Fibonacci numbers in nature and human life. Fibonacci golden ratio

Have you ever heard that mathematics is called "the queen of all sciences"? Do you agree with this statement? As long as mathematics remains a boring textbook puzzle for you, you can hardly feel the beauty, versatility and even humor of this science.

But there are topics in mathematics that help to make curious observations on things and phenomena that are common to us. And even try to penetrate the veil of the mystery of the creation of our universe. There are curious patterns in the world that can be described with the help of mathematics.

Introducing Fibonacci Numbers

Fibonacci numbers name the elements number sequence. In it, each next number in the series is obtained by summing two previous numbers.

Sample 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-sided (i.e. covers negative and positive numbers) and tends to infinity in both directions.

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

The formula in this case looks like this:

F n = F n+1 - F n+2 or otherwise you can do it like this: F-n = (-1) n+1 Fn.

What we now know as "Fibonacci numbers" was known to ancient Indian mathematicians long before they were used in Europe. And with this name, in general, one solid 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 the recognition of his descendants as the first major mathematician of Europe during the Middle Ages. Not in last turn thanks to the Fibonacci numbers (which then, we recall, were not yet called that way). which he is in early XIII century described in his work "Liber abaci" ("The Book of the Abacus", 1202).

Traveling with his father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were in this business, and in many other sciences, one of the best specialists). Works of mathematicians of Antiquity and ancient india he read in Arabic translations.

Having properly comprehended everything he read and connected his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the “Book of the Abacus” already mentioned above. In addition to her, he created:

"Practica geometriae" ("Practice of Geometry", 1220);
"Flos" ("Flower", 1225 - a study on cubic equations);
"Liber quadratorum" ("The Book of Squares", 1225 - problems on indefinite quadratic equations).

He was a great lover of mathematical tournaments, so in his treatises he paid much attention to the analysis of various mathematical problems.

Little is known about Leonardo's life. biographical information. As for the name Fibonacci, under which he entered the history of mathematics, it was fixed to him only in the 19th century.

Fibonacci and his problems

After Fibonacci left big number problems that were very popular among mathematicians in the following centuries. We will consider the problem of rabbits, in the solution of which the Fibonacci numbers are used.

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 a new pair rabbits. Also, as you might guess, male and female.

These conditional rabbits are placed in a closed space and breed enthusiastically. It is also stipulated that no 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 - 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 couple gives birth to a new couple, the second couple does not lose time and also gives birth to a new couple, the third couple is just 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 2 months before now). And all this is described by the formula that we have already given above: F n \u003d F n-1 + F n-2.

Thus, we obtain a recurrent (explanation of recursion- below) numerical 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 is in the problem: 377 rabbits will be obtained if all the stated conditions are met.

One of the properties of the Fibonacci sequence is very curious. If we take two consecutive pairs from a row and divide more to less, the result will gradually approach golden ratio(You can read more about it later in the article).

In the language of mathematics, "relationship limit a n+1 to a n equal to the golden ratio.

More problems in number theory

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 square number. It is known about him that if you add 5 to it or subtract 5, you again get a square number.

We invite you to find answers to these questions on your own. You can leave us your options in the comments to this article. And then we will tell you if your calculations were correct.

An explanation about recursion

recursion- definition, description, image of an object or process, which contains this object or process itself. That is, in fact, an object or process is a part of itself.

Recursion finds wide application in mathematics and computer science, and even in art and popular culture.

Fibonacci numbers are defined using recurrent relation. For number n>2 n- e number is (n - 1) + (n - 2).

Explanation of the golden ratio

golden ratio - division of a whole (for example, a segment) into such parts that are correlated according to following principle: most of refers to the smaller one in the same way as the whole value (for example, the sum of two segments) to the larger part.

The first mention of the golden ratio can be found in Euclid's treatise "Beginnings" (about 300 BC). In the context of building a regular rectangle.

The term familiar to us in 1835 was introduced by the German mathematician Martin Ohm.

If you describe the golden ratio approximately, it is a proportional division into two unequal parts: approximately 62% and 38%. AT in numerical terms the golden ratio is a number 1,6180339887 .

The golden ratio finds practical use in fine arts(paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (The Battleship Potemkin by S. Ezenstein) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This view is still popular today. Although, according to the results of research, visually, most people do not perceive such a proportion as the most successful option and consider it too elongated (disproportionate).

