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.. In this case, the sequence in this case is two-sided (i.e. covers negative and positive numbers) and tends to infinity in both directions.

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

The formula in this case looks like this:

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

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

1. 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.
2. 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 correlate according to the following principle: Most relates to the smaller the same 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 .

The golden cross section finds practical use in the visual arts (paintings by Leonardo da Vinci and other painters of the Renaissance), architecture, cinema ("Potemkin's armadapole" 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 building from squares with a side 1. To which, guided by the voiced principle, the figures with the parties equal to Fibonacci numbers are completed. 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 what amazing worlds can open 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 Fibonacci sequences in nature similar to the numbers 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, such an unusual phenomenon, like Fibonacci numbers, cannot but 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 of 2009, "Mr. Nobody" in one of the episodes, the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode, the main character should 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-game developers for Mobile Phones Doom RPG placed on one of the levels of the 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.

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

In the universe there are still many unsolved secrets, some of which scientists have already been able to determine and describe. Fibonacci numbers and a golden section make up the basis of the surrounding world, building its shape and optimal visual perception by a person with which it can feel beauty and harmony.

Golden cross section

The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technique. The teaching of the gold proportion was laid as a result of research by ancient scientists of the nature of numbers.

It is based on the theory of proportions and relationships of segments divisions, which was made by another ancient philosopher and mathematician Pythagorea. He proved that when dividing a segment into two parts: X (smaller) and y (greater), the ratio of greater to a smaller will be equal to the ratio of their sum (total segment):

As a result, an equation is obtained: x 2 - x - 1 \u003d 0,which is solved as x \u003d (1 ± √5) / 2.

If we consider the ratio of 1 / x, it is equal 1,618…

The evidence of the use of the ancient thinkers of the golden proportion is given in the book of Evklida "Beginning", written in 3rd. BC, who applied this rule to build the right 5-kalons. In Pythagoreans, this figure is considered sacred, since it is simultaneously symmetric and asymmetric. Pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber Abaci Mathematics from Italy Leonardo Pisansky, who later became known as Fibonacci, saw the light in 1202. In it, the scientist first leads the pattern of numbers, in a number of which each number is the sum of 2 previous numbers. The sequence of Fibonacci numbers is as follows:

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

Also, the scientist led a number of patterns:

• Any number from a series, divided by the subsequent, will be equal to a value that seeks to 0.618. Moreover, the first numbers of Fibonacci do not give such a number, but as it turns out from the beginning of the sequence, this ratio will be increasingly accurate.
• If you divide the number from a number to the previous one, the result will rush to 1.618.
• One number divided by the next one will show the value seeking to 0.382.

The use of communication and patterns of the golden section, the number of Fibonacci (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction and in many other sciences.

Spiral Archimedes and Golden Rectangle

Spirals, very common in nature, were investigated by Archimema, who even brought its equation. The form of the helix is \u200b\u200bbased on the laws of the golden section. When it is spinning, the length is obtained to which the proportions and numbers of fibonacci can be applied, increasing the step occurs evenly.

Parallel between the numbers of Fibonacci and the golden section can be seen and building the "golden rectangle", in which the parties are proportional to as 1,618: 1. It is built by moving from a larger rectangle to small so that the lengths of the parties will be equal to the numbers from the row. Building it can be done in reverse order, starting with the square "1". When connecting the corners of this rectangle in the center of their intersection, the fibonacci helix is \u200b\u200bobtained or logarithmic.

History of application of gold proportions

Many ancient monuments of Egypt architecture are elevated using golden proportions: the famous peyramids of Heops and others. Architects of ancient Greece widely used them when erecting architectural facilities such as temples, amphitherators, stadiums. For example, such proportions were applied during the construction of the ancient temple of Parfenon, (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical patterns.

In the later century, interest in the golden cross section of the clouds, and the patterns were forgotten, but again resumed in the Renaissance era, along with the book of the Franciscan monk L. Pacheli di Borgo "Divine proportion" (1509). There were illustrations of Leonardo da Vinci, which secured the new name "Golden section". The 12 properties of the golden proportion were also scientifically proved, and the author told about how she manifests itself in nature, in art and called it the "principle of building peace and nature".

