Golden cross section of snail. Fibonacci numbers: Nescable mathematical facts

Fibonacci numbers - elements of numerical sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, in which each subsequent number is equal to the sum of the two previous numbers. The name by the name of the medieval mathematician Leonardo Pisansky (or Fibonacci), who lived and worked as a merchant and mathematician in the Italian city of Pisa. He is one of the most famous European scientists of his time. Among his greatest achievements is the introduction of Arabic figures, replacing Roman. Fn \u003d Fn - 1 + Fn-2

The mathematical series is asymptotically (that is, approaching everything slower and slower) tends to a permanent relation. However, this ratio is irrational; It has an infinite, unpredictable sequence of decimal values \u200b\u200barising from it. It can never be expressed for sure. If each number that is part of the row is divided into the preceding value (for example, 13- ^ 8 or 21 -), the result of the action will be expressed in relation to which fluctuates around the irrational number 1,61803398875, slightly more or a little less than the neighboring relationships of the row. The attitude is never to infinity, will not be accurate to the last digit (even when using the most powerful computers created in our time). For the sake of brevity, we will use the number of 1.618 as the Fibonacci relationship and ask readers not to forget about this error.

Fibonacci numbers are essential and during the analysis of the Euclidean algorithm to determine the largest total divider of two numbers. Fibonacci numbers occur in the formula on the diagonal triangle of Pascal (binomial coefficients).

Fibonacci numbers were associated with the "golden cross section".

Ancient Egypt and Babylon, in India and China, knew about the golden section, in India and China. What is the "golden section"? The answer is unknown so far. Fibonacci numbers are really relevant for practice theory in our time. The rise of significance occurred in the 20th century and continues until now. The use of Fibonacci numbers in economics and computer science and attracted the masses of people to their study.

Methods of my research was to study specialized literature and summarizing the information received, as well as conducting its own research and identification of the properties of the numbers and the scope of their use.

In the course scientific research Defined the concept of Fibonacci numbers, their properties. Also, I found out interesting patterns in wildlife, directly in the structure of sunflower seeds.

On the sunflower seeds are built into the spiral, and the number of spirals going to the other side are different - they are consecutive numbers of Fibonacci.

On this sunflower 34 and 55.

The same is also observed at the fruits of pineapple, where the spirals are 8 and 14. The leaves of corn are connected with the unique property of Fibonacci numbers.

The fraction of the form A / B corresponding to the screw-like layout of the legs of the plant's steel, are often relations of consecutive Fibonacci numbers. For nuts, this ratio is 2/3, for oak-3/5, for a poplar 5/8, for willow 8/13, etc., etc.

Considering the location of the leaves on the plants stem, it can be noted that there is a third in the place of the golden section between each pair of leaves (A and C)

Another interesting feature of Fibonacci is that the work and private two of any different numbers of Fibonacci, other than the unit, are never a number of fibonacci.

As a result of the study, I came to the following conclusions: Fibonacci numbers - unique arithmetic progression, appeared in the 13th century of our era. This progression does not lose its relevance, which was confirmed during my research. The number of fibonacci is not in programming and economic forecasts, in painting, architecture and music. Pictures of such famous artists like Leonardo da Vinci, Michelangelo, Rafael and Botticelli hide in themselves the magic of the golden cross section. Even I. I. Shishkin used a golden cross section in his picture "Pine Grove".

It is difficult to believe, but the golden cross section is also found in the musical works of such great composers as Mozart, Beethoven, Chopin, etc.

Fibonacci numbers are found in architecture. For example, a gold cross section was used in the construction of Parfenon and the Cathedral of the Paris Mother

I found that Fibonacci numbers are used in our territories. For example, platbands of houses, frontones.

Have you ever heard that mathematics call the "queen of all sciences"? Do you agree with this statement? While mathematics remains for you a set of boring tasks in the textbook, you can hardly feel beauty, versatility and even humor of this science.

But there is such topics in mathematics that help to make curious observations of things ordinary for us and phenomena. And even try to penetrate the curtain of the mystery of the creation of our universe. There are curious patterns in the world that can be described using mathematics.

We present you the numbers of Fibonacci

Fibonacci numbers Called the elements of the numerical sequence. In it, each next number in a row is obtained by the summation of the two previous numbers.

