase thumbs only 8 fingers are created according to the principle of the golden section. While all these numbers 2, 3, 5 and 8 are the numbers of Fibonacci sequence.

Golden proportion in the structure of light man

American physicist B.D.Uest and Dr. A.L. Goldberger during physico-anatomical studies found that a golden cross section also exists in the structure of human lungs.

The peculiarity of the bronchi, components of human lungs, is enclosed in their asymmetry. Bronchi consist of two main respiratory tract, one of which (left) is longer, and the other (right) is shorter.

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

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches, based on the ratio of the golden section. They use measurements from the human body created also on the principle of the golden section. Leonardo da Vinci and Le Corbusier before creating their masterpieces took the parameters of the human body created under the law of the golden proportion.
There is another, more proserate application of human body proportions. For example, using these relations, criminal analysts and archaeologists on fragments of parts of the human body restore the appearance of the whole.

Gold proportions in the structure of the DNA molecule.

All information about the physiological features of living beings, whether it is a plant, an animal or person, is stored in a DNA microscopic molecule, the structure of which also contains the law of the golden proportion. The DNA molecule consists of two vertically twisted spirals. The length of each of these spirals is 34 angstroms, width 21 angstrom. (1 angstrom - one velomillion share of centimeter).

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

Not only spinning, but all floating, crawling, flying and jumping did not avoid fate to obey the number of fi. The heart muscle is reduced to 0, 618 of its volume. The structure of the snail shell corresponds to the proportions of Fibonacci. And such examples can be found plenty - there would be a desire to explore natural objects and processes. The world is so permeated by Fibonacci numbers that sometimes it seems: only the universe and can be explained.

Spiral Fibonacci.

In mathematics there is no other form that would have 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, repeat what is a golden cross section.

The golden section is such a proportional division of the segment on unequal parts, in which the entire segment belongs to the most part, as most of them belongs to the smaller, or, in other words, the smaller segment belongs to more as greater than everything.

That is (A + B) / A \u003d A / B

The rectangle with just such an attitude of the sides began to be called a golden rectangle. Its long sides correlate with short parties in the ratio of 1,168: 1.
Golden rectangle has many unusual properties. Cutting off from the golden rectangle square, the side of which is equal to the smaller side of the rectangle,

we again get a golden rectangle of smaller sizes.

This process can be continued to infinity. Continuing to cut the squares, we will receive all the smaller and smaller gold rectangles. Moreover, they will be located along the logarithmic spiral, which is important in mathematical models of natural objects.

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

We are surprising and delighting the spiral structure of seashells.

Most snails that have sinks, shell grows in the form of a spiral. However, there is no doubt that these unreasonable creatures have no idea not only about the helix, but do not even possess the simplest mathematical knowledge in order to create a spiral sink themselves.
But when these unreasonable creatures were able to determine and elect an ideal form of growth and existence in the form of a spiral shell? Could these living beings, whom the world scientists call primitive forms of life, calculate that the spiral form of the shell is ideal for their existence?

Trying to explain the origin of such a primitive form of life with a random coach of some natural circumstances at least absurd. It is quite clear that this project is aware of creation.

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

Also, in the inner ear of a person there is a cochlea ("snail") authority, which performs the function of transmitting sound vibration. This boniform structure is filled with liquid and created in the form of a snail having a gold proportion.

Spirals are on our palms and fingers:

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

In the form of a spiral, horns and animal tails are developing, claws of lions and cliques of parrots are logarithmic forms and resemble the shape of the axis, prone to contact the spiral.

Interestingly, a hurricane coil, cyclone clouds twist and it is clearly seen from the space:

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

Such a "household" and "prose" spiral will also learn everything.

After all, water runs out of the bathroom in the spiral:

Yes, and we live with you in the spiral, because the galaxy is a spiral corresponding to the formula of the Golden section!

So, we found out that if you take a golden rectangle and split it into smaller rectangles In the exact sequence of Fibonacci, and then each of them is divided into such proportions yet, it will turn out a system that is called Fibonacci Spiral.

We found this spiral in the most unexpected subjects and phenomena. Now it is clear why the spiral is called the "curve of life."
The spiral became a symbol of evolution, because it develops everything precisely.

Fibonacci numbers in human inventions.

Sewing naturally the law expressed by the sequence of Fibonacci numbers, scientists and people of art try to imitate him to embody this law in their creations.

The proportion of fi allows you to create masterpieces of painting, to fit the architectural structures in the space.

Not only science figures, but also architects, designers and artists are amazed by this flawless spiral at the rocushal Nautilus,

having a smallest space and ensuring the smallest heat loss. American and Thai architects inspired by an example of "Nautilus with cameras" in the issue of placing a maximum in a minimum of space, are engaged in the development of relevant projects.

