Fibonacci lived a long life, especially for his time, which he devoted to solving a number of mathematical problems, formulating them in the voluminous work “The Book of Abacus” (early 13th century). He was always interested in the mysticism of numbers - he was probably no less brilliant than Archimedes or Euclid. Problems related to quadratic equations were posed and partially solved earlier, for example by the famous Omar Khayyam, a scientist and poet; however, Fibonacci formulated the problem of the reproduction of rabbits, the conclusions from which did not allow his name to be lost in the centuries.
Briefly, the task is as follows. A pair of rabbits was placed in a place fenced on all sides by a wall, and each pair gives birth to another every month, starting from the second month of its existence. The reproduction of rabbits in time will be described by the following series: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. This series is called the Fibonacci sequence, also called the Fibonacci formula or numbers. From a mathematical point of view, the sequence turned out to be simply unique, since it had a number of outstanding properties:
the sum of any two consecutive numbers is the next number in the sequence
the ratio of each number in the sequence, starting from the fifth, to the previous one is 1.618
the difference between the square of any number and the square of the number two positions to the left will be the Fibonacci number
the sum of the squares of adjacent numbers will be the Fibonacci number, which is two positions after the largest of the squared numbers
Fibonacci Golden Ratio
Of these findings, the second is the most interesting because it uses the number 1.618, known as the “golden ratio.” This number was known to the ancient Greeks, who used it during the construction of the Parthenon (by the way, according to some sources it served as the Central Bank). No less interesting is that the number 1.618 can be found in nature on both micro and macro scales - from the coils on a snail’s shell to the large spirals of cosmic galaxies.
The pyramids at Giza, created by the ancient Egyptians, also contained several parameters of the Fibonacci series during construction. A rectangle, one side of which is 1.618 times larger than the other, looks most pleasing to the eye - this ratio was used by Leonardo da Vinci for his paintings, and in a more everyday sense it was intuitively used when creating windows or doorways. Even a wave can be represented as a Fibonacci spiral.
In living nature, the Fibonacci sequence appears no less often - it can be found in claws, teeth, sunflowers, spider webs and even the growth of bacteria. If desired, consistency is found in almost everything, including the human face and body. And yet, many of the claims that find the Fibonacci golden ratio in natural and historical phenomena are clearly false - it is a common myth that turns out to be an inaccurate fit to the desired result. There are comic drawings inscribing the Fibonacci spiral into scoliosis or the hairstyles of famous people.
Fibonacci numbers in financial markets
One of the first who was most closely involved in the application of Fibonacci numbers to the financial market was R. Elliot. His work was not in vain in the sense that market descriptions using the Fibonacci series are often called “Elliott waves”. The basis for his search for market patterns was a model of human development from supercycles with three steps forward and two steps back. Below is an example of how you can try to use Fibonacci levels:
The fact that humanity is developing nonlinearly is obvious to everyone - for example, the atomistic teaching of Democritus was completely lost until the end of the Middle Ages, i.e. forgotten for 2000 years. However, even if we accept the theory of steps and their number as truth, the size of each step remains unclear, which makes Elliott waves comparable to the predictive power of heads and tails. The starting point and the correct calculation of the number of waves were and apparently will be the main weakness of the theory.
Nevertheless, the theory had local successes. Bob Pretcher, who can be considered a student of Elliott, correctly predicted the bull market of the early 1980s and saw 1987 as the turning point. This actually happened, after which Bob obviously felt like a genius - at least in the eyes of others, he certainly became an investment guru. Global interest in Fibonacci levels has increased.
Subscriptions to Prechter's Elliott Wave Theorist grew to 20,000 that year, but declined in the early 1990s as the "doom and gloom" predictions for the American market were put on hold. However, it worked for the Japanese market, and a number of supporters of the theory, who were “late” there for one wave, lost either their capital or the capital of their companies’ clients.
Elliott waves cover a variety of trading periods - from weekly, which makes it similar to standard technical analysis strategies, to calculations for decades, i.e. gets into the territory of fundamental predictions. This is possible by varying the number of waves. The weaknesses of the theory, which are mentioned above, allow its adherents to speak not about the inconsistency of the waves, but about their own miscalculations among them and an incorrect definition of the starting position.
It's like a labyrinth - even if you have the right map, you can only follow it if you understand exactly where you are. Otherwise the card is of no use. In the case of Elliott waves, there is every sign of doubting not only the correctness of your location, but also the accuracy of the map as such.
conclusions
The wave development of humanity has a real basis - in the Middle Ages, waves of inflation and deflation alternated with each other, when wars gave way to a relatively calm peaceful life. The observation of the Fibonacci sequence in nature, at least in some cases, also does not raise doubts. Therefore, everyone has the right to give their own answer to the question of who God is: a mathematician or a random number generator. My personal opinion: although all of human history and markets can be represented in the wave concept, the height and duration of each wave cannot be predicted by anyone.
Fibonacci numbers are elements of a number 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 is named after the medieval mathematician Leonardo of Pisa (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 was the introduction of Arabic numerals, which replaced Roman numerals. Fn =Fn-1 +Fn-2
A mathematical series asymptotically (that is, approaching more and more slowly) tends to a constant ratio. However, this attitude is irrational; it has an endless, unpredictable sequence of decimal values lining up after it. It can never be expressed exactly. If each number that is part of a series is divided by its predecessor (for example, 13-^8 or 21 -IZ), the result of the action is expressed in a ratio that fluctuates around the irrational number 1.61803398875, slightly more or slightly less than the neighboring ratios of the series. The ratio will never, ad infinitum, be accurate down to the last digit (even using the most powerful computers created in our time). For the sake of brevity, we will use 1.618 as the Fibonacci ratio and ask readers to be aware of this error.