Vitruvian man Leonardo

The drawing, which Leonardo da Vinci illustrated the Book of Vitruvia in 1492, depicts a person's figure in 2 positions with his hands, divorced to the sides. The figure is inscribed in a circle and square. This drawing is considered to be canonical proportions of the human body (male) described by Leonardo based on the study of them in the treatises of the Roman architect Vitruvia.

The center of the body as an equidistant point from the end of the hands and feet is the navel, the length of the hands is equal to the growth of the person, the maximum width of the shoulders \u003d 1/8 growth, the distance from the top of the chest to the hair \u003d 1/7, from the top of the chest to the top of the head \u003d 1/6 etc.

Since then, the drawing is used as a symbol showing the inner symmetry of the human body.

The term "golden section" Leonardo used to designate proportional relations in the human figure. For example, the distance from the belt to the feet feet correlates to the same distance from the navel to the macushk, as well as growth to the first length (from the belt down). These calculation is made similarly to the ratio of segments when calculating the gold proportion and tends to 1,618.

All these harmonious proportions are often used by artists to create beautiful and impressive works.

Golden section studies in 16-19 centuries

Using the golden section and the number of Fibonacci, research work on the proportions continues not to one century. In parallel with Leonardo da Vinci, the German artist Albrecht Durer also developed the development of the theory of the correct proportions of the human body. For this, they even created a special circus.

In the 16th century The issue of the number of Fibonacci and the Golden section was devoted to the work of Astronomom I. Kepler, who for the first time applied these rules for Botany.

The new "discovery" was waiting for a golden cross section in 19 V. With the publication of the "aesthetic study" of the German scientist Professor Tseyziga. He erected these proportions to Absolut and announced that they were universal for all natural phenomena. They conducted studies of a huge number of people, rather of their bodily proportions (about 2 thousand), according to the results of which conclusions were made about statistical confirmed patterns in the ratios of various parts of the body: shoulder lengths, forearms, brushes, fingers, etc.

Art objects were also investigated (vases, architectural structures), musical tones, sizes when writing poems - all this Tseyzig brought through lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that a series of fibonacci was obtained.

Fibonacci number and golden cross section in nature

In the vegetation and animal world there is a tendency to form formation in the form of symmetry, which is observed in the direction of growth and movement. Decision on symmetric parts in which gold proportions are observed - such a pattern inherent in many plants and animals.

Nature around us can be described using Fibonacci numbers, for example:

• the location of the leaves or branches of any plants, as well as the distance correlated with a number of above numbers 1, 1, 2, 3, 5, 8, 13, and below;
• sunflower seeds (scales on cones, pineapple cells), located two rows of twisted spirals in different directions;
• the ratio of the length of the tail and the entire body of the lizard;
• the shape of the egg, if you hold the line conditionally through the wide part of it;
• the ratio of the size of the fingers on the hand of a person.

And, of course, the most interesting forms represent the spirals spiral snails, patterns on the web, the wind movement inside the hurricane, the double helix in DNA and the structure of galaxies - they all include the sequence of Fibonacci numbers.

Using a golden cross section in art

Researchers engaged in the art of examples of the use of a golden section in detail the various architectural objects and painting works. Famous sculptural works are known, the creators of which adhered to gold proportions, - Statues of Zeus Olympic, Apollo Belvedere and

One of the works of Leonardo da Vinci is the "Portrait of Mona Lisa" - for many years it is the subject of studies of scientists. They found that the composition of the work of the whole consists of "golden triangles", combined together in the right pentagon-star. All the works of Da Vinci are evidence of how deeply his knowledge is in the structure and proportions of the body of a person, so that he was able to catch an incredibly mysterious smile of Joconda.

Golden section in architecture

As an example, scientists explored the masterpieces of architecture created according to the rules of the Golden section: Egyptian pyramids, Pantheon, Parfenon, Cathedral of Notre Dame de Paris, Vasily's Church of Blessed, etc.

Parthenon is one of the most beautiful buildings in ancient Greece (5 century BC) - has 8 columns and 17 on different sides, the ratio of its height to the length of the parties is 0.618. The protrusions on its facades were made according to the "golden section" (photo below).

One of the scientists who came up with and successfully applied the improvement of the modular system of proportions for architectural objects (the so-called "modulor") was the French architect Le Corbusier. The module is based on a measuring system associated with conditional division into parts of the human body.