Cut length with = 1, a = 0,618, b = 0,382.
Attitude with to a = 1, 618.
Attitude with to b = 2,618

Now back to the Fibonacci numbers. Take two successive terms from its sequence. Divide the larger number by the smaller and get approximately 1.618. And now let's 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 is 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 almost not respected for the beginning of the sequence. But on the other hand, as you move along the row and the numbers increase, it works fine.

And in order to calculate the entire series of Fibonacci numbers, it is enough to know three members of the sequence, following each other. You can see for yourself!

Golden Rectangle and Fibonacci Spiral

Another curious parallel between the Fibonacci numbers and the golden ratio allows us to draw the so-called "golden rectangle": its sides are related in the proportion of 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 build a rectangle with the following parameters: width = 8, length = 13.

And then we break 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 should be equal to the sum sides of 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 from squares with side 1. To which, guided by the principle voiced above, figures with sides are completed, equal numbers Fibonacci. 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. Rather, her special case- Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.

Such a spiral is often found in nature. Mollusk shells are one of the most clear 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 shooting them from satellites.

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

Such amazing “coincidences” cannot but excite the minds and give rise to talk about a certain single algorithm that all phenomena in the life of the Universe obey. Now do you understand why this article is called that way? And what doors amazing worlds can mathematics open up for you?

Fibonacci numbers in nature

The connection between Fibonacci numbers and the golden ratio suggests curious patterns. So curious that it is tempting to try to find sequences similar to Fibonacci numbers in nature and even in the course of historical events. And nature indeed gives rise to such assumptions. But can everything in our life be explained and described with the help of mathematics?

Examples of wildlife that can be described using the Fibonacci sequence:

the order of arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);

the location of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row is clockwise, the other is counterclockwise);

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

Problems in combinatorics

Fibonacci numbers are widely used in solving problems in combinatorics.

Combinatorics- this is a branch of mathematics that deals with the study of a selection of a given number of elements from a designated set, enumeration, etc.

Let's look at examples of combinatorics problems calculated for the level high school(source - http://www.problems.ru/).

Task #1:

Lesha climbs a ladder of 10 steps. He jumps up either one step or two steps at a time. In how many ways can Lesha climb the stairs?

The number of ways that Lesha can climb the stairs from n steps, denote and n. Hence 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. Suppose he jumped two steps the first time. So, according to the condition 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 for the first time Lesha jumped only one step, then we will 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 there are 10 steps according to the condition of the problem, calculate in order all a n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.

Answer: 89 ways.

Task #2:

It is required to find the number of words with a length of 10 letters, which consist only of the letters "a" and "b" and should not contain two letters "b" in a row.

Denote by a n number of words long 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 next term in terms of the previous ones. Therefore, the number of words of length n letters that also do not contain a doubled letter "b" and begin with the letter "a", this a n–1. And if the word is long n letters begins 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 condition of the problem). Therefore, the number of words of length n letters in this case, denoted as a n-2. In both the first and second cases, any word (of length n - 1 and n - 2 letters respectively) without doubled "b".

We were able to explain why a n = a n–1 + a n–2.

Let's calculate now a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 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 cell of the ribbon. On whichever of the cells of the tape he is, he can only move to the right: either one cell, or two. How many ways are there for a grasshopper to jump from the beginning of the ribbon to n th cell?

Let us denote the number of ways the grasshopper moves along the tape up to n th cell as a n. In this case a 1 = a 2= 1. Also in n + 1-th cell the grasshopper can get either from n th cell, or by jumping over it. From here n + 1 = a n – 1 + a n. Where a n = F n – 1.

Answer: F n – 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 unusual phenomenon, like the 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 somehow "lit up" in many works of modern mass culture a wide variety of genres.