Fibonacci numbers are also important when performing analysis of the Euclidean algorithm to determine the greatest common divisor of two numbers. Fibonacci numbers come from the formula for the diagonal of Pascal's triangle (binomial coefficients).
Fibonacci numbers turned out to be related to the “golden ratio”.
The golden ratio was known back in ancient Egypt and Babylon, in India and China. What is the “golden ratio”? The answer is still unknown. Fibonacci numbers are really relevant to the theory of practice in our time. The rise in importance occurred in the 20th century and continues to this day. The use of Fibonacci numbers in economics and computer science attracted masses of people to their study.
The methodology of my research consisted of studying specialized literature and summarizing the information received, as well as conducting my own research and identifying the properties of numbers and the scope of their use.
In the course of scientific research, she defined the very concepts of Fibonacci numbers and their properties. I also found out interesting patterns in living nature, directly in the structure of sunflower seeds.
On a sunflower, the seeds are arranged in spirals, and the numbers of spirals going in the other direction are different - they are successive Fibonacci numbers.
This sunflower has 34 and 55.
The same is observed on pineapple fruits, where there are 8 and 14 spirals. Corn leaves are associated with the unique property of Fibonacci numbers.
Fractions of the form a/b, corresponding to the helical arrangement of the leaves of the legs of the plant stem, are often ratios of successive Fibonacci numbers. For hazel this ratio is 2/3, for oak 3/5, for poplar 5/8, for willow 8/13, etc.
Looking at the arrangement of leaves on a plant stem, you can notice that between each pair of leaves (A and C), the third is located at the place of the golden ratio (B)
Another interesting property of the Fibonacci number is that the product and quotient of any two different Fibonacci numbers other than one is never a Fibonacci number.
As a result of the research, I came to the following conclusions: Fibonacci numbers are a unique arithmetic progression that appeared in the 13th century AD. This progression does not lose its relevance, which was confirmed during my research. Fibonacci numbers are also found in programming and economic forecasts, in painting, architecture and music. Paintings by such famous artists as Leonardo da Vinci, Michelangelo, Raphael and Botticelli hide the magic of the golden ratio. Even I. I. Shishkin used the golden ratio in his painting “Pine Grove”.
It’s hard to believe, but the golden ratio is also found in the musical works of such great composers as Mozart, Beethoven, Chopin, etc.
Fibonacci numbers are also found in architecture. For example, the golden ratio was used in the construction of the Parthenon and Notre Dame Cathedral
I discovered that Fibonacci Numbers are used in our area as well. For example, house trims, pediments.
Have you ever heard that mathematics is called the “queen of all sciences”? Do you agree with this statement? As long as mathematics remains for you a set of boring problems in a textbook, you can hardly experience the beauty, versatility and even humor of this science.
But there are topics in mathematics that help make interesting observations about things and phenomena that are common to us. And even try to penetrate the veil of mystery of the creation of our Universe. There are interesting patterns in the world that can be described using mathematics.
Introducing Fibonacci numbers
Fibonacci numbers name the elements of a number sequence. In it, each next number in a series is obtained by summing the two previous numbers.
Example sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
You can write it like this:
F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2
You can start a series of Fibonacci numbers with negative values n. Moreover, the sequence in this case is two-way (that is, it covers negative and positive numbers) and tends to infinity in both directions.
An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The formula in this case looks like this:
F n = F n+1 - F n+2 or else you can do this: F -n = (-1) n+1 Fn.
What we now know as “Fibonacci numbers” was known to ancient Indian mathematicians long before they began to be used in Europe. And this name is generally one continuous historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only several centuries after his death. But let's talk about everything in order.
Leonardo of Pisa, aka Fibonacci
The son of a merchant who became a mathematician, and subsequently received recognition from posterity as the first major mathematician of Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which, let us remember, were not called that yet). Which he described at the beginning of the 13th century in his work “Liber abaci” (“Book of Abacus”, 1202).
I traveled with my father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were among the best specialists in this matter, and in many other sciences). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.
Having thoroughly comprehended everything he had read and using his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the above-mentioned “Book of Abacus.” In addition to this I created:
- "Practica geometriae" ("Practice of Geometry", 1220);
- "Flos" ("Flower", 1225 - a study on cubic equations);
- "Liber quadratorum" ("Book of Squares", 1225 - problems on indefinite quadratic equations).
He was a big fan of mathematical tournaments, so in his treatises he paid a lot of attention to the analysis of various mathematical problems.
There is very little biographical information left about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was assigned to him only in the 19th century.
Fibonacci and his problems
After Fibonacci there remained a large number of problems that were very popular among mathematicians in subsequent centuries. We will look at the rabbit problem, which is solved using Fibonacci numbers.
Rabbits are not only valuable fur
Fibonacci set the following conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, a male and a female.
These conditional rabbits are placed in a confined space and breed with enthusiasm. It is also stipulated that not a single rabbit dies from some mysterious rabbit disease.
We need to calculate how many rabbits we will get in a year.
- At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
- The second month - we already have 2 pairs of rabbits (a pair has parents + 1 pair is their offspring).
- Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
- Fourth month: The first pair gives birth to a new pair, the second pair does not waste time and also gives birth to a new pair, the third pair is still only mating. Total - 5 pairs of rabbits.