Russian Architect M. Cossacks, built several residential buildings in Moscow, as well as the building of the Senate in the Kremlin and the Golitsyn hospital (now the 1st clinical name. N. I. Pirogov), - was one of the architects that were used in designing and building laws About the golden section.

Application of proportions in design

In the design of clothing, all fashion designers make new images and models, taking into account the proportions of the human body and the rules of the Golden section, although from nature not all people have perfect proportions.

When planning a landscape design and creating bulk park compositions with plants (trees and shrubs), fountains and small architectural objects can also be applied by the patterns of "divine proportions". After all, the composition of the park should be focused on creating an impression on a visitor who can freely navigate in it and find a composite center.

All elements of the park are in such relations so that with the help of geometric structure, interpretation, lighting and light, make the impression of harmony and perfection on a person.

Application of a golden section in cybernetics and technique

The patterns of the golden section and Fibonacci numbers are also manifested in the transitions of energy, in processes occurring with elementary particles that make up chemical compounds in space systems in the DNA gene structure.

Similar processes occur in the human body, manifesting themselves in the biorhythms of his life, in the action of organs, for example, a brain or vision.

Algorithms and regularities of gold proportions are widely used in modern cybernetics and computer science. One of the simple tasks, which is given to solve novice programmers, is to write a formula and determine the sum of the Fibonacci numbers to a certain number using programming languages.

Modern studies of the theory of gold proportion

Starting from the mid-20th century, interest in the problems and influence of the patterns of gold proportions to the human life increases sharply, and from many scientists of various professions: mathematicians, researchers of ethnic groups, biologists, philosophers, medical workers, economists, musicians, etc.

In the US, the Fibonacci Quarterly magazine begins to be published from the 1970s, where work on this topic is published. The press appears in which the generalized rules of the golden section and a number of fibonacci are used in various branches of knowledge. For example, for encoding information, chemical research, biological, etc.

All this confirms the conclusions of the ancient and modern scientists that the golden proportion of multilaterally is associated with fundamental issues of science and manifests itself in the symmetry of many creations and phenomena of the world around us.

Sacred geometry. Energy Codes of Harmony Prokopenko Iolaant

The number "FI" \u003d 1,618

The number "FI" \u003d 1,618

To connect two parts with the third perfect manner, a proportion that would brought them into a single integer. At the same time, one part of the whole should be treated like this, as a whole to most.

The number of fi is considered the most beautiful number in the world, the basis of the foundations of the whole living. One of the sacred places of ancient Egypt hides in his title this is the dizes. This number has many titles, it is known to mankind more than 2500 years.

For the first time, the mention of this number is found in the work of the ancient Greek mathematics Euclida "Beginning" (about 300 years BC). There this number is used to build a regular pentagon, which is based on the ideal "Platonic Body" - dodecahedron, the symbol of the perfect universe.

The number of fi - the trazane number and is expressed by an infinite decimal fraction. Leonardo Pisa, Contemporary Leonardo da Vinci, more famous as Fibonacci, called this the number "divine proportion". Later, the "FI" constant was founded "Golden section". The term "golden section" was introduced in 1835 by Martin Ohm.

The proportion of "FI" in the statue of the dorifera spear

Fibonacci row (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.) in ancient times was considered a unique key to the laws of the universe. You can find a private between two next numbers and approach the "fi", but it is impossible to achieve it.

The permanent constant "FI" was used in constructing a peyramid of Heops, as well as to create bas-reliefs, household goods and jewelry from Tutankhamon's tomb. The proportion of the "golden section" is used everywhere to this day in the works of artists, sculptors, architects and even choreographers and musicians.

The French architect Le Corbusier found the meaning of the "fi" constant in the relief from the temple in Abidos, the relief of Pharaoh Ramses, the facade of Greek Parfenon. Gold proportions are also hidden in the Circle of the Ancient Roman city of Pompei. The proportion of "FI" is also present in the architecture of the human body. (For details, see the section "Golden Section".)

