The Italian mathematician Leonardo Fibonacci lived in the 13th century and was one of the first in Europe to use Arabic (Indian) numerals. He came up with a somewhat artificial problem about rabbits that are raised on a farm, with all of them being considered females, males are ignored. Rabbits start breeding after they are two months old and then give birth to a rabbit every month. Rabbits never die.

It is necessary to determine how many rabbits will be on the farm in n months, if at the initial moment of time there was only one newborn rabbit.

Obviously, the farmer has one rabbit in the first month and one rabbit in the second month. In the third month there will be two rabbits, in the fourth month there will be three, and so on. Let us denote the number of rabbits in n month like . Thus,
,
,
,
,
, …

We can construct an algorithm to find for any n.

According to the condition of the problem, the total number of rabbits
in n+1 month is decomposed into three components:

one-month-old rabbits, not capable of reproduction, in the amount

;

Thus, we get

. (8.1)

Formula (8.1) allows you to calculate a series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...

The numbers in this sequence are called Fibonacci numbers .

If accept
and
, then with the help of formula (8.1) one can determine all other Fibonacci numbers. Formula (8.1) is called recurrent formula ( recurrence - "return" in Latin).

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

If a n= 1, there is only one solution to the problem. For n= 2 there are 2 options: two single steps or one double step. For n= 3 there are 3 options: three single steps, or one single and one double, or one double and one single.

In the next case n= 4, we have 5 possibilities (1+1+1+1, 2+1+1, 1+2+1, 1+1+2, 2+2).

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

. (8.2)

The resulting formula, like a twin, resembles formula (8.1). However, this does not allow one to identify the number of combinations with Fibonacci numbers . We see, for example, that
, but
. However, there is the following relationship:

.

This is true for n= 1, 2, and is also valid for each n. Fibonacci numbers and number of combinations are calculated using the same formula, but the initial values
,
and
,
they differ.

Example 8.2. This example is of practical importance for problems of error-correcting coding. Find the number of all binary words of length n, not containing multiple zeros in a row. Let's denote this number by . Obviously,
, and the words of length 2 that satisfy our constraint are: 10, 01, 11, i.e.
. Let be
- a word from n characters. If the symbol
, then
can be arbitrary (
)-literal word that does not contain multiple zeros in a row. So the number of words with a unit at the end is
.

If the symbol
, then necessarily
, and the first
symbol
can be arbitrary, taking into account the considered restrictions. Therefore, there is
word length n with zero at the end. Thus, the total number of words of interest to us is

.

Taking into account the fact that
and
, the resulting sequence of numbers is the Fibonacci numbers.

Example 8.3. In Example 7.6 we found that the number of binary words of constant weight t(and length k) equals . Now let's find the number of binary words of constant weight t, not containing multiple zeros in a row.

You can reason like this. Let be
the number of zeros in the words under consideration. Every word has
gaps between the nearest zeros, each of which contains one or more ones. It is assumed that
. Otherwise, there is not a single word without adjacent zeros.

If we remove exactly one unit from each interval, then we get a word of length
containing zeros. Any such word can be obtained in the specified way from some (and only one) k-literal word containing zeros, no two of which are adjacent. Hence, the required number coincides with the number of all words of length
containing exactly zeros, i.e. equals
.

Example 8.4. Let us prove that the sum
equals Fibonacci numbers for any integer . Symbol
stands for smallest integer greater than or equal to . For example, if
, then
; and if
, then
ceil("ceiling"). There is also a symbol
, which stands for largest integer less than or equal to . In English, this operation is called floor ("floor").

If a
, then
. If a
, then
. If a
, then
.

Thus, for the considered cases, the sum is indeed equal to the Fibonacci numbers. We now give a proof for the general case. Since the Fibonacci numbers can be obtained using the recursive equation (8.1), the equality must hold:

.

And it actually does:

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

Sum of Fibonacci Numbers

Let us determine the sum of the first n Fibonacci numbers.

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

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

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

It is easy to see that by adding one to the right side of each equation, we again get the Fibonacci number. The general formula for determining the sum of the first n Fibonacci numbers has the form:

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

This amount must be equal to
.

Reducing the left and right sides of the equation by –1, we obtain equation (6.1).

Formula for Fibonacci numbers

Theorem 8.1. Fibonacci numbers can be calculated using the formula

.