Number of rabbits in n th month = number of pairs of rabbits from the previous month + number of newborn pairs (there are the same number of pairs of rabbits as there were pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: F n = F n-1 + F n-2.
Thus, we obtain a recurrent (explanation about recursion– below) number 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 <…>
You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we have set a specific period - a year, we are interested in the result obtained on the 12th “move”. Those. 13th member of the sequence: 377.
The answer to the problem: 377 rabbits will be obtained if all stated conditions are met.
One of the properties of the Fibonacci number sequence is very interesting. If you take two consecutive pairs from a series and divide the larger number by the smaller number, the result will gradually approach golden ratio(you can read more about it later in the article).
In mathematical terms, "the limit of relationships a n+1 To a n equal to the golden ratio".
More number theory problems
- Find a number that can be divided by 7. Also, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
- Find the square number. It is known about it that if you add 5 to it or subtract 5, you again get a square number.
We suggest you search for answers to these problems yourself. You can leave us your options in the comments to this article. And then we will tell you whether your calculations were correct.
Explanation of recursion
Recursion– definition, description, image of an object or process that contains this object or process itself. That is, in essence, an object or process is a part of itself.
Recursion is widely used in mathematics and computer science, and even in art and popular culture.
Fibonacci numbers are determined using a recurrence relation. For number n>2 n- e number is equal (n – 1) + (n – 2).
Explanation of the golden ratio
Golden ratio- dividing a whole (for example, a segment) into parts that are related according to the following principle: the larger part is related to the smaller one in the same way as the entire value (for example, the sum of two segments) is to the larger part.
The first mention of the golden ratio can be found in Euclid in his treatise “Elements” (about 300 BC). In the context of constructing a regular rectangle.
The term familiar to us was introduced into circulation in 1835 by the German mathematician Martin Ohm.
If we describe the golden ratio approximately, it represents a proportional division into two unequal parts: approximately 62% and 38%. In numerical terms, the golden ratio is the number 1,6180339887 .
The golden ratio finds practical application in fine arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (“Battleship Potemkin” by S. Esenstein) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This opinion is still popular today. Although, according to research results, visually most people do not perceive this proportion as the most successful option and consider it too elongated (disproportionate).
- Section length With = 1, A = 0,618, b = 0,382.
- Attitude With To A = 1, 618.
- Attitude With To b = 2,618
Now let's get back to Fibonacci numbers. Let's take two consecutive terms from its sequence. Divide the larger number by the smaller number and get approximately 1.618. And now we use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.
Here's an example: 144, 233, 377.
233/144 = 1.618 and 233/377 = 0.618
By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is hardly followed for the beginning of the sequence. But as you move along the series and the numbers increase, it works great.
And in order to calculate the entire series of Fibonacci numbers, it is enough to know three terms of the sequence, coming one after another. You can see this for yourself!
Golden Rectangle and Fibonacci Spiral
Another interesting parallel between the Fibonacci numbers and the golden ratio is the so-called “golden rectangle”: its sides are in proportion 1.618 to 1. But we already know what the number 1.618 is, right?
For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and construct a rectangle with the following parameters: width = 8, length = 13.
And then we will divide the large rectangle into smaller ones. Mandatory condition: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. The side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.
The way it is done in this figure (for convenience, the figures are signed in Latin letters).
By the way, you can build rectangles in reverse order. Those. start building with squares with a side of 1. To which, guided by the principle stated above, figures with sides equal to the Fibonacci numbers are completed. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.
If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Or rather, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.
A similar spiral is often found in nature. Clam shells are one of the most striking examples. Moreover, some galaxies that can be seen from Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when photographed from satellites.
It is curious that the DNA helix also obeys the rule of the golden section - the corresponding pattern can be seen in the intervals of its bends.
Such amazing “coincidences” cannot but excite minds and give rise to talk about some single algorithm to which all phenomena in the life of the Universe obey. Now do you understand why this article is called this way? And what kind of amazing worlds can mathematics open for you?
Fibonacci numbers in nature
The connection between Fibonacci numbers and the golden ratio suggests interesting patterns. So curious that it is tempting to try to find sequences similar to Fibonacci numbers in nature and even during historical events. And nature really gives rise to such assumptions. But can everything in our life be explained and described using mathematics?
Examples of living things that can be described using the Fibonacci sequence:
- the arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);
- arrangement of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row clockwise, the other counterclockwise);
- arrangement of pine cone scales;
- flower petals;
- pineapple cells;
- ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.
Combinatorics problems
Fibonacci numbers are widely used in solving combinatorics problems.
Combinatorics is a branch of mathematics that studies the selection of a certain number of elements from a designated set, enumeration, etc.
Let's look at examples of combinatorics problems designed for high school level (source - http://www.problems.ru/).
Task #1:
Lesha climbs a staircase of 10 steps. At one time he jumps up either one step or two steps. In how many ways can Lesha climb the stairs?
The number of ways in which Lesha can climb the stairs from n steps, let's denote and n. It follows that a 1 = 1, a 2= 2 (after all, Lesha jumps either one or two steps).
It is also agreed that Lesha jumps up the stairs from n> 2 steps. Let's say he jumped two steps the first time. This means, according to the conditions of the problem, he needs to jump another n – 2 steps. Then the number of ways to complete the climb is described as a n–2. And if we assume that the first time Lesha jumped only one step, then we describe the number of ways to finish the climb as a n–1.
From here we get the following equality: a n = a n–1 + a n–2(looks familiar, doesn't it?).