From the book the number of life. Fate code. Read this book if you were born 3rd, 12th, 21st or 30th Author Hardy Titania

From the book the number of life. Fate code. Read this book if you were born on the 4th, 13th, on the 22nd or 31st Author Hardy Titania

Number of day if your birthday is a two-digit number, fold its numbers to make a clear number. Birth rate - 22rd: 2 + 2 \u003d 4.Dime - 13th Number: 1 + 3 \u003d

From the book the number of life. Fate code. Read this book if you were born on the 5th, 14th or 23rd Author Hardy Titania

Number of day if your birthday is a two-digit number, fold its numbers to get an unambiguous number. Examples Birthday - February 14: 1 + 4 \u003d 5.The birth - August 23: 2 + 3 \u003d

From the book Mystery named Author Zgur Maria Pavlovna

Number of name and number of birth (fate) With the help of numbers, you can define a cipher of your name, to correlate it with the number that denotes the code of birth, look into the secret of your character and fate and learn the compatibility of "like a loved one" with the people around you in business, family,

From the book plot of the Siberian healer. Release 09. Author Stepanova Natalia Ivanovna

The number of three numbers three is an amazing, unusually strong number because it marks the Holy Trinity (Father, Son and the Holy Spirit). This is the number of holiness, the number of true faith, strong and unshakable. This is what the troika allocates from all other numbers. How much the effect of the troika on

From the book Yoga and sexual practices author Douglas Nik.

From the book sacred geometry. Energy codes of harmony Author Prokopenko Iolaanta

The number "FI" \u003d 1.618 for connecting two parts with the third perfect manner is necessary proportion that would bore them into a single integer. At the same time, one part of the whole should be treated like this, as a whole to most. Plato The number of fi is considered the most beautiful number in

From the book a numerical code of birth and its influence on fate. How to calculate luck Author Mikheeva Irina Firsovna

The number 12 on the energies of the earth channel number 12 has like a triple (12 \u003d 1 + 2 \u003d 3), yellow color, but this is the third digit of a new reality, its double sign. Troika is a sprout of his variety, a triangle, a sign of immutability and unshakable . Psychological plan is a sign of hardness and

From the book how to call a child to be happy Author Stephanie Sister

The number 13 on the energies of the earth channel 13, like the Four, has a green color - the sound level and information. This is the fourth digit of a new reality, its double sign. The amount of 13 gives in the amount of Figure 4, the fourth point of reality. In natural understanding is a flower waiting for pollination

From the book Eternal Horoscope by the author Kuchin Vladimir

The number 14 on the energies of the earth channel number 14 is manifested in representatives of the new, not yet mastered by our civilization of the first intellectual level of heavenly blue color. The codenal numeric sign 14 people born on the last day of the year come. These people are ne.

From the book of the author

Number 11 on the energies of the space channel Number 11 manifests the energy of two worlds: manifested and uncompressed. And this is the sun, reflected in the water, two Suns: in the sky and in water, two units. This is a game sign, a sign of creativity. Man of this sign - a mirror that

From the book of the author

Number 12 on the energies of the space channel Number 12 personifies the harmony and the end of the space at the new level of reality, which includes three basic concepts of life: the past, present and future. 6 contains a unit - a sign of the leader and a two - the owner sign

From the book of the author

Number 13 on the energies of the space channel Number 13 manifests the wind energy of all four sides of the light, mobility, society at a new level of development. Symmetically energy of the number 13 looks like the same wind rose as in the number 4, but without limiting space.

From the book of the author

Number 14 on the energies of the space channel Number 14 is a messenger of space. Royal number 13 is not the last in the levels of development of our civilization. There is another day in a year, when missionaries come from the very cosmos, these people do not have a clear body code (earthly channel), they do not have

From the book of the author

Step one. Calculate the number of birth, or the number of personality the number of birth reveals the natural characteristics of the person, it, as we have already spoken, remains unchanged for life. If only we are talking about numbers 11 and 22, which can be "simplified" to 2 and 4

From the book of the author

5th. Bor Bor is often lucky at birth, and he inherits some capital, "factories" and "steamboats". Perhaps he does not bother the inheritance, and will give it to his heirs. His personal preferences are undefined - whether he loves harmony and feels, or loves power and

The number of phi fi or Latin letters is a number that means everything beautiful in the universe. What is this unusual number, and what other names do it exist?

Why is this number called a golden cross section?

In ancient Greece there was one sculptor fidi, who had an amazing talent. Everyone admired his sculptures and tried to solve how this Creator managed to make a real work of art every time. Later it became known that in each of its sculpture fidi adheres to a certain number in proportions.