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

You can write it like this:

F 0 \u003d 0, f 1 \u003d 1, f n \u003d f n-1 + f n-2, n ≥ 2

You can begin a number of Fibonacci numbers and with negative values. n.. At the same time, the sequence in this case is bilateral (i.e. covers negative and positive numbers) And strives for 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 \u003d f n + 1 - f n + 2 Or otherwise you can: F -n \u003d (-1) n + 1 Fn.

What we now know under the name "Number of Fibonacci" was known to the Old Indian mathematicians long before they began to use in Europe. And with this name is generally one solid historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci - this name began to apply to Leonardo to Pisansky only after a few centuries after his death. But let's go about everything in order.

Leonardo Pisa, he fibonacci

The son of a merchant who became a mathematician, and later received the recognition of descendants as the first major mathematics of Europe of the middle ages. Not least due to the numbers of Fibonacci (which, then, we will not remember, have not yet been called). Which in the early XIII century he described in his work "Liber Abaci" ("Abaca Book", 1202 years old).

Traveling along with the Father to the East, Leonardo studied mathematics from Arab teachers (and they were in this time in this matter, and in many other sciences, one of the best specialists). Proceedings of antiquity mathematicians and Ancient India He read in Arab translations.

As it should be comprehended, all read and connecting his own intentional mind, Fibonacci wrote several scientific treatises in mathematics, including the above-mentioned "Book of Abaka". Besides her created:

"Practica Geometria" ("Geometry Practice", 1220);
"FLOS" ("Flower", 1225 - a study on cubic equations);
"Liber Quadratorum" ("Book of Squares", 1225 year - Objectives of indefinite square equations).

There was a big lover of mathematical tournaments, so in his treatises a lot of attention paid to the analysis of various mathematical problems.

Leonardo's life remains extremely little biographical information. As for Fibonacci's name, under which he entered the history of mathematics, it consolidated only in the XIX century.

Fibonacci and his tasks

After Fibonacci, a large number of tasks remained, which were very popular among mathematicians and in subsequent centuries. We will consider the task of rabbits, in the solution of which the numbers of Fibonacci are used.

Rabbits are not only valuable fur

Fibonacci asked such conditions: There is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (since the second month) produce offspring - always one new pair of rabbits. Also, as you can guess, male and female.

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

It is necessary to calculate how many rabbits we get in a year.

At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
For the second month - we already have 2 pairs of rabbits (a couple - parents + 1 pair are their offspring).
The third month: the first couple gives rise to a new pair, the second pair falls. Total - 3 pairs of rabbits.
Fourth month: The first pair gives rise to a new pair, the second pair of time does not lose and also gives rise to a new pair, the third pair is only pairing. Total - 5 pairs of rabbits.

Number of rabbits B. n.-Mime month \u003d number of rabbit pairs from the previous month + the number of newborn pairs (they are as much as the rabbit pairs were 2 months before the present moment). And all this is described by the formula that we have already led to above: F n \u003d f n-1 + f n-2.

Thus, we get a recurrent (explanation of recursions - Below) numeric sequence. In which each next number is equal to the sum of the previous two:

1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
13 + 8 = 21
21 + 13 = 34
34 + 21 = 55
55 + 34 = 89
89 + 55 = 144
144 + 89 = 233
233+ 144 = 377 <…>

Continue sequence Long: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we asked a specific period - a year, we are interested in the result obtained on the 12th "go". Those. 13th sequence member: 377.

The answer in the task: 377 rabbits will be obtained by complying with all stated conditions.

One of the properties of the sequence of Fibonacci numbers is very curious. If you take two consecutive pairs from the row and divide the larger number to the smaller, the result will gradually approach golden cross section (Read about it in more detail you can further in the article).

Talking to the language of mathematics "Limit of relations a n + 1to A N.equal to the golden section ".

More tasks on the theory of numbers

Find a number that can be divided into 7. In addition, if it is divided into 2, 3, 4, 5, 6, a unit will be in the residue.
Find a square number. It is known about him that if you add 5 or take it out 5, the square number will again.