Since we know a 1 And a 2 and remember that according to the conditions of the problem there are 10 steps, calculate all in order and n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.
Answer: 89 ways.
Task #2:
You need to find the number of words 10 letters long that consist only of the letters “a” and “b” and must not contain two letters “b” in a row.
Let's denote by a n number of words length n letters that consist only of the letters “a” and “b” and do not contain two letters “b” in a row. Means, a 1= 2, a 2= 3.
In sequence a 1, a 2, <…>, a n we will express each of its next members through the previous ones. Therefore, the number of words of length is n letters that also do not contain a double letter “b” and begin with the letter “a” are a n–1. And if the word is long n letters begin with the letter “b”, it is logical that the next letter in such a word is “a” (after all, there cannot be two “b” according to the conditions of the problem). Therefore, the number of words of length is n in this case we denote the letters as a n–2. In both the first and second cases, any word (length of n – 1 And n – 2 letters respectively) without double “b”.
We were able to justify why a n = a n–1 + a n–2.
Let us now calculate a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 144. And we get the familiar Fibonacci sequence.
Answer: 144.
Task #3:
Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first square of the tape. Whatever cell of the tape he is on, he can only move to the right: either one cell, or two. How many ways are there in which a grasshopper can jump from the beginning of the tape to n-th cells?
Let us denote the number of ways to move a grasshopper along the belt to n-th cells like a n. In this case a 1 = a 2= 1. Also in n+1 The grasshopper can enter the -th cell either from n-th cell, or by jumping over it. From here a n + 1 = a n – 1 + a n. Where a n = Fn – 1.
Answer: Fn – 1.
You can create similar problems yourself and try to solve them in math lessons with your classmates.
Fibonacci numbers in popular culture
Of course, such an unusual phenomenon as Fibonacci numbers cannot but attract attention. There is still something attractive and even mysterious in this strictly verified pattern. It is not surprising that the Fibonacci sequence has somehow “lit up” in many works of modern popular culture of various genres.
We will tell you about some of them. And you try to search for yourself again. If you find it, share it with us in the comments – we’re curious too!
- Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code used by the book's main characters to open a safe.
- In the 2009 American film Mr. Nobody, in one episode the address of a house is part of the Fibonacci sequence - 12358. In addition, in another episode the main character must call a phone number, which is essentially the same, but slightly distorted (extra digit after number 5) sequence: 123-581-1321.
- In the 2012 series “Connection”, the main character, a boy suffering from autism, is able to discern patterns in events occurring in the world. Including through Fibonacci numbers. And manage these events also through numbers.
- The developers of the java game for mobile phones Doom RPG placed a secret door on one of the levels. The code that opens it is the Fibonacci sequence.
- In 2012, the Russian rock band Splin released the concept album “Optical Deception.” The eighth track is called “Fibonacci”. The verses of the group leader Alexander Vasiliev play on the sequence of Fibonacci numbers. For each of the nine consecutive terms there is a corresponding number of lines (0, 1, 1, 2, 3, 5, 8, 13, 21):
0 The train set off
1 One joint snapped
1 One sleeve trembled
2 That's it, get the stuff
That's it, get the stuff
3 Request for boiling water
The train goes to the river
The train goes through the taiga<…>.
- A limerick (a short poem of a specific form - usually five lines, with a specific rhyme scheme, humorous in content, in which the first and last lines are repeated or partially duplicate each other) by James Lyndon also uses a reference to the Fibonacci sequence as a humorous motif:
The dense food of Fibonacci's wives
It was only for their benefit, nothing else.
The wives weighed, according to rumor,
Each one is like the previous two.
Let's sum it up
We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now look for the Fibonacci spiral in the nature around you. Maybe you will be the one who will be able to unravel “the secret of life, the Universe and in general.”
Use the formula for Fibonacci numbers when solving combinatorics problems. You can rely on the examples described in this article.
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Leonardo of Pisa (lat. Leonardus Pisanus, Italian. Leonardo Pisano, around 1170, Pisa - around 1250, ibid.) - the first major mathematician of medieval Europe. He is best known by his nickname Fibonacci.
More details here: http://ru.wikipedia.org/wiki/%D4%E8%E1%EE%ED%E0%F7%F7%E8
The Fibonacci sequence, known to everyone from the movie “The Da Vinci Code,” is a series of numbers described in the form of a riddle by the Italian mathematician Leonardo of Pisa, better known as Fibonacci, in the 13th century. Briefly the essence of the riddle:
Someone placed a pair of rabbits in a certain enclosed space in order to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that every month a pair of rabbits gives birth to another pair, and they become capable of producing offspring when they reach two months of age.
Fibonacci Sequence and Rabbits
The result is the following series of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, where the number of pairs of rabbits in each of the twelve months is shown, separated by commas. It can be continued indefinitely. Its essence is that each next number is the sum of the two previous ones.
This series has several mathematical features that definitely need to be touched upon. It asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is irrational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.
Thus, the ratio of any member of the series to the one preceding it fluctuates around the number 1.618, sometimes exceeding it, sometimes not reaching it. The ratio to the next one similarly approaches the number 0.618, which is inversely proportional to 1.618. If we divide the elements through one, we will get the numbers 2.618 and 0.382, which are also inversely proportional. These are the so-called Fibonacci ratios.
What is all this for?
This is how we approach one of the most mysterious natural phenomena. The savvy Leonardo essentially did not discover anything new; he simply reminded the world of such a phenomenon as the Golden Ratio, which is not inferior in importance to the Pythagorean theorem.