Then it turned out that not only this creator used in his art is an extraordinary number. It was found in the artworks of the artist of Rafael, the Russian artist Shishkin, the number nest in the musical works of Beethoven, Chopin and Tchaikovsky. The famous "Jokonda" Leonardo da Vinci also contains this number. It is also called a golden cross section.

Fibonacci numbers Amazing pattern [Number FI and Golden section]

The mystery of the number 1.618034 - the most important number in the world

Golden cross section

According to mathematical standards, the number of FI is 1.618, he received the Fibonacci researcher. This scientist as a result of his research came to the fact that all numbers have a clear sequence. Each next member starting from the third number carries the amount of two past members. And the private two adjacent numbers is as close as possible to the number 1.618, that is, to the very number of fi.

Golden section and proportions of the human body

Probably, everyone saw the famous picture of Leonardo da Vinci, where the human body is drawn. It is with the help of this famous scheme, Leonardo proved that the human body is created according to the principle of the golden section. The proportions of the human body always give that the very number of beauty fi.

If desired, such a theory can be easily checked in practice. It is necessary to measure with a centimeter length from the shoulder to the tip of the longest finger, and then divide it on the length of the elbow to the tip of the same finger. Amazing, but as a result you will get just 1.618! The very number of beauty. This is not the only example. Measure the distance from the top of the hip, divide it on the length of the knee to the floor, you will get the same value. Thus, it is easy to prove, a person fully consists of a divine proportion.

In addition, on the human body, it is easy to detect a sign of the most golden cross section. This is our navel. It is interesting to note that men's body measurements are slightly more close to the cherished number. This is approximately 1.625. Women's proportions are more suitable for 1.6.

Pyramid secrets

For many years, people have tried to open a pyramid pyramid in Giza. But this time the pyramid was interested in humanity not as a crypt, but as a unique combination of numerical values. This pyramid was erected by a master who possesses amazing ingenuity, he did not regret labor and time for this work. The best architects who managed to find were put on her creation. Long modern scientists perplexed as an ancient Egyptians who had no written written, managed to come up with such a complex geometro-mathematical key. After long miscalculations, it turned out that in this case it did not cost without a golden section and the number FI. Just in this principle, this pyramid is based. Some modern scientists believe that through this work the ancient Egyptians tried to transfer the secret of natural beauty and harmony to their contemporaries.

Not exclusively in Giza there are pyramids, which are built, the pyramids that are located in Mexico are also built in this way. That is why modern researchers come to the conclusion that the pyramids in these territories were built by the people who have common roots.

The number fi in space

Astronomer from Germany Titius in the XVIII century noticed that a number of fibonacci numeric values \u200b\u200bis present in the distance between the planets of the entire solar system. This would not be anything surprising if such a regularity did not follow in confrontation with one law. The fact is that there is no planet between Mars and Jupiter, as astronomers thought. However, after the elimination of this pattern, they carefully investigated this area of \u200b\u200bthe galaxy and found a number of asteroids there. Unfortunately, such an important discovery occurred when the very Titius had already passed away.

Now in astronomy, with the help of numerical relations, Fibonacci represent the structure of galaxies. This fact indicates the independence of these numerical relations on the conditions of manifestation, thereby proves their versatility.

Examples of Nature Nature

Here are interesting examples of the number fi from the nature itself:

• If you take the bees of the breath, to recalculate in it the number of bees boys and bees-girls, then boys to divide on girls, then every time you get 1,618.
• Seeds in the sunflower are located on the principle of spiral, against the direction of the clockwise. The diameter of each helix in the sunflower is equal to the next spiral too 1.618.
• The same principle with spirals acts on a snail shell.
• If each plant is drawn to the sky, then it can be noted that a small sprout makes a large jerk up, then the stop and release of one sheet, which will be somewhat shorter than the first sprout. Then again the throw up, but with less power. If all this is translated into mathematical value, then the first throw will be equal to 100, the second 62, the third 38 units, the fourth 24, and so on. This means that growing jerks are reduced by the same principle of the golden section.
• Vivoric lizard. In such an amazing creature, as a lizard, you can even notice the divine proportions in the unarmed look. The ratio of the tail length of this animal is equal to the length of the remaining body of this creature, as 62 refers to 38.

Based on all these examples, there are actually much more scientists conclude that in the world of plants and animal world there are symmetry in relation to growth and movement. The golden cross section is manifested here perpendicular to the direction of growth.

Golden section and chaos theory

Some scientists noticed that everything in the world is chaotic. And others summed up that even in chaos, which is subject to the whole world, you can find your specific patterns. These same patterns are also expressed in the numerical values \u200b\u200bof Fibonacci. In each natural phenomenon there is its golden ratio of numbers. In this sense, nature cannot compete with dry and boring geometry.

Geometry with all its accuracy and constructiveness is not capable of describing the form of the cloud, tree or mountain. The cloud cannot be represented by the sphere, the Mountain cone, the seashore can not find its expression in the geometric circumference. The bark of the tree cannot be expressed by this science, because it is not smooth, and zipper will never move in a straight line. Natural phenomena are not only a higher degree, and a completely new level of complexity. In nature, the scale sets, different lengths of objects are presented, so they are able to close the innumerable amount of needs. Such a set of scales and measurements is the name of the fractal. It is with fractals that scientists do not leave attempts to make a description of objects that are not available linear geometry. This is a fractal geometry. Each person is also a fractal.

And it is also interesting that the number of FI has an endless nature, it means that we can infinitely make new discoveries in the universe and in themselves.

1,6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269 5486262963 1361443814 9758701220 3408058879 5445474924 6185695364 8644492410 4432077134 4947049565 8467885098 7433944221 2544877066 4780915884 6074998871 2400765217 0575179788 3416625624 9407589069 7040002812 1042762177 1117778053 1531714101 1704666599 1466979873 1761356006 7087480710 1317952368 9427521948 4353056783 0022878569 9782977834 7845878228 9110976250 0302696156 1700250464 3382437764 8610283831 2683303724 2926752631 1653392473 1671112115 8818638513 3162038400 5222165791 2866752946 5490681131 7159934323 5973494985 0904094762 1322298101 7261070596 1164562990 9816290555 2085247903 5240602017 2799747175 3427775927 7862561943 2082750513 1218156285 5122248093 9471234145 1702237358 0577278616 0086883829 5230459264 7878017889 9219902707 7690389532 1968198615 1437803149 9741106926 0886742962 2675756052 3172777520 3536139362

Fibonacci numbers and golden section They constitute the basis of the surrounding world, building its shape and the optimal visual perception by a person with the help of which it can feel beauty and harmony.

The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technique. The teaching of the gold proportion was laid as a result of research by ancient scientists of the nature of numbers.

The evidence of the use of the ancient thinkers of the golden proportion is given in the book of Evklida "Beginning", written in 3rd. BC, who applied this rule to build the right 5-kalons. In Pythagoreans, this figure is considered sacred, since it is simultaneously symmetric and asymmetric. Pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber Abaci Mathematics from Italy Leonardo Pisansky, who later became known as Fibonacci, saw the light in 1202. In it, the scientist first leads the pattern of numbers, in a number of which each number is the sum of 2 previous numbers. The sequence of Fibonacci numbers is as follows:

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

Also, the scientist led a number of patterns:

Any number from a series, divided by the subsequent, will be equal to a value that seeks to 0.618. Moreover, the first numbers of Fibonacci do not give such a number, but as it turns out from the beginning of the sequence, this ratio will be increasingly accurate.

If you divide the number from a number to the previous one, the result will rush to 1.618.

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

The use of communication and patterns of the golden section, the number of Fibonacci (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction and in many other sciences.

For practical purposes, limited to an approximate value φ \u003d 1.618 or φ \u003d 1.62. In the percentage rounded value, the golden cross section is dividing any value in relation to 62% and 38%.

Historically, the division of the segment of the segment with into two parts (a smaller segment of the AU and a larger segment of the Sun) was called historically in a golden cross section (a smaller segment of the speaker and a larger segment) so that for the lengths of the segments it was right AC / BC \u003d BC / AV. Speaking with simple words, the golden section of the segment is dissected into two unequal parts so that a smaller part refers to a greater, as large to the whole segment. Later, this concept was distributed to arbitrary values.

The number φ is also called Golden number.

The golden cross section has many wonderful properties, but, in addition, many fictional properties are attributed to him.

Now details:

The definition of the CP is the division of the segment into two parts in such a relation, in which most relates to the smaller, as their sum (the entire segment) to the greater.

That is, if we take the entire segment C for 1, then the segment A will be 0.618, the segment B is 0.382. Thus, if you take the structure, for example, a temple built on the principle of the CP, then when it is height, we say 10 meters, the height of the drum with the dome will be equal to 3.82 cm, and the height of the structure of the structure will be 6, 18 cm. (It is clear that the numbers taken smooth for clarity)

And what about the connection between the zs and the numbers of Fibonacci?

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

The pattern of numbers is that each subsequent number is equal to the sum of the two previous numbers.
0 + 1 = 1;
1 + 1 = 2;
2 + 3 = 5;
3 + 5 = 8;
5 + 8 = 13;
8 + 13 \u003d 21, etc.,

and the relationship of adjacent numbers is approaching the ratio of the ZS.
So, 21: 34 \u003d 0.617, and 34: 55 \u003d 0.618.

That is, the basis of the CC is the number of Fibonacci sequences.

It is believed that the term "golden section" introduced Leonardo da Vinci, who said, "Let no one, without a mathematician, will not bother reading my work" and showed the proportions of the human body on his famous picture "Vitruvian man." "If we are a human figure - the most perfect creation of the universe - the belt to the belt and one then, then the distance from the belt to the feet, then this value will refer to the distance from the same belt to the macushkin, as the entire human growth to the length of the belt to the feet."

A number of Fibonacci numbers are clearly simulated (materialized) in the form of a helix.

And in nature Spiral ZS looks like this:

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

Seeds in most plants are spiral
- Spider weave the web on the spiral
- Hurricane spiral twists
- A frightened flock of reindeer is running around the spiral.
- DNK molecule is twisted with a double helix. The DNA molecule is two vertically intertwined spirals 34 animals and a width of 21 angstroms. Numbers 21 and 34 follow each other in Fibonacci sequence.
- The embryo develops in the form of a spiral
- Spiral "Snails in the inner ear"
- Water goes into the drained spiral
- Spiral dynamics shows the development of the personality of man and its values \u200b\u200bon the helix.
- And of course, the galaxy itself has the form of a spiral

In this way, it can be argued that the nature itself is built on the principle of the golden section, because this proportion is harmoniously perceived by the human eye. It does not require "corrections" or additions to the resulting picture of the world.

Film. The number of God. Irrefutable proof of God; The Number of God. The Incontrovertible Proof of God.

Gold proportions in the structure of the DNA molecule

All information about the physiological features of living beings is stored in the microscopic DNA 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).

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

Golden section in the structure of micromirov

Geometric shapes are not limited to a triangle, square, five or hexagon. If you connect these figures in a different way among themselves, we will get new three-dimensional geometric shapes. Examples of this are such figures as a cube or pyramid. However, in addition to them, there are also other three-dimensional figures with which we did not have to meet in everyday life, and whose names we hear may be for the first time. Among such three-dimensional figures, a tetrahedron can be called (the right four-sided figure), octahedron, dodecahedron, Ikosahedron, etc. Dodecahedron consists of 13 pentagons, Ikosahedron from 20-triangles. Mathematics note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden section.

In the microworld, three-dimensional logarithmic forms built on gold proportions are common everywhere. For example, many viruses have a three-dimensional geometric shape of the Ikosahedron. Perhaps the most famous of these viruses is the ADENO virus. The protein sheath of the adeno virus is formed from 252 units of protein cells located in a specific sequence. In each corner of the Ikosahedron, 12 units of protein cells are located in the form of a pentagonal prism and from these angles are shi-like structures.

For the first time, the golden cross section in the structure of viruses was found in the 1950s. Scientists from London Birkbek College A. Klug and D.Kaspar. 13 The first logarithmic form revealed the Polyo virus. The form of this virus turned out to be similar to the form of the Rhino 14 virus.

The question arises how the viruses form so complex three-dimensional forms, the device of which contains a golden cross section, which even our human mind construct quite difficult? The discoverer of these forms of viruses, Virologist A. Klug gives such a comment:

"Dr. Kaspar and I have shown that for the spherical shell of the virus, the most optimal form is the symmetry of the type of the shape of the ikoshedron. Such an order minimizes the number of binding elements ... Most of the geodesic hemispherical cubes of the wagers of fuller are built on a similar geometric principle. 14 Installation of such cubes requires an extremely accurate and detailed explanation scheme. Whereas the unconscious viruses themselves build a complex shell of elastic, flexible protein cellular units. "