Replies to these tasks We suggest you search for yourself. You can leave our options in the comments to this article. And then we will tell you whether your calculations were true.

Explanation of recursion

Recursion - Definition, description, image of an object or process in which this object itself is contained or process. Those., In fact, the object or process is part of itself.

Recursion is widely used in mathematics and computer science, and even in art and mass culture.

Fibonacci numbers are determined using a recurrent ratio. For numbers n\u003e 2 N-e number equal (n - 1) + (n - 2).

Explanation of the Golden section

Golden cross section - division of a whole (for example, a segment) to such parts that relate by next principle: Most relates to a smaller way as the entire value (for example, the sum of two segments) to the most part.

The first mention of the golden section can be found in Euclidea in his starting treatise (approximately 300 years BC). In the context of building a correct rectangle.

Our usual term in 1835 introduced into circulation of the German mathematician Martin Ohm.

If the golden section is described approximately, it is a proportional division into two unequal parts: approximately 62% and 38%. In numerical expression, the gold cross section is a number 1,6180339887 .

Golden cross section finds practical use in fine art (Pictures of Leonardo da Vinci and other painters of the Renaissance), architecture, cinema ("Potemkin's armadiole" S. Ezenstein) and other areas. For a long time it was believed that the golden cross section is the most aesthetic proportion. This opinion is popular today. Although, according to the results of research, visually most people do not perceive such a proportion to the most successful option and are considered too extended (disproportionate).

Length Cut from = 1, but = 0,618, b. = 0,382.
Attitude from to but = 1, 618.
Attitude fromto b. = 2,618

And now back to the numbers of Fibonacci. Take the two member next to each other from its sequence. We divide the larger number to the smaller and obtain approximately 1.618. And now we use the same number and the next member of the row (i.e. even more) - their ratio is early 0.618.

Here is an example: 144, 233, 377.

233/144 \u003d 1.618 and 233/377 \u003d 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 happen. Almost. The golden section rule is almost no compliance with the sequence. But as it moves along a row and increasing the numbers is perfect.

And in order to calculate the entire number of Fibonacci numbers, it is enough to know three members of the sequence, walking on each other. You can make sure that yourself!

Golden Rectangle and Spiral Fibonacci

Another curious parallel between the numbers of fibonacci and the golden section allows you to carry out the so-called "golden rectangle": its parties relate in the proportion of 1.618 K 1. But we already know that in number 1,618, right?

For example, take two consecutive member of the Fibonacci series - 8 and 13 - and we construct a rectangle with the following parameters: width \u003d 8, length \u003d 13.

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

So, as it is done in this picture (for convenience, the figures are signed by Latin letters).

By the way, it is possible to build rectangles in reverse order. Those. Start a construction from squares from the side 1. To which, guided by the above-mentioned principle, drawing figures with the parties, equal numbers Fibonacci. Theoretically, it is possible to continue so if you can endlessly - after all, the Fibonacci row is formally infinite.

If you combine the smooth line of the corners of the rectangles obtained in the figure, we get a logarithmic spiral. Rather, its private event is Fibonacci Spiral. It is characterized, in particular, in that it does not have borders and does not change the forms.

Such a spiral is often found in nature. Mollusc shells are one of the most vivid examples. Moreover, some galaxies that can be seen from the ground have a spiral form. If you pay attention to weather forecasts on TV, it could notice that the cyclones have a similar spiral form when shooting them from satellites.

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

Such amazing "coincidences" cannot not disturb the minds and do not generate conversations about a certain single algorithm, which is subject to all phenomena in the life of the Universe. Now you understand why this article is called this? And doors in which amazing worlds Is it capable of opening mathematics for you?

Fibonacci numbers in wildlife

The relationship between Fibonacci numbers and the golden section suggests the thought of curious laws. So curious that there is a temptation to try to find such numbers of fibonacci sequences in nature and even during historical events. And nature really gives a reason for this kind of assumptions. But is everything in our life can be explained and described with mathematics?

Examples of wildlife, which can be described using Fibonacci sequence:

the order of the leaves (and branches) in plants - the distances between them are relations with Fibonacci numbers (philloaxis);

the location of the seeds of the sunflower (seeds are located two rows of spirals twisted in different directions: one row clockwise, the other - against);

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

Combinatorics tasks

Fibonacci numbers are widely used when solving problems on combinatorics.

Combinatorics - This is a section of mathematics, which is engaged in the selection of a certain specified number of elements from the designated set, listing, etc.

Let's consider examples of tasks on the combinatorics designed to level the high school (source - http://www.prblems.ru/).

Task number 1:

Lesha rises the stairs out of 10 steps. At one time he jumps up either one step or two steps. How many ways is Lesha can climb the stairs?

The number of ways to which Lesha can climb the stairs from n. Steps, denotation a n.Hence it follows that a 1. = 1, a 2. \u003d 2 (after all, Lesha jumps either one or two steps).

Stipulated also that Lesha jumps on the stairs from n\u003e 2 Steps. Suppose the first time he jumped into two steps. So, by the condition of the task, he needs to jump on n - 2. Stairs. Then the number of ways to finish the rise is described as a N-2. And if we assume that for the first time, Lesha jumped only on one step, then the number of ways to finish the rise we describe how a N-1.

From here we get such equality: a n \u003d a n-1 + a n-2 (Looks familiar, is it?).

Once we know a 1.and A 2.and remember that the steps under the condition of task 10, calculated 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 number 2:

It is required to find the amount of words in 10 letters long, which consist only of letters "A" and "B" and should not contain two letters "B" in a row.

Denote by a N. The number of words in length in n.letters that consist only of letters "A" and "B" and do not contain two letters "B" in a row. It means a 1.= 2, a 2.= 3.

In sequence a 1., a 2., <…>, a N.we express each next one member through the previous ones. Consequently, the number of words in length in n.letters that also do not contain double letters "b" and begin with the letter "A", this a N-1. And if the word is long in n.letters begins with the letter "b", it is logical that the next letter in such a word is "a" (after all, two "b" cannot be under the condition of the task). Consequently, the number of words in length in n.letters in this case denote as a N-2. And in the first, and in the second case, it can follow any word (long in n - 1.and N - 2. Letters, respectively) without doubled "b".

We were able to justify why a n \u003d a n-1 + a n-2.

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.\u003d 144. And we get familiar to us Fibonacci sequence.

Answer: 144.

Task number 3:

Imagine that there is a tape, broken into the cells. It goes to the right and lasts indefinitely for a long time. On the first tape cell, put a grasshopper. For whatever the tape cells, it can only move to the right: or one cell, or two. How many methods that the grasshopper can surparate from the beginning of the tape to n.Cells?

Denote the number of ways to move the grasshopper on the ribbon to n.Cell as a N.. In this case a 1. = a 2. \u003d 1. Also in n + 1.cage grasshopper can get either from n.Cell, or jumping over it. From here a n + 1 = a N - 1 + a N.. From a N. = F n - 1.

Answer: F n - 1.

You can and make up such tasks yourself and try to solve them in mathematics lessons with classmates.

Fibonacci numbers in mass culture

Of course, that unusual phenomenonAs numbers Fibonacci, can not not attract attention. There is still in this strictly verified pattern of something attractive and even mysterious. It is not surprising that the Fibonacci sequence is somehow "lit up" in many works of modern mass culture of various genres.

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

Fibonacci numbers are referred to in the bestseller Dan Brown "Da Vinci Code": Fibonacci sequence serves as a code, with which the main characters of the book open the safe.
In the American film 2009, "Mr. Nobody" in one of the episodes of the house's address is part of the Fibonacci sequence - 12358. In addition, in another episode the main character Must call the telephone number, which is essentially the same, but slightly distorted (excessive digit after the figure 5) sequence: 123-581-1321.
In the 2012 TV series "Communication", the main character, a boy suffering from autism, is able to distinguish between the laws in the events occurring in the world. Including through Fibonacci numbers. And manage these events also through numbers.
Java-Games for Mobile Phones Doom RPG Placed at one level secret door. The code opening is the Fibonacci sequence.
In 2012, the Russian rock band "Spleen" released a conceptual album "Illusion". The eighth track is called Fibonacci. In verses of the leader of Alexander Vasilyeva, the sequence of Fibonacci numbers beat. For each of the nine consecutive members accounts for the corresponding number of rows (0, 1, 1, 2, 3, 5, 8, 13, 21):

0 Touched on the path

1 Closed one joint

1 Fucked one sleeve

2 All, get stuff

All, get stuff

3 Asking for boiling water

Train goes to the river

Train goes in Taiga<…>.

limerick (short poem of a certain form - usually it is five lines, with a specific rhyme scheme, comic in content in which the first and last line are repeated or partially duplicated each other) James Lyndon also uses a reference to the Fibonacci sequence as a humorous motive:

Dense Food Fibonacci

Only for the benefit of them was not different.

Weighed wives, according to Molve,

Each - as the previous two.

Let's sum up

We hope that you can tell you today a lot of interesting and useful. You, for example, now you can search for a Spiral Fibonacci in the nature around you. Suddenly it will be possible to solve the "secret of life, the universe and in general."

Use the formula for Fibonacci numbers when solving tasks by combinatorics. You can rely on the examples described in this article.

the site, with full or partial copying of the material reference to the original source is required.

Recently, working in individual and group processes with people, I returned to thoughts on the combination of all processes (karmic, mental, physiological, spiritual, transformational, etc.) to one.

Friends behind the veil more and more widely revealed the image of a multidimensional person and the relationship of everything in everything.

The inner motivation pushed me back to the old studies with numbers and once again view the book Drunvalo Melchizedek " Ancient mystery Flower of life. "

At this time, the film "Code Da Vinci" 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 in this film for me.

Intuition suggested me that it is worth paying attention to the numerical sequence of Fibonacci and a golden cross section. If you look into the Internet in order to find anything about Fibonacci, avalanche of information will be collapsed. You will learn that they knew about this sequence 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 came to the belief that the key events in the life of a person, the state, civilization are also subject to the law of the Golden Section.

It seems that a person is given a fundamental tip.

Then the idea arises that a person can consciously apply the principle of the golden section to restore the health and correction of fate, i.e. Organizing the processes in its own universe, expanding consciousness, return to welfare.

Together recall Fibonacci sequence:

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 the addition of the previous two:

1 + 1 \u003d 2, 1 + 2 \u003d 3, 2 + 3 \u003d 5, etc.

Now I offer every number of rows to lead 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)…

That's what we did:

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…

the sequence of 24 numbers that is repeated 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…

Do not seem to you strange or natural that

in days - 24 hours,
space houses - 24,
DNA threads - 24,
24 elder with sirius bogo stars,
Repeating sequence in a number of fibonacci - 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,

we will see that the 1st and 13th number of sequences, the 2nd and 14th, 3rd and 15th, 4th and 16th ... 12th and 24th in the amount give 9 .

3 3 6 9 6 6 3 9

When testing these numeric rows, we turned out:

Children's principle;
Father's principle;
Maternal 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

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

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 a row of Fibonacci

One of my friend expressed his intention to individually work with him on the topic of developing its capabilities and abilities.

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

We began to climb a friend inside the Divine Monad and, coming out of it through the causal body, were in another reality at the level of the space house.

Who studied the works of Mark and Elizabeth Claire of Profrets, know the doctrine of space clock, which Mother Maria handed over to them.

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

The old man, who met us at this level, said that the divine clocks and the 12th arrows were personifolding 12 (24) manifestations of divine aspects ... (possibly creators).

As for the cosmic hours, they were located under Divine on the principle of energy eight.

- In what mode is the divine clock in relation to you?

- The arrows at the clock stand, no movement.Now the thoughts come to me that I refused the divine consciousness years ago and went to another way, by magician. All my magical artifacts and amulets that I have accumulated in me for many incarnations, at this level look like children's rattles. On a thin plan, they are an image of magical energy clothing.

- Completed.Nevertheless, I bless my magical experience.The accommodation of this experience sincerely prompted me to return to the original source, to integrity.I am proposed to take off my magic artifacts and get up in the center of the clock.

- What needs to be done to activate the Divine Watch?

- Appeared again by Sai Baba and suggests expressing the intention about connecting a silver string with a clock. He also says that you have some kind of numerical row. It is the key to activation. Before the inner gaze, the image of a man Leonard da Vinci occurs.

- 12 times.

- I ask to moisten the whole process and send the effect of energy numeric row to activate divine clocks.

I read out loud 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 arrows on the clock went.

The silver string went to the energy that connected all the levels of the lawn monad, as well as the earthly and heavenly energy ...

The most unexpected in this process was that four entities appeared on the clock, which are some united parts with Yura.

During communication, it turned out that the central soul was once occurred, and each part chose its area in the universe for implementation.

It was decided to integrate, which happened in the center of the Divine Watch.

The result of this process was the creation of a general crystal at this level.

After that, I remembered that Sai Baba somehow spoke about some kind of plan, which implies a compound first two entities in one, then four and so on for a binary principle.

Of course, this numeric number is not a panacea. This is just a tool that allows you to quickly make the necessary work with a person, to keep it vertically with different levels Genesis.

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You, of course, are familiar with the idea that mathematics is the most important of all sciences. But many can disagree with it, because Sometimes it seems that mathematics is only tasks, examples, and the like of the boring. However, mathematics can easily show us familiar things with a completely unfamiliar side. Moreover, she can even reveal the secrets of the universe. How? Let's turn to Fibonacci numbers.

What is Fibonacci numbers?

Fibonacci numbers are elements of a numerical sequence, where each subsequent by summing up two previous ones, for example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... As a rule, such a sequence is written: f 0 \u003d 0, f 1 \u003d 1, f n \u003d f n - 1 + f n-2, n ≥ 2.

Fibonacci numbers can begin with negative values \u200b\u200bof "N", but in this case the sequence will be bilateral - it will cover and positive and negative numbers, striving for infinity in two directions. An example of such a sequence can serve: -34, -21, -13, -8, -5, -3, -2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 21, 34, and the formula will be: f n \u003d f n + 1 - f n + 2 or f -n \u003d (-1) n + 1 Fn.

The creator of Fibonacci numbers is one of the first mathematicians of the European Middle Ages named Leonardo Pisa, whom, actually know how Fibonacci is the nickname he received many years after his death.

During the lifetime of Leonardo, Pisansky loved mathematical tournaments, because of which in his works ("Liber Abaci" / "Abaca Book", 1202; "Practica Geometria" / "Practice of Geometry", 1220, "Flos" / "Flower", 1225 - Study on the theme of cubic equations and "Liber Quadratorum" / "Book of Squares", 1225 - Objectives about undefined square equations) Very often disassembled all sorts of mathematical tasks.

ABOUT life path Fibonacci itself is known to be extremely small. But it is reliably aware that his tasks enjoyed great popularity in mathematical circles in the following centuries. One of these we will look at one.

Fibonacci task with rabbits

To fulfill the task, the author was delivered to the author: there are a couple of newborns rabbit (female and male), distinguished by an interesting feature - from the second month of life they produce a new pair of rabbits - also female and male. Rabbits are in a closed space and constantly breed. And no rabbit dies.

A task: Determine the number of rabbits in a year.

Decision:

We have:

One pair of rabbits at the beginning of the first month, which mates at the end of the month
Two pairs of rabbits in the second month (first couple and offspring)
Three pairs of rabbits in the third month (first couple, the offspring of the first couple from last month and new offspring)
Five pairs of rabbits in the fourth month (the first pair, the first and second offspring of the first couple, the third offspring of the first couple and the first offspring of the second pair)

The number of rabbits per month "n" \u003d the number of rabbits of the last month + the number of new rabbit pairs, in other words, the above formula: F n \u003d f n-1 + f n-2. From here it turns out recurrent number sequence (We will follow the recursion further), where every new number corresponds to the sum of the two previous numbers:

1 month: 1 + 1 \u003d 2

2 month: 2 + 1 \u003d 3

3 month: 3 + 2 \u003d 5

4 month: 5 + 3 \u003d 8

5 month: 8 + 5 \u003d 13

6 month: 13 + 8 \u003d 21

7 month: 21 + 13 \u003d 34

8 month: 34 + 21 \u003d 55

9 month: 55 + 34 \u003d 89

10 month: 89 + 55 \u003d 144

11 month: 144 + 89 \u003d 233

12 month: 233+ 144 \u003d 377

And this sequence can continue indefinitely for a long time, but considering that the task is to know the number of rabbits after the expiration of the year, 377 pairs are obtained.