We distinguish all the objects around us by their shape. We like some more, some less, some are completely off-putting. Sometimes interest can be dictated by the life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape promotes the best visual perception and evokes a feeling of beauty and harmony. A complete image always consists of parts of different sizes that are in a certain relationship with each other and the whole. The golden ratio is the highest manifestation of the perfection of the whole and its parts in science, art and nature.
To use a simple example, the Golden Ratio is the division of a segment into two parts in such a ratio that the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.
Golden Ratio - Segment
If we take the entire segment c as 1, then segment a will be equal to 0.618, segment b - 0.382, only in this way will the condition of the Golden Ratio be met (0.618/0.382=1.618; 1/0.618=1.618). The ratio of c to a is 1.618, and c to b is 2.618. These are the same Fibonacci ratios that are already familiar to us.
Of course there is a golden rectangle, a golden triangle and even a golden cuboid. The proportions of the human body are in many respects close to the Golden Section.
The Golden Ratio and the Human Body 
Image: marcus-frings.de
Fibonacci Sequence - Animation 
But the fun begins when we combine the knowledge we have gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. Add a square of the second size on top. Draw a square next to it with a side equal to the sum of the sides of the previous two, third size. By analogy, a square of size five appears. And so on until you get tired, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the previous two. We see a series of rectangles whose side lengths are Fibonacci numbers, and, oddly enough, they are called Fibonacci rectangles.
If we draw smooth lines through the corners of our squares, we will get nothing more than an Archimedes spiral, the increment of which is always uniform.
Fibonacci Spiral 
Doesn't remind you of anything?
Photo: ethanhein on Flickr
And not only in the shell of a mollusk you can find Archimedes’ spirals, but in many flowers and plants, they’re just not so obvious.
Aloe multifolia: 
Photo: brewbooks on Flickr
Broccoli Romanesco: 
Photo: beart.org.uk
Sunflower: 
Photo: esdrascalderan on Flickr
Pine cone: 
Photo: mandj98 on Flickr
And now it’s time to remember the Golden Section! Are some of the most beautiful and harmonious creations of nature depicted in these photographs? And that's not all. If you look closely, you can find similar patterns in many forms.
Of course, the statement that all these phenomena are based on the Fibonacci sequence sounds too loud, but the trend is obvious. And besides, she herself is far from perfect, like everything in this world.
There is an assumption that the Fibonacci series is an attempt by nature to adapt to a more fundamental and perfect golden ratio logarithmic sequence, which is almost the same, only it starts from nowhere and goes to nowhere. Nature definitely needs some kind of whole beginning from which it can start; it cannot create something out of nothing. The ratios of the first terms of the Fibonacci sequence are far from the Golden Ratio. But the further we move along it, the more these deviations are smoothed out. To define any series, it is enough to know its three terms, coming one after another. But not for the golden sequence, two are enough for it, it is a geometric and arithmetic progression at the same time. One might think that it is the basis for all other sequences.
Each term of the golden logarithmic sequence is a power of the Golden Proportion (z). Part of the series looks something like this: ... z-5; z-4; z-3; z-2; z-1; z0; z1; z2; z3; z4; z5 ... If we round the value of the Golden Ratio to three digits, we get z = 1.618, then the series looks like this: ... 0.090 0.146; 0.236; 0.382; 0.618; 1; 1.618; 2.618; 4.236; 6.854; 11.090 ... Each next term can be obtained not only by multiplying the previous one by 1.618, but also by adding the two previous ones. Thus, exponential growth is achieved by simply adding two adjacent elements. It's a series without beginning or end, and that's what the Fibonacci sequence tries to be like. Having a very definite beginning, she strives for the ideal, never achieving it. That is life.
And yet, in connection with everything we have seen and read, quite logical questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Was everything ever the way he wanted? And if so, why did it go wrong? Mutations? Free choice? What will be next? Is the spiral curling or unwinding?
Having found the answer to one question, you will get the next one. If you solve it, you'll get two new ones. Once you deal with them, three more will appear. Having solved them too, you will have five unsolved ones. Then eight, then thirteen, 21, 34, 55...
Kanalieva Dana
In this work, we studied and analyzed the manifestation of the Fibonacci sequence numbers in the reality around us. We discovered an amazing mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the Fibonacci sequence numbers. We also saw strict mathematics in the human structure. The human DNA molecule, in which the entire development program of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical relationships.
We are convinced that Nature has its own laws, expressed using mathematics.
And mathematics is very important tool of cognition secrets of Nature.
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MBOU "Pervomaiskaya Secondary School"
Orenburg district, Orenburg region
RESEARCH
"The Mystery of Numbers"
Fibonacci"
Completed by: Kanalieva Dana
6th grade student
Scientific adviser:
Gazizova Valeria Valerievna
Mathematics teacher of the highest category
n. Experimental
2012
Explanatory note………………………………………………………………………………........ 3.
Introduction. History of Fibonacci numbers.……………………………………………………...... 4.
Chapter 1. Fibonacci numbers in living nature.........……. …………………………………... 5.
Chapter 2. Fibonacci Spiral.................................................... ..........……………..... 9.
Chapter 3. Fibonacci numbers in human inventions.........…………………………….. 13
Chapter 4. Our research……………………………………………………………....... 16.
Chapter 5. Conclusion, conclusions………………………………………………………………………………...... 19.
List of used literature and Internet sites…………………………………........21.
Object of study:
Man, mathematical abstractions created by man, human inventions, the surrounding flora and fauna.