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

Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code by which the main characters of the book open the safe.
AT American film 2009 "Mr. Nobody" in one of the episodes, the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode main character should call on phone number, which is essentially the same, but slightly distorted (an extra number after the number 5) sequence: 123-581-1321.
In the 2012 TV series The Connection, the main character, an autistic boy, is able to discern patterns in the events taking place in the world. Including through the Fibonacci numbers. And manage these events also through numbers.
Java game developers 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 a concept album called Illusion. The eighth track is called "Fibonacci". In the verses of the leader of the group Alexander Vasiliev, the sequence of Fibonacci numbers is beaten. For each of the nine consecutive members, there is a corresponding number of rows (0, 1, 1, 2, 3, 5, 8, 13, 21):

0 Set off on the road

1 Clicked one joint

1 One sleeve trembled

2 Everything, get the staff

Everything, get the staff

3 Request for boiling water

The train goes to the river

The train goes to the taiga<…>.

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

Dense food of the Fibonacci wives

It was only for their benefit, not otherwise.

The wives weighed, according to rumor,

Each is like the previous two.

Summing 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. Suddenly, it is you who will be able to unravel the "secret of life, the universe and in general."

Use the formula for Fibonacci numbers when solving problems in combinatorics. You can build on the examples described in this article.

site, with full or partial copying of the material, a link to the source is required.

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Fibonacci numbers and the golden ratio form the basis for unraveling the surrounding world, constructing its shape and optimal visual perception a person with the help of which he can feel beauty and harmony.

The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden ratio was founded as a result of research by ancient scientists on the nature of numbers.

Evidence of the use of the golden ratio by ancient thinkers is given in the book of Euclid's "Beginnings", written back in the 3rd century. BC, who used this rule to construct regular 5-gons. Among the Pythagoreans, this figure is considered sacred, since it is both symmetrical and asymmetrical. The pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber abaci by the Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time gives a pattern of numbers, in a series of which each number is the sum of the 2 previous digits. The sequence of Fibonacci numbers is as follows:

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

The scientist also cited a number of patterns:

Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will be more and more accurate.

If you divide the number from the series by the previous one, then the result will tend to 1.618.

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

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

For practical purposes, they are limited to an approximate value of Φ = 1.618 or Φ = 1.62. In a rounded percentage, the golden ratio is the division of any value in relation to 62% and 38%.

Historically, the division of segment AB by point C into two parts (a smaller segment AC and a larger segment BC) was originally called the golden section, so that AC / BC = BC / AB was true for the lengths of the segments. talking in simple words, the segment is divided by the golden section into two unequal parts so that the smaller part is related to the larger one, as the larger one is to the entire segment. Later this concept was extended to arbitrary quantities.

The number Φ is also called golden number.

The golden ratio has many wonderful properties, but in addition, many fictional properties are attributed to it.

Now the details:

The definition of ZS 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.

That is, if we take the entire segment c as 1, then segment a will be equal to 0.618, segment b - 0.382. Thus, if we take a building, for example, a temple built according to the principle of GS, then with its height, say, 10 meters, the height of the drum with the dome will be 3.82 cm, and the height of the base of the building will be 6.18 cm. (It is clear that the numbers taken equal for clarity)

And what is the relationship between GL and Fibonacci numbers?

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

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

and the ratio of adjacent numbers approaches the ratio of 3S.
So, 21:34 = 0.617, and 34:55 = 0.618.

That is, at the heart of the ZS are the numbers of the Fibonacci sequence.

It is believed that the term "Golden Ratio" was introduced by Leonardo Da Vinci, who said, "let no one, not being a mathematician, dare to read my works" and showed the proportions human body in his famous drawing "Vitruvian Man". “If we tie a human figure – the most perfect creation of the Universe – with a belt and then measure the distance from the belt to the feet, then this value will refer to the distance from the same belt to the top of the head, as the entire height of a person to the length from the belt to the feet.”

A series of Fibonacci numbers is visually modeled (materialized) in the form of a spiral.

And in nature, the 3S spiral looks like this:

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

Seeds in most plants are arranged in a spiral
- A spider weaves a web in a spiral
- A hurricane spirals
- A frightened herd of reindeer scatters in a spiral.
- The DNA molecule is twisted in a double helix. The DNA molecule consists of two vertically intertwined helices 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.
- The embryo develops in the form of a spiral
- Spiral "cochlea in the inner ear"
- Water goes down the drain in a spiral
- Spiral dynamics shows the development of a person's personality and his values in a spiral.
- And of course, the Galaxy itself has the shape of a spiral

Thus, it can be argued that nature itself is built on the principle of the Golden Section, which is why this proportion is more harmoniously perceived by the human eye. It does not require "fixing" or supplementing the resulting picture of the world.