Proof. Let us verify the validity of this formula for n= 0, 1, and then we prove the validity of this formula for an arbitrary n by induction. Let's calculate the ratio of the two closest Fibonacci numbers:

We see that the ratio of these numbers fluctuates around the value of 1.618 (if we ignore the first few values). This property of Fibonacci numbers resemble members of a geometric progression. Accept
, (
). Then the expression

converted to

which after simplification looks like this

.

We have obtained a quadratic equation whose roots are equal to:

Now we can write:

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

For n=0 this difference gives , i.e:
. However, when n=1 we have
. To obtain
must be accepted:
.

Now we have two sequences: and
, which start with the same two numbers and satisfy the same recursive formula. They must be equal:
. The theorem has been proven.

With increasing n member becomes very large while
, and the role of the member is reduced in difference. Therefore, at large n we can write approximately

.

We are ignoring 1/2 (because the Fibonacci numbers increase to infinity as n to infinity).

Attitude
called golden ratio, it is used outside of mathematics (for example, in sculpture and architecture). The golden ratio is the ratio between the diagonal and the side regular pentagon(Fig. 8.1).

Rice. 8.1. Regular pentagon and its diagonals

To denote the golden section, it is customary to use the letter
in honor of the famous Athenian sculptor Phidias.

prime numbers

All natural numbers, large ones, fall into two classes. The first includes numbers that have exactly two natural divisors, one and itself, the second includes all the rest. Numbers of the first class are called simple, and the second constituent. Prime numbers within the first three tens: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

The properties of prime numbers and their connection with all natural numbers were studied by Euclid (3rd century BC). If you write out prime numbers in a row, you can see that their relative density decreases. The first ten of them account for 4, i.e. 40%, for a hundred - 25, i.e. 25%, per thousand - 168, i.e. less than 17%, per million - 78498, i.e. less than 8%, etc. However, their total number is infinite.

Among prime numbers, there are pairs of such, the difference between which is equal to two (the so-called simple twins), but the finiteness or infinity of such pairs has not been proved.

Euclid considered it obvious that by multiplying only prime numbers, one can obtain all natural numbers, and each natural number can be represented as a product of prime numbers in a unique way (up to the order of factors). Thus, the prime numbers form a multiplicative basis of the natural series.

The study of the distribution of primes led to the creation of an algorithm that allows one to obtain tables of primes. Such an algorithm is sieve of Eratosthenes(3rd century BC). This method consists in sifting (for example, by crossing out) those integers of a given sequence
, which are divisible by at least one of the prime numbers less than
.

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

Proof. Euclid's theorem on the infinity of the number of primes will be proved by the method proposed by Leonhard Euler (1707–1783). Euler considered the product over all prime numbers p:

at
. This product converges, and if it is expanded, then, due to the uniqueness of the decomposition of natural numbers into prime factors, it turns out that it is equal to the sum of the series , whence the Euler identity follows:

.

Since at
series on the right diverges (harmonic series), then the Euler identity implies Euclid's theorem.

Russian mathematician P.L. Chebyshev (1821–1894) derived a formula that determines the limits within which the number of primes is contained
, not exceeding X:

,

where
,
.

If you look at the plants and trees around us, you can see how many leaves each of them has. From afar, it seems that the branches and leaves on the plants are arranged randomly, in an arbitrary order. However, in all plants it is miraculously, mathematically precisely planned which branch will grow from where, how 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 the appearance of the plant is already precisely programmed. How many branches will be on the future tree, where the branches will grow, how many leaves will be on each branch, and how, in what order the leaves will be arranged. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), in the number of turns on the stem, in the number of leaves in the cycle, the Fibonacci series manifests itself, and therefore, the law of the golden section also manifests itself.

If you set out to find numerical patterns in wildlife, you will notice that these numbers are often found in various spiral forms, which the plant world is so rich in. For example, leaf cuttings adjoin the stem in a spiral that runs between two adjacent leaves: a full turn - in hazel, - in oak, - in poplar and pear, - in 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.

A clear, symmetrical form of flowers is also subject to a strict law.

Many flowers have the number of petals - exactly the numbers from the Fibonacci series. For example:

iris, 3 lep. buttercup, 5 lep. golden flower, 8 lep. delphinium,

chicory, 21 lep. 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 the man himself is just a storehouse of the number phi.

The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built. The principle of calculating the golden measure on the human body can be depicted in the form of a diagram.

M/m=1.618

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

If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.

Human hand

It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula 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 golden ratio (with the exception of the thumb).