From time immemorial, the proportion of the golden section is considered the highest proportion of perfection, harmony and even divinity. Golden attitude can be detected in sculptures, and even in music. An example is musical works of Mozart. Even stock courses and alphabet of Hebrew contain a gold relationship.

But we want to stay on a unique example of creating an effective solar installation. An American schoolboy from New York Aidan Duyer has given together his knowledge of the trees and found that the efficiency of solar power plants can be enhanced if you attract mathematics. Being on a winter walk, Duyer thought, why the trees are such a "drawing" of branches and leaves. He knew that the branches on the trees are located according to Fibonacci sequence, and the leaves are carried out photosynthesis.

At some point, the wonderful boy decided to check whether the branch does not help the branches to collect more sunlight. Eydan built an experienced installation in his backyard with small solar panels instead of leaves and checked it in action. It turned out that in comparison with the usual flat solar panel, his "Tree" collects by 20% more energy and more effectively works for 2.5 hours.

Model of a dwayer solar tree and graphics built by a schoolboy.

"And such an installation takes less space than a flat panel, collects 50% more than the sun in winter even where it does not look at south, and the snow in that quantity it does not accumulate. In addition, the design in the form of a tree is much more suitable For urban landscape, "the young inventor notes.

Eidana recognized one of the best young natural scientists. Competition "2011 Young Naturalist" conducted a New York Natural Science Museum. Eidan filed a preliminary application for a patent of his invention.

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

Yu. Matyatsevich using Fibonacci numbers solves the 10th problem of Hilbert.

Elegant methods of solving a number of cybernetic tasks (search theory, games, programming) using Fibonacci and Golden Section are arising.

Even Mathematical Fibonachchi-Association is created in the USA, which since 1963 produces a special magazine.

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

Watching the phenomena occurring in nature, scientists made the striking conclusions that the entire sequence of events occurring in life, revolution, crash, bankruptcy, periods of prosperity, laws and waves of development in stock and foreign exchange markets, family life cycles, and so on are organized on the timeline in the form of cycles, waves. These cycles and waves are also distributed in accordance with the numeric number of Fibonacci!

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

4. Our research.

We continued our observations, and studied the structure

Pine cones

yarrow

moser

man

And they were convinced that in these, such different objects at first glance, it is invisibly present those the most numbers of Fibonacci sequences.

So, step 1.

Take a pine cone:

Consider it closer:

We notice two series of fibonacci spirals: one - clockwise, the other is against, their number8 and 13.

Step 2.

Take the yarrow:

Carefully consider the structure of stems and colors:

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

Step 3.

And do Fibonacci numbers in the morphology of various organisms manifest? Consider the well-known mosquito:

We see: 3. pairs of legs, head5 Masteries - antennas, the abdomen is divided into8 segments.

Output:

In our studies, we saw that in the plants around us, living organisms, and even in the structure of a person, there are numbers from Fibonacci's sequence, which reflects the harmony of their structure.

Pine bump, yarrow, mosquito, people are arranged with mathematical accuracy.

We were looking for an answer to the question: how does Fibonacci a number of fibonacci be reality? 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? Spiral twists or spinned?

How amazing a person knows this world !!!

Finding an answer to one question receives the following. Glinds it, gets two new ones. Smashed with them, three more will appear. Having decided and of them, will acquire five unresolved. Then eight, then thirteen, 21, 34, 55 ...

Recognize?

Conclusion.

Creator himself in all objects

Laid a unique code,

And one who friends with mathematics

He knows and understand!

We studied and analyzed the manifestation of the numbers of the Fibonacci sequence in the surrounding reality. We also learned that the patterns of this numerical series, including the patterns of "golden" symmetry, are manifested in the energy transitions of elementary particles, in planetary and space systems, in the gene structures of living organisms.

We found an amazing mathematical connection between the number of spirals in plants, the number of branches in any horizontal plane and numbers in the Fibonacci sequence. We saw the morphology of various organisms also obeys this mysterious law. We also saw strict mathematics in the structure of a person. The human DNA molecule, in which the entire program for the development of the human being, the respiratory system, the structure of the ear is encrypted, is all obeys certain numerical relations.

We learned that pine cones, snail shells, ocean waves, animal horns, cyclone clouds and galaxies - they all form logarithmic spirals. Even the human finger, which is composed of three phalanges in relation to each other in the golden proportion, takes a spiral shape when compressed.

The eternity of time and the light years of the cosmos share a pine conine and spiral galaxy, but the structure remains the same: the coefficient1,618 ! Perhaps this is a paramount law, managing 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 thoughtful and miscalculated by our most important designer - Nature!

We were convinced that nature has its own laws expressed withmathematics. And mathematics is a very important tool

for the knowledge of the secrets of nature.

List of Internet Literature and Websites:

1. Vorobyev N. N. Fibonacci numbers. - M., Science, 1984.
2. Gick 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

A life. - 1982. - № 10.
5. Malay Garmonia - the identity of paradoxes // MN. - 1982. - № 19.
6. Sokolov A. The secrets of the golden section // The technique of youth. - 1978.- № 5.
7. Stakhov A. P. Codes of the golden proportion. - M., 1984.
8. Urmansev Yu. A. Symmetry of nature and nature of symmetry. - M., 1974.
9. Urmansev Yu. A. Golden section // Nature. - 1968. - № 11.

10. Shevelev I.Sh., Marutaev MA, Shmelev I.P. Golden section / three

View of the nature of harmony. - m., 1990.

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

The world around the world, starting with the smallest invisible particles, and ending with the distant galaxies of the endless cosmos, pays a lot of unsolved secrets. However, some of them have been raised by the veil of mystery due to the inquisitive minds of a number of scientists.

One such example is Golden section and Fibonacci numbers The basis of its foundation. This pattern has been mapping in mathematical form and is often found in the human environment, once again excluding the likelihood that it arose as a result of the case.

Fibonacci numbers and their sequence

The sequence of Fibonacci numbers Call a number of numbers, each 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 …

A feature of this sequence is numerical values \u200b\u200bthat are obtained due to dividing the numbers of this series on each other.

The number of Fibonacci numbers has its own interesting patterns:

• In a number of Fibonacci numbers, each number divided into the following will show the value seeking to 0,618 . The farther the numbers from the beginning of the row, the more accurate the ratio. For example, figures taken at the beginning of the row 5 and 8 will show 0,625 (5/8=0,625 ). If you take numbers 144 and 233 then they will show the ratio 0.618 .
• In turn, if in a number of Fibonacci numbers divided the number to the previous one, the result of the division will seek 1,618 . For example, the same figures are specified above: 8/5=1,6 and 233/144=1,618 .
• The number divided into the following by one through one will show the value approaching 0,382 . And the farther from the beginning of the row, the numbers are taken, the more accurate the value of the relation: 5/13=0,385 and 144/377=0,382 . Decaling numbers in the reverse order will give results 2,618 : 13/5=2,6 and 377/144=2,618 .

Using the above calculation methods and increasing the gaps between the numbers can be displayed next series of values: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236, which is widely used in Fibonacci tools in the Forex market.

Golden section or Divine proportion

Very clearly represents the "golden section" and the number of fibonacci analogy with a segment. If the segment AV is divided by a point with in such a ratio to comply with the condition:

AC / Sun \u003d aircraft / av, then it will be a "golden section"

Read also the following articles:

Surprisingly, it is precisely this ratio traced in a number of Fibonacci numbers. Taking a few numbers from the row, you can calculate that it is so. For example, such a sequence of Fibonacci numbers ... 55, 89, 144 ... Let the number 144 be a whole segment of AB, which was mentioned above. Since 144 is the sum of the two previous numbers, then 55 + 89 \u003d ac + sun \u003d 144.

Segments Decision will show the following results:

AC / Sun \u003d 55/89 \u003d 0,618

Sun / AB \u003d 89/144 \u003d 0,618

If you take a segment of AB for an integer, or per unit, then AC \u003d 55 will be 0.382 from this whole, and the aircraft \u003d 89 will be equal to 0.618.

Where are Fibonacci numbers

The regular sequence of Fibonacci numbers knew the Greeks and the Egyptians long before Leonardo Fibonacci. This name has acquired this name after the famous mathematician has ensured the widespread spread of this mathematical phenomenon in scholars.

It is important to note that the gold numbers of Fibonacci are not just science, but a mathematical mapping of the surrounding world. Many natural phenomena, plant and animal world representatives have a "golden section" in their proportions. It is also spiral shell curls, and the location of the sunflower seeds, cacti, pineapples.

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

The length of the tail of the lizard to her torso has a ratio of 62 to 38. The process of chicory, before you release a piece of leaf, makes emission. After the first sheet is released, the second emission is released before the release of the second sheet, equal to 0.62 from the conditionally accepted unit of force of the first emission. The third emection is 0.38, and the fourth - 0.24.

For a trader, the fact that the price of price in the Forex market is often subject to the regularities of the Fibonacci gold numbers. Based on this sequence, a number of tools have been created that a trader can use in its arsenal

Frequently used by traders tool "" can with high accuracy to show the goals of price movement, as well as levels of its correction.