Movie. God number. Irrefutable Proof of God; The number of God. The incontrovertible proof of God.

Golden proportions in the structure of the DNA molecule

All information about physiological features living beings are stored in a microscopic DNA molecule, the structure of which also contains the law of the golden ratio. The DNA molecule consists of two vertically intertwined helices. Each of these spirals is 34 angstroms long and 21 angstroms wide. (1 angstrom is one hundred millionth of a centimeter).

21 and 34 are numbers following one after another in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic helix of the DNA molecule carries the golden section formula 1: 1.618

The golden section in the structure of microworlds

Geometric shapes are not limited to just a triangle, square, five- or hexagon. If we connect these figures in various ways with each other, then we will get new three-dimensional geometric figures. Examples of this are figures such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we did not have to meet in Everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures one can name a tetrahedron (a regular four-sided figure), an octahedron, a dodecahedron, an icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easy to transform, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden section.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. For example, many viruses have a three-dimensional geometric shape icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a certain sequence. In each corner of the icosahedron are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.

The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from London's Birkbeck College A.Klug and D.Kaspar. 13 The Polyo virus was the first to show a logarithmic form. The form of this virus was found to be similar to that of the Rhino 14 virus.

The question arises, how do viruses form such complex three-dimensional forms, the structure of which contains the golden section, which is quite difficult to construct even with our human mind? The discoverer of these forms of viruses, virologist A. Klug makes the following comment:

“Dr. Kaspar and I have shown that for a spherical shell of a virus, the most optimal shape is symmetry like the shape of an icosahedron. This order minimizes the number of connecting elements ... Most of the geodesic hemispherical cubes of Buckminster Fuller are built in a similar way geometric principle. 14 Installation of such cubes requires an extremely precise and detailed explanation scheme. Whereas unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units.

Fibonacci numbers... in nature and life

Leonardo Fibonacci is one of the greatest mathematicians Middle Ages. In one of his works, The Book of Calculations, Fibonacci described the Indo-Arabic calculus and the advantages of using it over the Roman one.

Definition
Fibonacci numbers or Fibonacci Sequence is a numerical sequence that has a number of properties. For example, the sum of two neighboring numbers in the sequence gives the value of the next one (for example, 1+1=2; 2+3=5, etc.), which confirms the existence of the so-called Fibonacci coefficients, i.e. constant ratios.

The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…

Complete definition of Fibonacci numbers

3.

Properties of the Fibonacci Sequence

1. The ratio of each number to the next more and more tends to 0.618 as it increases serial number. The ratio of each number to the previous one tends to 1.618 (reverse to 0.618). The number 0.618 is called (FI).

2. When dividing each number by the next one, the number 0.382 is obtained through one; vice versa - respectively 2.618.

3. Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: … 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

5.

Relationship between the Fibonacci sequence and the "golden section"

The Fibonacci sequence 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 cannot be expressed exactly.

If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13:8), the result will be a value that fluctuates around the irrational value 1.61803398875 ... and after a time either exceeding it or not reaching it. But even having spent Eternity on it, it is impossible to know the ratio exactly, to the last decimal digit. For the sake of brevity, we will give it in the form of 1.618. Special names for this ratio began to be given even before Luca Pacioli (a medieval mathematician) called it the Divine Proportion. Among its modern names are such as the Golden Ratio, the Golden Mean and the ratio of rotating squares. Kepler called this relation one of the "treasures of geometry". In algebra, it is commonly denoted by the Greek letter phi

Let's imagine the golden section on the example of a segment.

Consider a segment with ends A and B. Let point C divide segment AB so that,

AC/CB = CB/AB or

AB/CB = CB/AC.

You can imagine it like this: A-–C--–B

The golden section is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller section is related to the larger one as the larger one is to everything.

Segments of the golden ratio are expressed as an infinite irrational fraction 0.618 ..., if AB is taken as one, AC = 0.382 .. As we already know, the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.

Fibonacci proportions and the golden ratio in nature and history

10.

It is important to note that Fibonacci, as it were, reminded humanity of his sequence. It was known to the ancient Greeks and Egyptians. Indeed, since then, patterns described by Fibonacci coefficients have been found in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other areas. It is simply amazing how many constants can be calculated using the Fibonacci sequence, and how its terms appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a number game, but the most important mathematical expression. natural phenomena of all ever discovered.

11.

The examples below show some interesting applications of this mathematical sequence.

12.

1. The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. The shape of the spirally curled shell attracted the attention of Archimedes. The fact is that the ratio of measurements of the volutes of the shell is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn by this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.

2. Plants and animals. Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf. The process 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 less force, releases a leaf of an even smaller size and ejection again. If the first outlier is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section.

The lizard is viviparous. In the lizard, at first glance, proportions that are pleasing to our eyes are caught - the length of its tail relates to the length of the rest of the body as 62 to 38.

Both in the plant and animal worlds, the shaping tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.

Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry environment. 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. These patterns, as indicated above, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, using this series (Fibonacci) found regularity and order in the distances between the planets of the solar system

However, one case that seemed to be against the law: there was no planet between Mars and Jupiter. Focused observation of this area of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in early XIX in.

The Fibonacci series is widely used: it is used to represent architectonics and living beings, and man-made structures, and the structure of the galaxies. These facts are evidence of independence number series on the conditions of its manifestation, which is one of the signs of its universality.

4. Pyramids. Many have tried to unravel the secrets of the Giza pyramid. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was pre-literate, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the Giza pyramid, so long a mystery to mankind, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of each of its faces was equal to the square of its height.

Triangle area

356 x 440 / 2 = 78320

square area

280 x 280 = 78400

The length of the edge of the base of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the edge of the base, divided by the height, leads to the ratio Ф=1.618. The height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from the Fibonacci sequence. These interesting observations suggest that the construction of the pyramid is based on the proportion Ф=1.618. Some modern scholars tend to interpret that the ancient Egyptians built it for the sole purpose of passing on the knowledge they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.

Pyramids in Mexico. Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids. The idea arises that both Egyptian and Mexican pyramids were erected at approximately the same time by people of a common origin.

There are many more in the universe unsolved mysteries, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for unraveling the world around us, building its shape and optimal visual perception by a person, with the help of which he can feel beauty and harmony.

golden ratio

It is based on the theory of the proportions and ratios of segment divisions, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (of the entire segment):

The result is an equation: x 2 - x - 1=0, which is solved as x=(1±√5)/2.

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

Fibonacci numbers

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

The scientist also cited a number of patterns:

Any number from the series, divided by the next, will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as you move from the beginning of the sequence, this ratio will be more and more accurate.
If you divide the number from the series by the previous one, then the result will tend to 1.618.
One number divided by the next one will show a value tending to 0.382.

Spiral of Archimedes and golden rectangle

Spirals, very common in nature, were explored by Archimedes, who even derived her equation. The shape of the spiral is based on the laws of the golden ratio. When it is untwisted, a length is obtained to which proportions and Fibonacci numbers can be applied, the step increase occurs evenly.

The parallel between the Fibonacci numbers and the golden ratio can also be seen by constructing a "golden rectangle" whose sides are proportional as 1.618:1. It is built by moving from a larger rectangle to smaller ones so that the lengths of the sides will be equal to the numbers from the row. Its construction can be done in the reverse order, starting with the square "1". When connecting the corners of this rectangle with lines in the center of their intersection, a Fibonacci or logarithmic spiral is obtained.

The history of the use of golden proportions

Many ancient architectural monuments of Egypt were built using golden proportions: the famous pyramids of Cheops and others. Architects Ancient Greece they were widely used in the construction architectural objects such as temples, amphitheatres, stadiums. For example, such proportions were used in the construction of the ancient Parthenon temple (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical regularity.

In later centuries, interest in the golden ratio subsided, and the patterns were forgotten, but again resumed in the Renaissance, along with the book of the Franciscan monk L. Pacioli di Borgo "Divine Proportion" (1509). It included illustrations by Leonardo da Vinci, who fixed the new name "golden section". Also, 12 properties of the golden ratio were scientifically proven, and the author talked about how it manifests itself in nature, in art and called it "the principle of building the world and nature."

Vitruvian Man Leonardo

The drawing by which Leonardo da Vinci illustrated the book of Vitruvius in 1492 depicts a figure of a man in 2 positions with arms extended to the sides. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions of the human body (male), described by Leonardo based on their study in the treatises of the Roman architect Vitruvius.

The center of the body as an equidistant point from the end of the arms and legs is the navel, the length of the arms is equal to the height of a person, the maximum width of the shoulders = 1/8 of the height, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.

Since then, the drawing has been used as a symbol showing the internal symmetry of the human body.

The term "Golden Ratio" was used by Leonardo to denote proportional relationships in the human figure. For example, the distance from the waist to the feet is related to the same distance from the navel to the top of the head in the same way as the height to the first length (from the waist down). This calculation is done similarly to the ratio of the segments when calculating the golden ratio and tends to 1.618.

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

Studies of the golden ratio in the 16th-19th centuries

Using the golden ratio and Fibonacci numbers, research work on the issue of proportions have been going on for more than one century. In parallel with Leonardo da Vinci, the German artist Albrecht Dürer was also developing the theory of the correct proportions of the human body. For this, he even created a special compass.

In the 16th century the question of the connection between the Fibonacci number and the golden section was devoted to the work of the astronomer I. Kepler, who first applied these rules to botany.

A new "discovery" awaited the golden ratio in the 19th century. with the publication of "Aesthetic Research" by the German scientist Professor Zeisig. He raised these proportions to the absolute and announced that they are universal for all natural phenomena. They have done research huge amount people, or rather their bodily proportions (about 2 thousand), as a result of which conclusions were drawn about statistically confirmed patterns in the ratios various parts body: lengths of shoulders, forearms, hands, fingers, etc.

Objects of art (vases, architectural structures), musical tones, sizes when writing poems - Zeisig displayed all this through the lengths of segments and numbers, he also introduced the term "mathematical aesthetics". After receiving the results, it turned out that the Fibonacci series is obtained.

Fibonacci number and golden ratio in nature

In the plant and animal world, there is a tendency to form in the form of symmetry, which is observed in the direction of growth and movement. The division into symmetrical parts in which golden proportions are observed is a pattern inherent in many plants and animals.

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

the arrangement of leaves or branches of any plants, as well as the distances, are related to the series of given numbers 1, 1, 2, 3, 5, 8, 13 and so on;
sunflower seeds (scales on cones, pineapple cells), arranged in two rows in twisted spirals in different directions;
the ratio of the length of the tail and the entire body of the lizard;
the shape of the egg, if you draw a line conditionally through its wide part;
the ratio of the size of the fingers on the human hand.

And of course the most interesting shapes represent spiraling snail shells, patterns on the web, the movement of wind inside a hurricane, double helix in DNA and the structure of galaxies - they all include a sequence of Fibonacci numbers.

The use of the golden ratio in art

Researchers looking for examples of the use of the golden section in art examine in detail various architectural objects and paintings. Famous sculptural works are known, the creators of which adhered to golden proportions - the statues of Olympian Zeus, Apollo Belvedere and

One of the creations of Leonardo da Vinci - "Portrait of Mona Lisa" - has been the subject of research by scientists for many years. They found that the composition of the work entirely consists of "golden triangles", united together into a regular pentagon-star. All the works of da Vinci are evidence of how deep his knowledge of the structure and proportions of the human body was, thanks to which he was able to catch the incredibly mysterious smile of the Mona Lisa.

The golden ratio in architecture

As an example, scientists studied the masterpieces of architecture created according to the rules of the "golden section": Pyramids of Egypt, Pantheon, Parthenon, Notre Dame de Paris Cathedral, St. Basil's Cathedral, etc.

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

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

The Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the buildings of the Senate in the Kremlin and the Golitsyn Hospital (now the 1st Clinical named after N.I. Pirogov), was one of the architects who used laws in the design and construction about the golden ratio.

Applying proportions in design

In fashion design, all fashion designers make new images and models, taking into account the proportions of the human body and the rules of the golden ratio, although by nature not all people have ideal proportions.

When planning landscape design and creating voluminous park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the laws " divine proportions". After all, the composition of the park should be focused on creating an impression on the visitor, who will be able to freely navigate in it and find the compositional center.

All elements of the park are in such proportions that, with the help of geometric structure, mutual arrangement, lighting and light, they give the impression of harmony and perfection on a person.

Application of the golden section in cybernetics and technology

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

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

Algorithms and patterns of golden proportions are widely used in modern cybernetics and informatics. One of the simple tasks that beginner programmers are given to solve is to write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.

Modern research on the theory of the golden ratio

Since the middle of the 20th century, interest in the problems and influence of the laws of the golden proportions on human life has increased dramatically, and from many scientists of various professions: mathematicians, ethnos researchers, biologists, philosophers, medical workers, economists, musicians, etc.

Since the 1970s, The Fibonacci Quarterly has been published in the United States, where works on this topic are published. Works appear in the press in which the generalized rules of the golden section and the Fibonacci series are used in various branches of knowledge. For example, to encode information, chemical research, biological, etc.

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

About numbers and formulas that are found in nature. Well, a few words about these same numbers and formulas.

Numbers and formulas in nature are a stumbling block between those who believe in the creation of the universe by someone and those who believe in the creation of the universe by itself. For the question: “If the universe arose by itself, then wouldn’t practically all living and non-living objects be built according to the same scheme, according to the same formulas?”

Well, for this philosophical question we will not answer here (the format of the site is not the same 🙂), but we will announce the formulas. And let's start with the Fibonacci numbers and the Golden Spiral.

So, Fibonacci numbers are elements of a numerical sequence in which each subsequent number is equal to the sum of the two previous numbers. That is, 0 +1=1, 1+1=2, 2+1=3, 3+2=5 and so on.

In total, a series is obtained: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946

Another example of a Fibonacci series: 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178 and so on. You can experiment yourself 🙂

How do Fibonacci numbers show up in nature? Very simple:

Leaf arrangement in plants is described by the Fibonacci sequence. Sunflower seeds, pine cones, flower petals, pineapple cells are also arranged according to the Fibonacci sequence.
The lengths of the phalanges of human fingers are approximately the same as the Fibonacci numbers.
The DNA molecule consists of two vertically intertwined helices 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.

With the help of Fibonacci numbers, you can build a Golden Spiral. So, let's draw a small square with a side of, say, 1. Next, remember the school. How much is 1 2 ? This will be 1. So, let's draw another square next to the first one, close. Next, the next Fibonacci number is 2 (1+1). What is 2 2 ? This will be 4. Let's draw another square close to the first two squares, but now with a side of 2 and an area of 4. Next number is the number 3 (1+2). The square of the number 3 is 9. Draw a square with a side of 3 and an area of 9 next to those already drawn. Next we have a square with a side of 5 and an area of 25, a square with a side of 8 and an area of 64, and so on, ad infinitum.

It's time for the golden spiral. Let's connect the border points between the squares with a smooth curved line. And we will get the same golden spiral, on the basis of which many living and non-living objects in nature are built.

And before moving on to the golden ratio, let's think. Here we have built a spiral based on the squares of the Fibonacci sequence (sequence 1, 1, 2, 3, 5, 8 and squares 1, 1, 4, 9, 25, 64). But what happens if we use not the squares of numbers, but their cubes? The cubes will look like this from the center:

And on the side like this:

Well, when building a spiral, it turns out voluminous golden spiral:

This is how this voluminous golden spiral looks from the side:

But what if we take not the cubes of Fibonacci numbers, but go to the fourth dimension?.. This is a puzzle, right?

However, I have no idea how the volumetric golden ratio manifests itself in nature based on the cubes of Fibonacci numbers, and even more so numbers to the fourth degree. Therefore, we return to the golden section on the plane. So, let's look at our squares again. Mathematically speaking, it looks like this:

That is, we get the golden ratio - where one side is divided into two parts in such a ratio that the smaller part is related to the larger one, as the larger one is to the entire value.

That is, a: b = b: c or c: b = b: a.

On the basis of such a ratio of magnitudes, among other things, a regular pentagon and a pentagram are built:

For reference: to build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471…1528). Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, raised at point O, intersects with the circle at point D. Using a compass, mark the segment CE = ED on the diameter. The length of a side of a regular pentagon inscribed in a circle is DC. We set aside segments DC on the circle and get five points for drawing a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

In general, these are the patterns. Moreover, there are much more diverse patterns than have been described. And now, after all these boring numbers - the promised video clip, where everything is simple and clear:

As you can see, mathematics is indeed present in nature. And not only in the objects listed in the video, but also in many other areas. For example, when a wave hits the shore and twists, it twists along the Golden Spiral. Well, and so on 🙂