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

A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 found that in the structure of the human lungs there is also a golden ratio.

The peculiarity of the bronchi that make up the lungs of a person lies in their asymmetry. The bronchi are made up of two main airways, one (left) is longer and the other (right) is shorter.

It was found that this asymmetry continues in the branches of the bronchi, in all smaller airways. Moreover, the ratio of the length 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, also created according to the principle of the golden ratio. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.
There is another, more prosaic application of the proportions of the human body. For example, using these ratios, criminal analysts and archaeologists restore the appearance of the whole from fragments of parts of the human body.

The Fibonacci sequence, made famous by the film and book The Da Vinci Code, is a series of numbers deduced by the Italian mathematician Leonardo of Pisa, better known by his pseudonym Fibonacci, in the thirteenth century. The scientist's followers noticed that the formula to which this series of numbers is subject finds its reflection in the world around us and echoes other mathematical discoveries, thereby opening the door to the secrets of the universe for us. In this article, we will explain what the Fibonacci sequence is, consider examples of how this pattern is displayed in nature, and also compare it with other mathematical theories.

Formulation and definition of the concept

The Fibonacci series is a mathematical sequence, each element of which is equal to the sum of the previous two. Let's denote a certain member of the sequence as x n. Thus, we obtain a formula that is valid for the entire series: x n + 2 \u003d x n + x n + 1. In this case, the sequence order will look like this: 1, 1, 2, 3, 5, 8, 13, 21, 34. The next number will be 55, since the sum of 21 and 34 is 55. And so on according to the same principle.

Examples in the environment

If we look at the plant, in particular, at the crown of leaves, we will notice that they bloom in a spiral. Angles are formed between adjacent leaves, which, in turn, form the correct mathematical Fibonacci sequence. Thanks to this feature, each individual leaf that grows on a tree receives the maximum amount of sunlight and heat.

Fibonacci math puzzle

A famous mathematician presented his theory in the form of a riddle. It sounds like this. You can put a pair of rabbits in a closed space in order to find out how many pairs of rabbits will be born in one year. Considering the nature of these animals, the fact that every month a pair is able to produce a new pair, and they become ready for reproduction when they reach two months, as a result, he received his famous series of numbers: 1, 1, 2, 3, 5, 8 , 13, 21, 34, 55, 89, 144 - which shows the number of new pairs of rabbits in each month.

Fibonacci Sequence and Proportional Ratio

This series has several mathematical nuances that must be considered. He, approaching more slowly and more slowly (asymptotically), tends to a certain proportional relationship. But it is irrational. In other words, it is a number with an unpredictable and infinite sequence of decimal numbers in the fractional part. For example, the ratio of any element of the series varies around the figure 1.618, sometimes surpassing it, sometimes reaching it. The next by analogy approaches 0.618. Which is inversely proportional to the number 1.618. If we divide the elements through one, we get 2.618 and 0.382. As you already understood, they are also inversely proportional. The resulting numbers are called Fibonacci ratios. Now let's explain why we performed these calculations.

golden ratio

We distinguish all the objects around us according to certain criteria. One of them is form. Some attract us more, some less, and some do not like at all. It has been noticed that a symmetrical and proportional object is much easier for a person to perceive and evokes a sense of harmony and beauty. A whole image always includes parts of different sizes, which are in a certain ratio with each other. From this follows the answer to the question of what is called the Golden Ratio. This concept means the perfection of the ratio of the whole and parts in nature, science, art, etc. From a mathematical point of view, consider the following example. Take a segment of any length and divide it into two parts in such a way that the smaller part is related to the larger one as the sum (the length of the entire segment) to the larger one. So let's take a cut with for the size of one. part of it a will be equal to 0.618, the second part b, it turns out, is equal to 0.382. Thus, we observe the condition of the Golden Ratio. Segment ratio c to a equals 1.618. And the relation of the parts c and b- 2.618. We get the Fibonacci coefficients already known to us. The golden triangle, the golden rectangle and the golden cuboid are built according to the same principle. It is also worth noting that the proportional ratio of human body parts is close to the Golden Ratio.

Is the Fibonacci sequence the basis of everything?