It is also important here to note that one of the properties of Fibonacci numbers is that if you compare two consecutive pairs, and then divided large to smaller, the result will move towards the golden section, which we also say below.

In the meantime, we offer you two more tasks in Fibonacci numbers:

Determine the square number that it is only known that if you take 5 from it or add 5 to it, the square number will come out again.
Determine the number divided by 7, but under the condition that it will take it to 2, 3, 4, 5 or 6 in the residue.

Such tasks will not only become an excellent way to develop mind, but also entertaining pastime. About how these tasks are solved, you can also find out the search for information on the Internet. We will not sharpen attention on them, but will continue our story.

What is Recursion and Golden Section?

Recursion

Recursion is a description, definition or image of a object or process in which there is a given object or process. In other words, the object or process can be called part of itself.

Recursion is widely used not only in mathematical science, but also in computer science, mass culture and art. Applicable to Fibonacci numbers, it can be said that if the number is "n\u003e 2", then "n" \u003d (n - 1) + (N-2).

Golden cross section

The gold cross section is a division of the whole part, correlated according to the principle: more relates to a smaller similar to how total amount refers to most.

For the first time, the Golden section mentions Euclide (Treatise "Beginning" approx. 300 years BC), speaking and build a right rectangle. However, the more familiar concept was introduced by the German mathematician Martin Ohm.

Approximately the gold cross section can be represented as proportional division into two different parts, for example, by 38% and 68%. The numerical expression of the golden section is approximately 1,6180339887.

In practice, the golden cross section is used in architecture, visual art (look at work), cinema and other directions. For a long time, however, as now, the golden cross section was considered aesthetic proportion, although most people are perceived by disproportionate - elongated.

You can try to evaluate the golden section yourself, guided by the following proportions:

Cut length a \u003d 0,618
Cut length B \u003d 0.382
C \u003d 1 length length
The ratio C and a \u003d 1,618
The ratio C and B \u003d 2,618

Now we will apply a golden section to Fibonacci: We take two neighboring member of its sequence and divide more to the smaller. We get about 1.618. If we take the same number and share it for the next larger behind it, we will get about 0.618. Try: "Play" with numbers 21 and 34 or some others. If you spend this experience with the first numbers of Fibonacci sequence, there will be no such result, because The golden section "does not work" at the beginning of the sequence. By the way, to determine all the numbers of Fibonacci, you need to know only the first three consecutive numbers.

And in conclusion, some more food for the mind.

Golden Rectangle and Spiral Fibonacci

"Golden Rectangle" is another relationship between the golden section and the numbers of Fibonacci, because The ratio of its parties is 1.618 K 1 (remember the number 1,618!).

Here is an example: we take two numbers from Fibonacci sequence, for example 8 and 13, and blacks are a rectangle with a width of 8 cm and a long 13 cm. Next, we divide the main rectangle into small, but their length and width must correspond to the Fibonacci numbers - the length of one face of a large rectangle must Return two lengths of the face of smaller.

After that, we combine the smooth line of the angles of all rectangles we have and we get a special case of a logarithmic spiral - Spiral Fibonacci. Its main properties are the lack of boundaries and changes in forms. Such a spiral can often be found in nature: the brighter examples are molluscs, cyclones on images from a satellite and even a number of galaxies. But it is more interesting that the DNA of living organisms is subject to the same rule, because you remember that it has a spiral shape?

These and many other "random" coincidences even today excite the consciousness of scientists and suggest that everything in the universe is subordinated to a single algorithm, and it is the mathematical one. And this science carries a huge amount of completely mischievous secrets and mysteries.

Fibonacci numbers - the numerical sequence where each subsequent member of the series equal to sum Two Previous, That is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 17711, 28657, 46368, .. 75025, .. 3478759200, 5628750625, .. 260993908980000, .. 422297015649625, .. 19581068021641812000, .. learning complex and amazing properties The numbers of the Fibonacci series were engaged in a variety of professional scientists and mathematics lovers.

In 1997, several strange features of the series described the researcher Vladimir Mikhailov, who was convinced that nature (including a person) develops according to the laws that are laid in this numerical sequence.

The remarkable property of the numerical series of Fibonacci is that as the number of rows increases the ratio of two neighboring members of this series asymptotically approaches the exact proportion of the golden section (1: 1.618) - the basis of beauty and harmony in the nature around us, including in human relations.

Note that Fibonacci himself opened his famous row, reflecting on the task of the number of rabbits, which for one year should be born from one pair. It turned out that in each subsequent month after the second number of pairs of rabbits is exactly the digital row, which now wears its name. Therefore, it is not by chance that the person himself is arranged for a number of Fibonacci. Each body is arranged in accordance with the inner, or external duality.

Fibonacci numbers attracted mathematicians with their peculiarity to occur in the most unexpected places. It is noticed, for example, that the ratios of Fibonacci numbers taken through one correspond to the corner between the adjacent leaves on the plant's stem, more precisely, they say what kind of turnover is this angle: 1/2 - for ebvious and linden, 1/3 - for beech, 2/5 - for oak and apple, 3/8 - for poplar and roses, 5/13 - for willow and almonds, etc. The same numbers can be found when counting seeds in sunflower spirals, in the amount of rays that reflect from two Mirrors, in the number of options for closing the bee bee from one cell to another, in many mathematical games and focus.

What is the difference between the spirals of the golden section and the spiral of Fibonacci? The spiral of the golden section is ideal. It corresponds to the original source of harmony. This helix has no beginning, no end. She is infinite. Spiral Fibonacci has the beginning from which it starts "promotion." This is a very important property. It allows nature after another closed cycle to build a new helix with "zero".

It should be said that the fibonacci spiral can be double. There are numerous examples of these double spirals found everywhere. So, the helix of sunflowers will always relate to Near Fibonacci. Even in a conventional pine chish, you can see this double spiral Fibonacci. The first spiral goes in one direction, the second one - to another. If you calculate the number of scales in the spiral rotating in the same direction, and the number of scales in another helix can be seen that it is always two consecutive numbers of Fibonacci row. The number of these spirals 8 and 13. In sunflowers there are couples of spirals: 13 and 21, 21 and 34, 34 and 55, 55 and 89. And there is no deviations from these pairs! ..

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

But why in nature exactly this series plays a decisive role? This question can give an exhaustive response concept of triple, determining the conditions for its self-preservation. If the "balance of interests" is violated, the triads are one of its "partners", "opinions" of two other "partners" must be adjusted. Especially clearly, the concept of tripod is manifested in physics, where all elementary particles built from quarks. If we recall that the rating of fractional charge charges of quark particles make up a number, and these are the first members of the Fibonacci series, which are necessary for the formation of other elementary particles.

It is possible that the fibonacci spiral can play a decisive role in the formation of the patterns of limited and closetness of hierarchical spaces. Indeed, imagine that at some stage of the evolution of the Spiral Fibonacci reached perfection (it became indistinguishable from the spiral of the golden section) and for this reason the particle should be transformed into the following "category".

These facts again confirm that the law on duality gives not only high-quality, but also quantitative results. They are forced to think about the fact that the macromir surrounding us and the microme is evolving according to the same laws - the laws of the hierarchy, and that these laws are united for living and for inanimate matter.

All this indicates that the number of Fibonacci numbers is a certain encrypted law of nature.

The digital development code of civilization can be determined using various methods in numerology. For example, by bringing complex numbers to unambiguous (for example, there are 1 + 5 \u003d 6, etc.). Conducting a similar procedure for addition with all the complex numbers of a number of Fibonacci, Mikhailov received the following series of these numbers: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 8, 1, 9, then everything is repeated 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 4, 8, 8, 2, .. and repeats again And again ... This series also has the properties of a row of Fibonacci, each infinitely subsequent member is equal to the amount of the previous ones. For example, the amount of the 13th and 14th members is 15, i.e. 8 and 8 \u003d 16, 16 \u003d 1 + 6 \u003d 7. It turns out that this series is periodic, with a period of 24 member, after which, the entire order of numbers is repeated. Having received this period, Mikhailov put forward an interesting assumption - is it not a set of 24 digits a kind of digital code for the development of civilization? Published