Subject of study:
form and structure of the objects and phenomena being studied.
Purpose of the study:
study the manifestation of Fibonacci numbers and the associated law of the golden ratio in the structure of living and non-living objects,
find examples of using Fibonacci numbers.
Job objectives:
Describe a method for constructing the Fibonacci series and Fibonacci spiral.
See mathematical patterns in the structure of humans, flora and inanimate nature from the point of view of the Golden Ratio phenomenon.
Novelty of the research:
Discovery of Fibonacci numbers in the reality around us.
Practical significance:
Using acquired knowledge and research skills when studying other school subjects.
Skills and abilities:
Organization and conduct of the experiment.
Use of specialized literature.
Acquiring the ability to review collected material (report, presentation)
Design of work with drawings, diagrams, photographs.
Active participation in discussions of your work.
Research methods:
empirical (observation, experiment, measurement).
theoretical (logical stage of cognition).
Explanatory note.
“Numbers rule the world! Number is the power that reigns over gods and mortals!” - this is what the ancient Pythagoreans said. Is this basis of Pythagoras’ teaching still relevant today? When studying the science of numbers at school, we want to make sure that, indeed, the phenomena of the entire Universe are subject to certain numerical relationships, to find this invisible connection between mathematics and life!
Is it really in every flower,
Both in the molecule and in the galaxy,
Numerical patterns
This strict “dry” mathematics?
We turned to a modern source of information - the Internet and read about Fibonacci numbers, about magic numbers that are fraught with a great mystery. It turns out that these numbers can be found in sunflowers and pine cones, in dragonfly wings and starfish, in the rhythms of the human heart and in musical rhythms...
Why is this sequence of numbers so common in our world?
We wanted to know about the secrets of Fibonacci numbers. This research work was the result of our activities.
Hypothesis:
in the reality around us, everything is built according to amazingly harmonious laws with mathematical precision.
Everything in the world is thought out and calculated by our most important designer - Nature!
Introduction. History of the Fibonacci series.
Amazing numbers were discovered by the Italian medieval mathematician Leonardo of Pisa, better known as Fibonacci. Traveling around the East, he became acquainted with the achievements of Arab mathematics and contributed to their transfer to the West. In one of his works, entitled “The Book of Calculations,” he introduced Europe to one of the greatest discoveries of all time - the decimal number system.
One day, he was racking his brains over solving a mathematical problem. He was trying to create a formula to describe the breeding sequence of rabbits.
The solution was a number series, each subsequent number of which is the sum of the two previous ones:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...
The numbers that form this sequence are called “Fibonacci numbers”, and the sequence itself is called the Fibonacci sequence.
"So what?" - you say, “Can we really come up with similar number series ourselves, increasing according to a given progression?” Indeed, when the Fibonacci series appeared, no one, including himself, had any idea how close he managed to come to solving one of the greatest mysteries of the universe!
Fibonacci led a reclusive lifestyle, spent a lot of time in nature, and while walking in the forest, he noticed that these numbers began to literally haunt him. Everywhere in nature he encountered these numbers again and again. For example, the petals and leaves of plants strictly fit into a given number series.
There is an interesting feature in Fibonacci numbers: the quotient of dividing the next Fibonacci number by the previous one, as the numbers themselves grow, tends to 1.618. It was this constant division number that was called the Divine proportion in the Middle Ages, and is now referred to as the golden section or golden proportion.
In algebra, this number is denoted by the Greek letter phi (Ф)
So, φ = 1.618
233 / 144 = 1,618
377 / 233 = 1,618
610 / 377 = 1,618
987 / 610 = 1,618
1597 / 987 = 1,618
2584 / 1597 = 1,618
No matter how many times we divide one by another, the number adjacent to it, we will always get 1.618. And if we do the opposite, that is, divide the smaller number by the larger one, we will get 0.618, this is the inverse of 1.618. also called the golden ratio.
The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden division.
Scientists, analyzing the further application of this number series to natural phenomena and processes, discovered that these numbers are contained in literally all objects of living nature, in plants, animals and humans.
The amazing mathematical toy turned out to be a unique code embedded in all natural objects by the Creator of the Universe himself.
Let's look at examples where Fibonacci numbers occur in living and inanimate nature.
Fibonacci numbers in living nature.
If you look at the plants and trees around us, you can see how many leaves there are on each of them. From a distance, it seems that the branches and leaves on the plants are located randomly, in no particular order. However, in all plants, in a miraculous, mathematically precise way, which branch will grow from where, how the branches and leaves will be located near the stem or trunk. From the first day of its appearance, the plant exactly follows these laws in its development, that is, not a single leaf, not a single flower appears by chance. Even before its appearance, the plant is already precisely programmed. How many branches will there be on the future tree, where will the branches grow, how many leaves will there be on each branch, and how and in what order the leaves will be arranged. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that the Fibonacci series manifests itself in the arrangement of leaves on a branch (phylotaxis), in the number of revolutions on the stem, in the number of leaves in a cycle, and therefore, the law of the golden ratio also manifests itself.
If you set out to find numerical patterns in living nature, you will notice that these numbers are often found in various spiral forms, which are so rich in the plant world. For example, leaf cuttings are adjacent to the stem in a spiral that runs betweentwo adjacent leaves:full rotation - at the hazel tree,- by the oak tree, - at the poplar and pear trees,- at the willow.
The seeds of sunflower, Echinacea purpurea and many other plants are arranged in spirals, and the number of spirals in each direction is the Fibonacci number.