Let's try to combine the theory of the Golden Section and the well-known series of the Italian mathematician. Let's start with two squares of the first size. Then add another square of the second size on top. Let's draw next to the same figure with a side length equal to the sum of the two previous sides. Similarly, we draw a square of the fifth size. And so you can continue indefinitely, until you get bored. The main thing is that the size of the side of each subsequent square is equal to the sum of the sides of the previous two. We get a series of polygons whose side lengths are Fibonacci numbers. These figures are called Fibonacci rectangles. Let's draw a smooth line through the corners of our polygons and get ... Archimedes' spiral! The increase in the step of this figure, as you know, is always uniform. If you turn on fantasy, then the resulting pattern can be associated with a clam shell. From here we can conclude that the Fibonacci sequence is the basis of proportional, harmonious ratios of elements in the surrounding world.

Mathematical sequence and the universe

If you look closely, then the spiral of Archimedes (somewhere explicitly, but somewhere veiled) and, therefore, the Fibonacci principle can be traced in many familiar natural elements surrounding a person. For example, the same shell of a clam, inflorescences of ordinary broccoli, a sunflower flower, a cone of a coniferous plant, and the like. If we look further, we will see the Fibonacci sequence in infinite galaxies. Even a person, inspired by nature and adopting its forms, creates objects in which the above-mentioned series can be traced. It's time to remember the Golden Section. Along with the Fibonacci pattern, the principles of this theory are traced. There is a version that the Fibonacci sequence is a kind of test of nature to adapt to the more perfect and fundamental logarithmic sequence of the Golden Ratio, which is almost identical, but has no beginning and is infinite. The pattern of nature is such that it must have its own starting point, from which to build on to create something new. The ratio of the first elements of the Fibonacci series is far from the principles of the Golden Ratio. However, the further we continue it, the more this discrepancy is smoothed out. To determine a sequence, you need to know its three elements that follow each other. For the Golden sequence, two are enough. Since it is both an arithmetic and a geometric progression.

Conclusion

Still, based on the foregoing, one can ask quite logical questions: “Where did these numbers come from? Who is this author of the device of the whole world who tried to make it ideal? Was everything always the way he wanted? If so, why did the failure occur? What will happen next?" Finding the answer to one question, you get the next. Solve it - two more appear. If you solve them, you get three more. Having dealt with them, you will receive five unresolved. Then eight, then thirteen, twenty-one, thirty-four, fifty-five...

Fibonacci numbers are elements of a numerical sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, in which each subsequent number is equal to the sum of the previous two 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 celebrated European scientists of his time. Among his greatest achievements is the introduction of Arabic numerals to replace Roman numerals. Fn=Fn-1+Fn-2

The 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 the series is divided by the previous value (for example, 13-^8 or 21-FROM), 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, indefinitely, be accurate to the last digit (even with the most powerful computers built in our time). For the sake of brevity, we will use the number 1.618 as the Fibonacci ratio and ask readers not to forget about this error.

Fibonacci numbers are also important when performing analysis. Euclid's algorithm for determining the greatest common divisor of two numbers. The Fibonacci numbers come from Pascal's triangle diagonal formula (binomial coefficients).

Fibonacci numbers have been linked to the Golden Ratio.

The golden ratio was known in ancient Egypt and Babylon, in India and China. What is the "golden section"? The answer is still unknown. Fibonacci numbers are really relevant for 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 in studying the 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 determined the very concept of Fibonacci numbers, their properties. I also found out interesting patterns in wildlife, directly in the structure of sunflower seeds.

On a sunflower, the seeds line up in spirals, and the number of spirals going in the other direction is different - they are consecutive Fibonacci numbers.

This sunflower has 34 and 55.

The same is observed on the fruits of pineapple, 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 stem legs of a plant, 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.

Considering the arrangement of leaves on the stem of plants, you can see that between each pair of leaves (A and C) the third is located in the place of the golden section (B)

Another interesting property of the Fibonacci number is that the product and the 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 in the course of my research. The Fibonacci number is also found in programming and economic forecasts, in painting, architecture and music. The paintings of 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 have found that Fibonacci numbers are being used in our area as well. For example, platbands of houses, gables.

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Introduction

THE HIGHEST PURPOSE OF MATHEMATICS IS TO FIND THE HIDDEN ORDER IN THE CHAOS THAT SURROUNDS US.

Viner N.

A person strives for knowledge all his life, tries to study the world around him. And in the process of observation, he has questions that need to be answered. Answers are found, but new questions appear. In archaeological finds, in the traces of civilization, distant from each other in time and space, one and the same element is found - a pattern in the form of a spiral. Some consider it a symbol of the sun and associate it with the legendary Atlantis, but its true meaning is unknown. What do the shapes of a galaxy and an atmospheric cyclone, the arrangement of leaves on a stem and seeds in a sunflower have in common? These patterns come down to the so-called "golden" spiral, the amazing Fibonacci sequence, discovered by the great Italian mathematician of the 13th century.

History of Fibonacci Numbers

For the first time about what Fibonacci numbers are, I heard from a mathematics teacher. But, besides, how the sequence of these numbers is formed, I did not know. This is what this sequence is actually famous for, how it affects a person, and I want to tell you. Little is known about Leonardo Fibonacci. There is not even an exact date of his birth. It is known that he was born in 1170 in the family of a merchant, in the city of Pisa in Italy. Fibonacci's father was often in Algiers on business, and Leonardo studied mathematics there with Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is the "Book of the abacus", which contains almost all the arithmetic and algebraic information of that time. 2

Fibonacci numbers are a sequence of numbers with a number of properties. Fibonacci discovered this numerical sequence by accident when he tried to solve a practical problem about rabbits in 1202. “Someone placed a pair of rabbits in a certain place, enclosed on all sides by a wall, in order to find out how many pairs of rabbits will be born during the year, if the nature of rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth. When solving the problem, he took into account that each pair of rabbits gives birth to two more pairs during their life, and then dies. This is how the sequence of numbers appeared: 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence, each next number is equal to the sum of the two previous ones. It's called the Fibonacci sequence. Mathematical properties of a sequence

I wanted to explore this sequence, and I identified some of its properties. This rule is of great importance. The sequence slowly approaches some constant ratio of about 1.618, and the ratio of any number to the next is about 0.618.

One can notice a number of curious properties of Fibonacci numbers: two neighboring numbers are coprime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you choose any 10 neighboring numbers from the Fibonacci sequence and add them together, you will always get a number that is a multiple of 11. But that's not all. Each sum is equal to the number 11 multiplied by the seventh member of the given sequence. And here is another interesting feature. For any n, the sum of the first n members of the sequence will always be equal to the difference of the (n + 2) -th and first member of the sequence. This fact can be expressed by the formula: 1+1+2+3+5+…+an=a n+2 - 1. Now we have the following trick: to find the sum of all terms

sequence between two given members, it suffices to find the difference of the corresponding (n+2)-x members. For example, a 26 + ... + a 40 \u003d a 42 - a 27. Now let's look for a connection between Fibonacci, Pythagoras and the "golden section". The most famous evidence of the mathematical genius of mankind is the Pythagorean theorem: in any right triangle, the square of the hypotenuse is equal to the sum of the squares of its legs: c 2 \u003d b 2 + a 2. From a geometric point of view, we can consider all the sides of a right triangle as the sides of three squares built on them. The Pythagorean theorem says that the total area of ​​the squares built on the legs of a right triangle is equal to the area of ​​the square built on the hypotenuse. If the lengths of the sides of a right triangle are integers, then they form a group of three numbers called Pythagorean triples. Using the Fibonacci sequence, you can find such triples. Take any four consecutive numbers from the sequence, for example, 2, 3, 5 and 8, and construct three more numbers as follows: 1) the product of the two extreme numbers: 2*8=16; 2) the double product of the two numbers in the middle: 2* (3 * 5) \u003d 30; 3) the sum of the squares of two average numbers: 3 2 +5 2 \u003d 34; 34 2 =30 2 +16 2 . This method works for any four consecutive Fibonacci numbers. Predictably, any three consecutive numbers of the Fibonacci series behave in a predictable way. If you multiply the two extremes of them and compare the result with the square of the average number, then the result will always differ by one. For example, for numbers 5, 8 and 13 we get: 5*13=8 2 +1. If we consider this property from the point of view of geometry, we can notice something strange. Divide the square

size 8x8 (total 64 small squares) into four parts, the lengths of the sides of which are equal to the Fibonacci numbers. Now from these parts we will build a rectangle measuring 5x13. Its area is 65 small squares. Where does the extra square come from? The thing is that a perfect rectangle is not formed, but tiny gaps remain, which in total give this additional unit of area. Pascal's triangle also has a connection with the Fibonacci sequence. You just need to write the lines of Pascal's triangle one under the other, and then add the elements diagonally. Get the Fibonacci sequence.

Now consider a "golden" rectangle, one side of which is 1.618 times longer than the other. At first glance, it may seem like an ordinary rectangle to us. However, let's do a simple experiment with two ordinary bank cards. Let's put one of them horizontally and the other vertically so that their lower sides are on the same line. If we draw a diagonal line in a horizontal map and extend it, we will see that it will pass exactly through the upper right corner of the vertical map - a pleasant surprise. Maybe this is an accident, or maybe such rectangles and other geometric shapes using the "golden ratio" are especially pleasing to the eye. Did Leonardo da Vinci think about the golden ratio while working on his masterpiece? This seems unlikely. However, it can be argued that he attached great importance to the connection between aesthetics and mathematics.

Fibonacci numbers in nature

The connection of the golden section with beauty is not only a matter of human perception. It seems that nature itself has allocated a special role to F. If squares are sequentially entered into the "golden" rectangle, then an arc is drawn in each square, then an elegant curve is obtained, which is called a logarithmic spiral. It is not a mathematical curiosity at all. 5

On the contrary, this wonderful line is often found in the physical world: from the shell of a nautilus to the arms of galaxies, and in the elegant spiral of the petals of a full-blown rose. The connections between the golden ratio and Fibonacci numbers are numerous and unexpected. Consider a flower that looks very different from a rose - a sunflower with seeds. The first thing we see is that the seeds are arranged in two kinds of spirals: clockwise and counterclockwise. If we count the clockwise spirals, we get two seemingly ordinary numbers: 21 and 34. This is not the only example when you can find Fibonacci numbers in the structure of plants.

Nature gives us numerous examples of the arrangement of homogeneous objects described by Fibonacci numbers. In the various spiral arrangements of small plant parts, two families of spirals can usually be seen. In one of these families, the spirals curl clockwise, and in the other - counterclockwise. Spiral numbers of one type and another often turn out to be neighboring Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from bottom left to right up. On many cones, the seeds are arranged in three spirals, gently winding around the stem of the cone. They are arranged in five spirals, winding steeply in the opposite direction. In large cones, it is possible to observe 5 and 8, and even 8 and 13 spirals. The Fibonacci spirals are also clearly visible on the pineapple: there are usually 8 and 13 of them.

The chicory shoot makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of lesser force, releases an even smaller leaf and ejection again. Its growth impulses gradually decrease in proportion to the "golden" section. To appreciate the huge role of Fibonacci numbers, just look at the beauty of the nature around us. Fibonacci numbers can be found in quantity

branches on the stem of each growing plant and in the number of petals.

Let's count the petals of some flowers - the iris with its 3 petals, the primrose with 5 petals, the ragweed with 13 petals, the daisy with 34 petals, the aster with 55 petals, and so on. Is this a coincidence, or is it the law of nature? Look at the stems and flowers of the yarrow. Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of the "Golden" numbers found in nature. These laws operate regardless of our consciousness and the desire to accept them or not. The patterns of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms, in the structure of individual human organs and the body as a whole, and also manifest themselves in biorhythms and the functioning of the brain and visual perception.

Fibonacci numbers in architecture

The Golden Ratio also manifests itself in many remarkable architectural creations throughout the history of mankind. It turns out that even ancient Greek and Egyptian mathematicians knew these coefficients long before Fibonacci and called them the "golden section". The principle of the "golden section" was used by the Greeks in the construction of the Parthenon, the Egyptians - the Great Pyramid of Giza. Advances in building technology and the development of new materials opened up new possibilities for 20th-century architects. American Frank Lloyd Wright was one of the main proponents of organic architecture. Shortly before his death, he designed the Solomon Guggenheim Museum in New York, which is an inverted spiral, and the interior of the museum resembles a nautilus shell. Polish-Israeli architect Zvi Hecker also used spiral structures in the design of the Heinz Galinski School in Berlin, completed in 1995. Hecker started with the idea of ​​a sunflower with a central circle, from where

all architectural elements diverge. The building is a combination

orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and controlled chaos of nature. Its architecture mimics a plant that follows the movement of the sun, so the classrooms are lit up throughout the day.

In Quincy Park, located in Cambridge, Massachusetts (USA), the "golden" spiral can often be found. The park was designed in 1997 by artist David Phillips and is located near the Clay Mathematical Institute. This institution is a well-known center for mathematical research. In Quincy Park, you can walk among the "golden" spirals and metal curves, reliefs of two shells and a rock with a square root symbol. On the plate is written information about the "golden" proportion. Even bike parking uses the F symbol.

Fibonacci numbers in psychology

In psychology, there are turning points, crises, upheavals that mark the transformation of the structure and functions of the soul on a person's life path. If a person has successfully overcome these crises, then he becomes able to solve problems of a new class, which he had not even thought about before.

The presence of fundamental changes gives reason to consider the time of life as a decisive factor in the development of spiritual qualities. After all, nature measures time for us not generously, “no matter how much it will be, so much will be,” but just enough so that the development process materializes:

in the structures of the body;

in feelings, thinking and psychomotor - until they acquire harmony necessary for the emergence and launch of the mechanism

creativity;

in the structure of human energy potential.

The development of the body cannot be stopped: the child becomes an adult. With the mechanism of creativity, everything is not so simple. Its development can be stopped and its direction changed.

Is there a chance to catch up with time? Undoubtedly. But for this you need to do a lot of work on yourself. What develops freely, naturally, does not require special efforts: the child develops freely and does not notice this enormous work, because the process of free development is created without violence against oneself.

How is the meaning of the life path understood in everyday consciousness? The inhabitant sees it like this: at the foot - the birth, at the top - the prime of life, and then - everything goes downhill.

The wise man will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth ... Why is that? Few people are able to answer, although everyone is sure that these are closed, integral stages of life.

To find out how the mechanism of creativity develops, V.V. Klimenko used mathematics, namely the laws of Fibonacci numbers and the proportion of the "golden section" - the laws of nature and human life.

Fibonacci numbers divide our life into stages according to the number of years lived: 0 - the beginning of the countdown - the child was born. He still lacks not only psychomotor skills, thinking, feelings, imagination, but also operational energy potential. He is the beginning of a new life, a new harmony;

1 - the child has mastered walking and masters the immediate environment;

2 - understands speech and acts using verbal instructions;

3 - acts through the word, asks questions;

5 - "age of grace" - the harmony of psychomotor, memory, imagination and feelings, which already allow the child to embrace the world in all its integrity;

8 - feelings come to the fore. They are served by imagination, and thinking, by the forces of its criticality, is aimed at supporting the internal and external harmony of life;

13 - the mechanism of talent begins to work, aimed at transforming the material acquired in the process of inheritance, developing one's own talent;

21 - the mechanism of creativity has approached a state of harmony and attempts are being made to perform talented work;

34 - harmony of thinking, feelings, imagination and psychomotor skills: the ability to brilliant work is born;

55 - at this age, subject to the preserved harmony of soul and body, a person is ready to become a creator. Etc…

What are Fibonacci serifs? They can be compared to dams on the path of life. These dams await each of us. First of all, it is necessary to overcome each of them, and then patiently raise your level of development, until one day it falls apart, opening the way to the next free flow.

Now that we understand the meaning of these nodal points of age development, let's try to decipher how it all happens.

At 1 year the child learns to walk. Before that, he knew the world with the front of his head. Now he knows the world with his hands - the exclusive privilege of man. The animal moves in space, and he, cognizing, masters the space and masters the territory on which he lives.

2 years understands the word and acts in accordance with it. It means that:

the child learns the minimum number of words - meanings and patterns of action;

yet does not separate itself from the environment and is merged into integrity with the environment,

Therefore, he acts on someone else's instructions. At this age, he is the most obedient and pleasant for parents. From a man of the senses, the child turns into a man of knowledge.

3 years- action with the help of one's own word. The separation of this person from the environment has already taken place - and he is learning to be an independently acting person. Hence he:

consciously opposes the environment and parents, kindergarten teachers, etc.;

is aware of its sovereignty and fights for independence;

tries to subjugate close and well-known people to his will.

Now for a child, a word is an action. This is where the acting person begins.

5 years- Age of Grace. He is the personification of harmony. Games, dances, dexterous movements - everything is saturated with harmony, which a person tries to master with his own strength. Harmonious psychomotor contributes to bringing to a new state. Therefore, the child is directed to psychomotor activity and strives for the most active actions.

Materialization of the products of the work of sensitivity is carried out through:

the ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all sense organs work for this process);

ability to design the outside world, including yourself

(creation of a second nature, hypotheses - to do both tomorrow, build a new machine, solve a problem), by the forces of critical thinking, feelings and imagination;

the ability to create a second, man-made nature, products of activity (implementation of the plan, specific mental or psychomotor actions with specific objects and processes).

After 5 years, the imagination mechanism comes forward and begins to dominate the rest. The child does a gigantic job, creating fantastic images, and lives in the world of fairy tales and myths. The hypertrophy of the child's imagination causes surprise in adults, because the imagination does not correspond to reality in any way.

8 years- feelings come to the fore and their own measurements of feelings (cognitive, moral, aesthetic) arise when the child unmistakably:

evaluates the known and the unknown;

distinguishes the moral from the immoral, the moral from the immoral;

beauty from what threatens life, harmony from chaos.

13 years old- the mechanism of creativity begins to work. But that doesn't mean it's working at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If even in this age period of development harmony is preserved, which almost all the time rebuilds its structure, then the child will painlessly get to the next dam, overcome it imperceptibly and will live at the age of a revolutionary. At the age of a revolutionary, the youth must take a new step forward: to separate from the nearest society and live in it a harmonious life and activity. Not everyone can solve this problem that arises before each of us.

21 years old If a revolutionary has successfully overcome the first harmonious peak of life, then his mechanism of talent is capable of fulfilling a talented

work. Feelings (cognitive, moral, or aesthetic) sometimes overshadow thinking, but in general, all elements work in harmony: feelings are open to the world, and logical thinking is able to name and find measures of things from this peak.

The mechanism of creativity, developing normally, reaches a state that allows it to receive certain fruits. He starts to work. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by feelings and thinking, antagonism arises between them. Feelings win. This ability is gradually gaining power, and the boy begins to use it.

34 years- balance and harmony, productive effectiveness of talent. Harmony of thinking, feelings and imagination, psychomotor skills, which is replenished with optimal energy potential, and the mechanism as a whole - an opportunity is born to perform brilliant work.

55 years- a person can become a creator. The third harmonious peak of life: thinking subdues the power of feelings.

Fibonacci numbers name the stages of human development. Whether a person goes through this path without stopping depends on parents and teachers, the educational system, and then on himself and on how a person will learn and overcome himself.

On the path of life, a person discovers 7 objects of relationships:

From birthday to 2 years - the discovery of the physical and objective world of the immediate environment.

From 2 to 3 years - the discovery of oneself: "I am Myself."

From 3 to 5 years - speech, the effective world of words, harmony and the "I - You" system.

From 5 to 8 years old - the discovery of the world of other people's thoughts, feelings and images - the "I - We" system.

From 8 to 13 years old - the discovery of the world of tasks and problems solved by the geniuses and talents of mankind - the system "I - Spirituality".

From 13 to 21 years old - the discovery of the ability to independently solve well-known tasks, when thoughts, feelings and imagination begin to work actively, the "I - Noosphere" system arises.

From 21 to 34 years old - the discovery of the ability to create a new world or its fragments - the realization of the self-concept "I am the Creator".

The life path has a space-time structure. It consists of age and individual phases, determined by many parameters of life. A person masters to a certain extent the circumstances of his life, becomes the creator of his history and the creator of the history of society. A truly creative attitude to life, however, does not appear immediately and not even in every person. There are genetic links between the phases of the life path, and this determines its natural character. It follows that, in principle, it is possible to predict future development on the basis of knowledge of its early phases.

Fibonacci numbers in astronomy

It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found regularity and order in the distances between the planets of the solar system. But one case seemed to be against the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the XIX century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.

Conclusion

In the process of research, I found out that Fibonacci numbers are widely used in the technical analysis of stock prices. One of the simplest ways to use Fibonacci numbers in practice is to determine the length of time after which an event will occur, for example, a price change. The analyst counts a certain number of Fibonacci days or weeks (13,21,34,55, etc.) from the previous similar event and makes a forecast. But this is too hard for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are the statue in front of the Leaning Tower of Pisa and two streets that bear his name, one in Pisa and the other in Florence. And yet, in connection with everything I have seen and read, quite natural questions arise. Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? What will be next? Finding the answer to one question, you get the next. If you solve it, you get two new ones. Deal with them, three more will appear. Having solved them, you will acquire five unresolved ones. Then eight, thirteen, and so on. Do not forget that there are five fingers on two hands, two of which consist of two phalanges, and eight of which consist of three.

Literature:

Voloshinov A.V. "Mathematics and Art", M., Enlightenment, 1992

Vorobyov N.N. "Fibonacci numbers", M., Nauka, 1984

Stakhov A.P. "The Da Vinci Code and the Fibonacci Series", Peter Format, 2006

F. Corvalan “The Golden Ratio. Mathematical language of beauty”, M., De Agostini, 2014

Maksimenko S.D. "Sensitive periods of life and their codes".

"Fibonacci numbers". Wikipedia