Sunflower, 21 and 34 spirals. Echinacea, 34 and 55 spirals.
The clear, symmetrical shape of flowers is also subject to a strict law.
For many flowers, the number of petals is precisely the numbers from the Fibonacci series. For example:
iris, 3p. buttercup, 5 lep. golden flower, 8 lep. delphinium,
13 lep.
chicory, 21lep. aster, 34 lep. daisies, 55 lep.
The Fibonacci series characterizes the structural organization of many living systems.
We have already said that the ratio of neighboring numbers in the Fibonacci series is the number φ = 1.618. It turns out that man himself is simply a storehouse of phi numbers.
The proportions of the various parts of our body are a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned. The principle of calculating the gold measure on the human body can be depicted in the form of a diagram.
M/m=1.618
The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.
Human hand
It is enough just to bring your palm closer to you and look carefully at your index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.
The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb).
In addition, the ratio between the middle finger and little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence.
The golden ratio in the structure of the human lungs
American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.
The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.
There is another, more prosaic application of the proportions of the human body. For example, using these relationships, crime analysts and archaeologists use fragments of parts of the human body to reconstruct the appearance of the whole.
Golden proportions in the structure of the DNA molecule.
All information about the physiological characteristics of living beings, be it a plant, an animal or a person, is stored in a microscopic DNA molecule, the structure of which also contains the law of the golden proportion. The DNA molecule consists of two vertically intertwined helices. The length of each of these spirals is 34 angstroms and the width is 21 angstroms. (1 angstrom is one hundred millionth of a centimeter).
So, 21 and 34 are numbers following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the golden ratio 1:1.618.
Not only erect walkers, but also all swimming, crawling, flying and jumping creatures did not escape the fate of being subject to the number phi. The human heart muscle contracts to 0.618 of its volume. The structure of a snail shell corresponds to the Fibonacci proportions. And such examples can be found in abundance - if there was a desire to explore natural objects and processes. The world is so permeated with Fibonacci numbers that sometimes it seems that the Universe can only be explained by them.
Fibonacci spiral.
There is no other form in mathematics that has the same unique properties as the spiral, because
The structure of the spiral is based on the Golden Ratio rule!
To understand the mathematical construction of a spiral, let us repeat what the Golden Ratio is.
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one, or, in other words, the smaller segment is related to the larger one as the larger one is to the whole.
That is (a+b) /a = a / b
A rectangle with exactly this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.
The golden rectangle has many unusual properties. Cutting a square from a golden rectangle whose side is equal to the smaller side of the rectangle,
we will again get a smaller golden rectangle.
This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects.
For example, the spiral shape can be seen in the arrangement of sunflower seeds, in pineapples, cacti, the structure of rose petals, and so on.
We are surprised and delighted by the spiral structure of shells.
In most snails that have shells, the shell grows in a spiral shape. However, there is no doubt that these unreasonable creatures not only have no idea about the spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.
But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientific world calls primitive life forms, calculate that the spiral shape of a shell would be ideal for their existence?
Trying to explain the origin of such even the most primitive form of life by a random combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.
Spirals also exist in humans. With the help of spirals we hear:
Also, in the human inner ear there is an organ called Cochlea (“Snail”), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and created in the shape of a snail with golden proportions.
There are spirals on our palms and fingers:
In the animal kingdom we can also find many examples of spirals.
The horns and tusks of animals develop in a spiral shape, the claws of lions and the beaks of parrots are logarithmic shapes and resemble the shape of an axis that tends to turn into a spiral.
It’s interesting that a hurricane and a cyclone’s clouds are twisting like a spiral, and this is clearly visible from space:
In ocean and sea waves, the spiral can be mathematically represented on a graph with points 1,1,2,3,5,8,13,21,34 and 55.
Everyone will also recognize such an “everyday” and “prosaic” spiral.
After all, the water escapes from the bathroom in a spiral:
Yes, and we live in a spiral, because the galaxy is a spiral corresponding to the formula of the Golden Ratio!
So, we found out that if we take the Golden Rectangle and break it into smaller rectanglesin the exact Fibonacci sequence, and then divide each of them in such proportions again and again, you get a system called the Fibonacci spiral.
We discovered this spiral in the most unexpected objects and phenomena. Now it’s clear why the spiral is also called the “curve of life.”
The spiral has become a symbol of evolution, because everything develops in a spiral.
Fibonacci numbers in human inventions.
Having observed a law in nature expressed by the sequence of Fibonacci numbers, scientists and artists try to imitate it and embody this law in their creations.
The phi proportion allows you to create masterpieces of painting and correctly fit architectural structures into space.
Not only scientists, but also architects, designers and artists are amazed by this perfect spiral of the nautilus shell,
occupying the least space and providing the least heat loss. American and Thai architects, inspired by the example of the “chambered nautilus” in the matter of placing the maximum in the minimum space, are busy developing corresponding projects.
Since time immemorial, the Golden Ratio proportion has been considered the highest proportion of perfection, harmony and even divinity. The golden ratio can be found in sculptures and even in music. An example is the musical works of Mozart. Even stock exchange rates and the Hebrew alphabet contain a golden ratio.
But we want to focus on a unique example of creating an efficient solar installation. An American schoolboy from New York, Aidan Dwyer, put together his knowledge of trees and discovered that the efficiency of solar power plants can be increased by using mathematics. While on a winter walk, Dwyer wondered why trees needed such a “pattern” of branches and leaves. He knew that branches on trees are arranged according to the Fibonacci sequence, and leaves carry out photosynthesis.
At some point, the smart boy decided to check whether this position of the branches helps to collect more sunlight. Aidan built a pilot plant in his backyard using small solar panels instead of leaves and tested it in action. It turned out that compared to a conventional flat solar panel, its “tree” collects 20% more energy and operates efficiently for 2.5 hours longer.
Dwyer solar tree model and graphs made by a student.
“This installation also takes up less space than a flat panel, collects 50% more sun in winter even where it does not face south, and it does not accumulate as much snow. In addition, a tree-shaped design is much more suitable for the urban landscape,” notes the young inventor.
Aidan was recognized one of the best young naturalists of 2011. The 2011 Young Naturalist competition was hosted by the New York Museum of Natural History. Aidan has filed a provisional patent application for his invention.
Scientists continue to actively develop the theory of Fibonacci numbers and the golden ratio.
Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers.
Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio.
In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.
So, we see that the scope of the Fibonacci sequence of numbers is very multifaceted:
Observing the phenomena occurring in nature, scientists have made striking conclusions that the entire sequence of events occurring in life, revolutions, crashes, bankruptcies, periods of prosperity, laws and waves of development in the stock and foreign exchange markets, cycles of family life, and so on , are organized on a time scale in the form of cycles and waves. These cycles and waves are also distributed according to the Fibonacci number series!
Based on this knowledge, a person will learn to predict and manage various events in the future.
4. Our research.
We continued our observations and studied the structure
pine cone
yarrow
mosquito
person
And we became convinced that in these objects, so different at first glance, the same numbers of the Fibonacci sequence were invisibly present.
So, step 1.
Let's take a pine cone:
Let's take a closer look at it:
We notice two series of Fibonacci spirals: one - clockwise, the other - counterclockwise, their number 8 and 13.
Step 2.
Let's take yarrow:
Let's carefully consider the structure of the stems and flowers:
Note that each new branch of the yarrow grows from the axil, and new branches grow from the new branch. By adding up the old and new branches, we found the Fibonacci number in each horizontal plane.
Step 3.
Do Fibonacci numbers appear in the morphology of various organisms? Consider the well-known mosquito:
We see: 3 pairs of legs, head 5 antennae, the abdomen is divided into 8 segments.
Conclusion:
In our research, we saw that in the plants around us, living organisms and even in the human structure, numbers from the Fibonacci sequence manifest themselves, which reflects the harmony of their structure.
The pine cone, the yarrow, the mosquito, and the human being are arranged with mathematical precision.
We were looking for an answer to the question: how does the Fibonacci series manifest itself in the reality around us? But, answering it, we received more and more questions.
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Is the spiral curling or unwinding?
How amazing it is for a person to experience this world!!!
Having found the answer to one question, he gets the next one. If he solves it, he gets two new ones. Once he deals with them, three more will appear. Having solved them too, he will have five unsolved ones. Then eight, then thirteen, 21, 34, 55...
Do you recognize?
Conclusion.
by the creator himself into all objects
A unique code is provided
And the one who is friendly with mathematics,
He will know and understand!
We have studied and analyzed the manifestation of the Fibonacci sequence numbers in the reality around us. We also learned that the patterns of this number series, including the patterns of “Golden” symmetry, are manifested in the energy transitions of elementary particles, in planetary and cosmic systems, in the gene structures of living organisms.
We discovered a surprising mathematical relationship between the number of spirals in plants, the number of branches in any horizontal plane, and the numbers in the Fibonacci sequence. We saw how the morphology of various organisms also obeys this mysterious law. We also saw strict mathematics in the human structure. The human DNA molecule, in which the entire development program of a human being is encrypted, the respiratory system, the structure of the ear - everything obeys certain numerical relationships.
We learned that pine cones, snail shells, ocean waves, animal horns, cyclone clouds and galaxies all form logarithmic spirals. Even the human finger, which is composed of three phalanges in the Golden Ratio relative to each other, takes on a spiral shape when squeezed.
An eternity of time and light years of space separate the pine cone and the spiral galaxy, but the structure remains the same: coefficient 1,618 ! Perhaps this is the primary law governing natural phenomena.
Thus, our hypothesis about the existence of special numerical patterns that are responsible for harmony is confirmed.
Indeed, everything in the world is thought out and calculated by our most important designer - Nature!
We are convinced that Nature has its own laws, expressed using mathematics. And mathematics is a very important tool
to learn the secrets of nature.
List of literature and Internet sites:
1.
Vorobiev N. N. Fibonacci numbers. - M., Nauka, 1984.
2. Ghika M. Aesthetics of proportions in nature and art. - M., 1936.
3. Dmitriev A. Chaos, fractals and information. // Science and Life, No. 5, 2001.
4. Kashnitsky S. E. Harmony woven from paradoxes // Culture and
Life. - 1982.- No. 10.
5. Malay G. Harmony - the identity of paradoxes // MN. - 1982.- No. 19.
6. Sokolov A. Secrets of the golden section // Youth technology. - 1978.- No. 5.
7. Stakhov A.P. Codes of the golden proportion. - M., 1984.
8. Urmantsev Yu. A. Symmetry of nature and the nature of symmetry. - M., 1974.
9. Urmantsev Yu. A. Golden section // Nature. - 1968.- No. 11.
10. Shevelev I.Sh., Marutaev M.A., Shmelev I.P. Golden Ratio/Three
A look at the nature of harmony.-M., 1990.
11. Shubnikov A. V., Koptsik V. A. Symmetry in science and art. -M.: