The rule of the golden ratio in the human body. Divine harmony: what is the golden ratio in simple words

This harmony is striking in its scale...

Hello, friends!

Have you heard anything about Divine Harmony or the Golden Ratio? Have you ever thought about why something seems ideal and beautiful to us, but something repels us?

If not, then you have successfully come to this article, because in it we will discuss the golden ratio, find out what it is, what it looks like in nature and in humans. Let's talk about its principles, find out what the Fibonacci series is and much more, including the concept of the golden rectangle and the golden spiral.

Yes, the article has a lot of images, formulas, after all, the golden ratio is also mathematics. But everything is described in fairly simple language, clearly. And at the end of the article, you will find out why everyone loves cats so much =)

What is the golden ratio?

To put it simply, the golden ratio is a certain rule of proportion that creates harmony?. That is, if we do not violate the rules of these proportions, then we get a very harmonious composition.

The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole.

But besides this, the golden ratio is mathematics: it has a specific formula and a specific number. Many mathematicians, in general, consider it the formula of divine harmony, and call it “asymmetrical symmetry”.

The golden ratio has reached our contemporaries since the times of Ancient Greece, however, there is an opinion that the Greeks themselves had already spied the golden ratio among the Egyptians. Because many works of art of Ancient Egypt are clearly built according to the canons of this proportion.

It is believed that Pythagoras was the first to introduce the concept of the golden ratio. The works of Euclid have survived to this day (he used the golden ratio to build regular pentagons, which is why such a pentagon is called “golden”), and the number of the golden ratio is named after the ancient Greek architect Phidias. That is, this is our number “phi” (denoted by the Greek letter φ), and it is equal to 1.6180339887498948482... Naturally, this value is rounded: φ = 1.618 or φ = 1.62, and in percentage terms the golden ratio looks like 62% and 38%.

What is unique about this proportion (and believe me, it exists)? Let's first try to figure it out using an example of a segment. So, we take a segment and divide it into unequal parts in such a way that its smaller part relates to the larger one, as the larger part relates to the whole. I understand, it’s not very clear yet what’s what, I’ll try to illustrate it more clearly using the example of segments:

So, we take a segment and divide it into two others, so that the smaller segment a relates to the larger segment b, just as the segment b relates to the whole, that is, the entire line (a + b). Mathematically it looks like this:

This rule works indefinitely; you can divide segments as long as you like. And, see how simple it is. The main thing is to understand once and that’s it.

But now let’s look at a more complex example, which comes across very often, since the golden ratio is also represented in the form of a golden rectangle (the aspect ratio of which is φ = 1.62). This is a very interesting rectangle: if we “cut off” a square from it, we will again get a golden rectangle. And so on endlessly. See:

But mathematics would not be mathematics if it did not have formulas. So, friends, now it will “hurt” a little. I hid the solution to the golden ratio under a spoiler; there are a lot of formulas, but I don’t want to leave the article without them.

Fibonacci series and golden ratio

We continue to create and observe the magic of mathematics and the golden ratio. In the Middle Ages there was such a comrade - Fibonacci (or Fibonacci, they spell it differently everywhere). He loved mathematics and problems, he also had an interesting problem with the reproduction of rabbits =) But that’s not the point. He discovered a number sequence, the numbers in it are called “Fibonacci numbers”.

The sequence itself looks like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... and so on ad infinitum.

In other words, the Fibonacci sequence is a sequence of numbers where each subsequent number is equal to the sum of the previous two.

What does the golden ratio have to do with it? You'll see now.

Fibonacci Spiral

To see and feel the whole connection between the Fibonacci number series and the golden ratio, you need to look at the formulas again.

In other words, from the 9th term of the Fibonacci sequence we begin to obtain the values of the golden ratio. And if we visualize this whole picture, we will see how the Fibonacci sequence creates rectangles closer and closer to the golden rectangle. This is the connection.

Now let's talk about the Fibonacci spiral, it is also called the “golden spiral”.

The golden spiral is a logarithmic spiral whose growth coefficient is φ4, where φ is the golden ratio.

In general, from a mathematical point of view, the golden ratio is an ideal proportion. But this is just the beginning of her miracles. Almost the entire world is subject to the principles of the golden ratio; nature itself created this proportion. Even esotericists see numerical power in it. But we will definitely not talk about this in this article, so in order not to miss anything, you can subscribe to site updates.

Golden ratio in nature, man, art

Before we begin, I would like to clarify a number of inaccuracies. Firstly, the very definition of the golden ratio in this context is not entirely correct. The fact is that the very concept of “section” is a geometric term, always denoting a plane, but not a sequence of Fibonacci numbers.

And, secondly, the number series and the ratio of one to the other, of course, have been turned into a kind of stencil that can be applied to everything that seems suspicious, and one can be very happy when there are coincidences, but still, common sense should not be lost.

However, “everything was mixed up in our kingdom” and one became synonymous with the other. So, in general, the meaning is not lost from this. Now let's get down to business.

You will be surprised, but the golden ratio, or rather the proportions as close as possible to it, can be seen almost everywhere, even in the mirror. Don't believe me? Let's start with this.

You know, when I was learning to draw, they explained to us how easier it is to build a person’s face, his body, and so on. Everything must be calculated relative to something else.

Everything, absolutely everything is proportional: bones, our fingers, palms, distances on the face, the distance of outstretched arms in relation to the body, and so on. But even this is not all, the internal structure of our body, even this, is equal or almost equal to the golden section formula. Here are the distances and proportions:

from shoulders to crown to head size = 1:1.618

from the navel to the crown to the segment from the shoulders to the crown = 1:1.618

from navel to knees and from knees to feet = 1:1.618

from the chin to the extreme point of the upper lip and from it to the nose = 1:1.618

Isn't this amazing!? Harmony in its purest form, both inside and outside. And that is why, at some subconscious level, some people do not seem beautiful to us, even if they have a strong, toned body, velvety skin, beautiful hair, eyes, etc., and everything else. But, all the same, the slightest violation of the proportions of the body, and the appearance already slightly “hurts the eyes.”

In short, the more beautiful a person seems to us, the closer his proportions are to ideal. And this, by the way, can be attributed not only to the human body.

Golden ratio in nature and its phenomena

A classic example of the golden ratio in nature is the shell of the mollusk Nautilus pompilius and the ammonite. But this is not all, there are many more examples:

in the curls of the human ear we can see a golden spiral;

its same (or close to it) in the spirals along which galaxies twist;

and in the DNA molecule;

According to the Fibonacci series, the center of a sunflower is arranged, cones grow, the middle of flowers, a pineapple and many other fruits.

Friends, there are so many examples that I’ll just leave the video here (it’s just below) so as not to overload the article with text. Because if you dig into this topic, you can go deeper into the following jungle: even the ancient Greeks proved that the Universe and, in general, all space is planned according to the principle of the golden ratio.

You will be surprised, but these rules can be found even in sound. See:

The highest point of sound that causes pain and discomfort in our ears is 130 decibels.

We divide the proportion 130 by the golden ratio number φ = 1.62 and we get 80 decibels - the sound of a human scream.

We continue to divide proportionally and get, let’s say, the normal volume of human speech: 80 / φ = 50 decibels.

Well, the last sound that we get thanks to the formula is a pleasant whispering sound = 2.618.

Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. I haven’t tested it, and I don’t know how true this theory is, but you must agree, it sounds impressive.

One can read the highest beauty and harmony in absolutely everything living and non-living.

The main thing is not to get carried away with this, because if we want to see something in something, we will see it, even if it is not there. For example, I paid attention to the design of the PS4 and saw the golden ratio there =) However, this console is so cool that I wouldn’t be surprised if the designer really did something clever there.

Golden ratio in art

This is also a very large and extensive topic that is worth considering separately. Here I will just note a few basic points. The most remarkable thing is that many works of art and architectural masterpieces of antiquity (and not only) were made according to the principles of the golden ratio.

Egyptian and Mayan pyramids, Notre Dame de Paris, Greek Parthenon and so on.

In the musical works of Mozart, Chopin, Schubert, Bach and others.

In painting (this is clearly visible): all the most famous paintings by famous artists are made taking into account the rules of the golden ratio.

These principles can be found in Pushkin’s poems and in the bust of the beautiful Nefertiti.

Even now, the rules of the golden ratio are used, for example, in photography. Well, and of course, in all other arts, including cinematography and design.

Golden Fibonacci cats

And finally, about cats! Have you ever wondered why everyone loves cats so much? They've taken over the Internet! Cats are everywhere and it's wonderful =)

And the whole point is that cats are perfect! Don't believe me? Now I’ll prove it to you mathematically!

Do you see? The secret is revealed! Cats are ideal from the point of view of mathematics, nature and the Universe =)

*I'm kidding, of course. No, cats are really ideal) But no one has measured them mathematically, probably.

That's basically it, friends! We'll see you in the next articles. Good luck to you!

P.S. Images taken from medium.com.

What figure is considered beautiful in women, and what figure is considered beautiful in men? This sounds surprising, but our perception of female or male beauty depends not on a person’s “taste,” but on numbers. Let us ask ourselves the question, why is a man with broad shoulders considered attractive, and a woman with rounded shapes? The male X-figure has always emphasized masculinity and strength. In women, the hourglass figure has been associated with fertility since ancient times. We look at people's appearance through the prism of many generations of human eyes, and our choice, it turns out, has already been proven by numbers.

Golden ratio of man is a number that describes the proportions of a person's entire body (for example, the length of the legs and arms compared to the length of the torso) and determines which of these proportions looks best.

Since the Middle Ages, sculptors and artists have known about the “golden ratio” and used it to depict the ideal body in their works. Today, plastic surgeons and dentists use this formula to reconstruct the face.

How to determine " human golden ratio».
Typically, the proportion looks like 1:1.618. Let's explain: if the length of your arm is 1, then the sum of the length of your arm plus your forearm should equal 1.618. Accordingly, if the leg equals 1, then the leg plus the lower leg is already 1.618.

The face is a part of the body where there are many examples of the “golden ratio”. The human head forms a so-called “golden rectangle”, in its center are the human eyes. The nose and mouth are in the "golden sections", between the chin and eyes.

All this is interesting for us from the point of view of physiology, but no less from the point of view of psychology. The human brain searches for symmetry and balance everywhere, or tries to create it. Hence the conclusion that we usually judge the beauty of the human body based on how similar it is to an ideally symmetrical body, and it is precisely this ideal symmetry that the “golden ratio” can describe.

How can we use this information to enhance everyday appeal?

First, you need to understand that training your body should be symmetrical. Let's say there are places that you cannot change. All beauty salons taken together cannot make a person’s body 100% perfect, and is this really necessary?

The most visible part that can be changed is the relationship of the shoulders and lower back. For a man, wider shoulders than the waist and hips indicate his strength and masculinity, making the body especially attractive to the female gaze. It is the “golden ratio” that allows us to determine how wide a man’s shoulders should be.

What to do:

First you need to decide on your goal: increasing muscle size or dieting.

If your goal is diet, then strictly measure and regulate the problematic part of the body that, in your opinion, should be wider. If the goal is to increase muscle mass, then it is necessary to measure the part that, according to the proportion, should be narrower.

Focus your attention on changing one or another part of your body. As a rule, for men, if you are dieting, you need to focus on changing your waist size, and when building muscle, put effort into changing your shoulder width.

From open spaces for educational purposes)

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's painting "Mona Lisa", a sunflower, a snail, a pine cone and human fingers have in common?

The answer to this question is hidden in the amazing numbers that have been discovered Italian medieval mathematician Leonardo of Pisa, better known by the name Fibonacci (born about 1170 - died after 1228), Italian mathematician . Traveling around the East, he became acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called after the famous mathematician. The amazing essence of the Fibonacci number sequence is that that each number in this sequence is obtained from the sum of the two previous numbers.

So, the numbers forming the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence. There is one very interesting feature about Fibonacci numbers. When dividing any number from the sequence by the number in front of it in the series, the result will always be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. (Approx. irrational number, i.e. a number whose decimal representation is infinite and non-periodic)

Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series... It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden ratio, the golden mean, or the golden proportion. . In algebra, this number is denoted by the Greek letter phi (Ф)

So, Golden ratio = 1:1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio.

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.

The most important book of all modern architects, E. Neufert's reference book "Building Design", contains basic calculations of the parameters of the human torso, which contain the golden proportion.

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

M/m=1.618

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.

In addition to this, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1:1.618;

* the distance from shoulder level to the top of the head and the size of the head is 1:1.618;

* the distance from the navel point to the crown of the head and from shoulder level to the crown of the head is 1:1.618;

* the distance of the navel point to the knees and from the knees to the feet is 1:1.618;

* the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1.618;

* the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1:1.618;

* the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618:

The golden ratio in human facial features as a criterion of perfect beauty.

In the structure of human facial features there are also many examples that are close in value to the golden ratio formula. However, do not immediately rush for a ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and artists, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.

For example, if we sum up the width of the two front upper teeth and divide this sum by the height of the teeth, then, having obtained the golden ratio number, we can say that the structure of these teeth is ideal.

There are other embodiments of the golden ratio rule on the human face. Here are a few of these relationships:

*Face height/face width;

* Central point of connection of the lips to the base of the nose / length of the nose;

* Face height / distance from the tip of the chin to the central point where the lips meet;

*Mouth width/nose width;

* Nose width / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand.

It is enough just to bring your palm closer to you and look carefully at your index finger, and you will immediately find the formula of the golden ratio in it. Each finger of our hand consists of three phalanges.

* The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb);

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

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:

The golden ratio in the structure of the human lungs.

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.

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

* It was found that this asymmetry continues in the branches of the bronchi, in all the smaller airways. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.

Structure of a golden orthogonal quadrangle and spiral.

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

In geometry, a rectangle with this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.

The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. By cutting a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again obtain a golden rectangle of smaller dimensions. This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located in a logarithmic spiral, which is important in mathematical models of natural objects (for example, snail shells).

The pole of the spiral lies at the intersection of the diagonals of the initial rectangle and the first vertical one to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also the golden triangle.

English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for thousands of years, explaining it this way:

“We like the look of the spiral because visually we can see it easily.”

In nature.

* The rule of the golden ratio, which underlies the structure of the spiral, is found in nature very often in creations of unparalleled beauty. The most obvious examples are that the spiral shape can be seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, the structure of rose petals, etc.;

* Botanists have found that in the arrangement of leaves on a branch, sunflower seeds or pine cones, the Fibonacci series is clearly manifested, and therefore the law of the golden ratio is manifested;

The Almighty Lord established a special measure for each of His creations and gave it proportionality, which is confirmed by examples found in nature. One can give a great many examples when the growth process of living organisms occurs in strict accordance with the shape of a logarithmic spiral.

All springs in the spiral have the same shape. Mathematicians have found that even with an increase in the size of the springs, the shape of the spiral remains unchanged. There is no other form in mathematics that has the same unique properties as the spiral.

The structure of sea shells.

Scientists who studied the internal and external structure of the shells of soft-bodied mollusks living at the bottom of the seas stated:

"The inner surface of the shells is impeccably smooth, and the outer surface is completely covered with roughness and irregularities. The mollusk was in the shell and for this the inner surface of the shell had to be impeccably smooth. The outer corners-curves of the shell increase its strength, hardness and thus increase its strength. Perfection and the amazing intelligence of the structure of the shell (snail) delights. The spiral idea of shells is a perfect geometric shape and is amazing in its refined beauty."

In most snails that have shells, the shell grows in the shape of a logarithmic spiral. However, there is no doubt that these unreasonable creatures not only have no idea about the logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.

But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living creatures, which the scientific world calls primitive life forms, calculate that the logarithmic shell shape would be ideal for their existence?

Of course not, because such a plan cannot be realized without intelligence and knowledge. But neither primitive mollusks nor unconscious nature possess such intelligence, which, however, some scientists call the creator of life on earth (?!)

Trying to explain the origin of such even the most primitive form of life by a random combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.

Biologist Sir D'arky Thompson calls this type of growth of sea shells "growth form of dwarves."

Sir Thompson makes this comment:

"There is no simpler system than the growth of seashells, which grow and expand in proportion, maintaining the same shape. The shell, most amazingly, grows, but never changes shape."

The Nautilus, measuring several centimeters in diameter, is the most striking example of the gnome growth habit. S. Morrison describes this process of nautilus growth as follows, which seems quite difficult to plan even with the human mind:

"Inside the nautilus shell there are many compartments-rooms with partitions made of mother-of-pearl, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in the front part of the shell, but this time larger than the previous one, and the partitions of the remaining behind the room are covered with a layer of mother-of-pearl. Thus, the spiral expands proportionally all the time."

Here are just some types of spiral shells with a logarithmic growth pattern in accordance with their scientific names:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa, Solarium Trochleare.

All discovered fossil remains of shells also had a developed spiral shape.

However, the logarithmic growth form is found in the animal world not only in mollusks. The horns of antelopes, wild goats, rams and other similar animals also develop in the form of a spiral according to the laws of the golden ratio.

Golden ratio in the human ear.

In the human inner ear there is an organ called Cochlea ("Snail"), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also shaped like a snail, containing a stable logarithmic spiral shape = 73º 43'.

Animal horns and tusks developing in a spiral shape.

The tusks of elephants and extinct mammoths, the claws of lions and the beaks of parrots are logarithmic in shape and resemble the shape of an axis that tends to turn into a spiral. Spiders always weave their webs in the form of a logarithmic spiral. The structure of microorganisms such as plankton (species globigerinae, planorbis, vortex, terebra, turitellae and trochida) also have a spiral shape.

The golden ratio in the structure of microcosms.

Geometric shapes are not limited to just a triangle, square, pentagon or hexagon. If we connect these figures with each other in different ways, we 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 have not encountered in everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures are the tetrahedron (regular four-sided figure), octahedron, dodecahedron, icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden ratio.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous . For example, many viruses have the three-dimensional geometric shape of an 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. At each corner of the icosahedron there are 12 units of protein cells in the shape 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 Birkbeck College London A. Klug and D. Kaspar. 13 The Polyo virus was the first to display a logarithmic form. The form of this virus turned out to be similar to the form of the Rhino 14 virus.

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

“Dr. Kaspar and I showed that for the spherical shell of the virus, the most optimal shape is symmetry such as the icosahedron shape. This order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. 14 The installation of such cubes requires extremely precise and a detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units."

From the history

"... If, from the point of view of the execution or function of an element, any form has proportionality and is pleasant, attractive to the eye, then in this case we can immediately look for in it any of the functions of the Golden Number... The Golden Number is not at all a mathematical invention. It is is in fact the product of a law of nature, based on the rules of proportionality."

A person distinguishes objects around him by their shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the shape. The form, the construction of which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a feeling of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's painting "Mona Lisa", a sunflower, a snail, a pine cone and human fingers have in common?

The answer to this question is hidden in the amazing numbers that were discovered by the Italian medieval mathematician Leonardo of Pisa, better known by the name Fibonacci (born about 1170 - died after 1228. After his discovery, these numbers began to be called after the famous mathematician. The amazing essence of the sequence Fibonacci numbers is that each number in this sequence is obtained from the sum of the two previous numbers.

The numbers that form the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ... are called "Fibonacci numbers" , and the sequence itself is the Fibonacci sequence. This is in honor of the 13th century Italian mathematician Fibonacci.

There is one very interesting feature about Fibonacci numbers. When dividing any number from the sequence by the number in front of it in the series, the result will always be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it.
(Approx. irrational number, i.e. a number whose decimal representation is infinite and non-periodic)

Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series. It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden section, the golden average, or the golden proportion.

It is no coincidence that the value of the golden ratio is usually denoted by the Greek letter F (phi) - this was done in honor of Phidias.

So, Golden ratio = 1:1.618

233 / 144 = 1,618
377 / 233 = 1,618
610 / 377 = 1,618
987 / 610 = 1,618
1597 / 987 = 1,618
2584 / 1597 = 1,618

Golden ratio- a relationship of proportions in which the whole is related to its larger part as the larger one is to the smaller one. (If we designate the whole as C, most of A, less of B, then the golden section rule appears as the ratio C:A=A:B.) Author of the Golden Rule- Pythagoras - considered perfect a body in which the distance from the crown to the waist was related to the total length of the body as 1:3. Deviations of body weight and volume from ideal norms depend primarily on the structure of the skeleton. It is important that the body is proportional.
In creating their creations, Greek masters (Phidias, Myron, Praxiteles, etc.) used this principle of the golden proportion. The center of the golden proportion of the human body structure was located exactly at the navel.

CANON
The canon - a system of ideal proportions of the human body - was developed by the ancient Greek sculptor Polycletus in the 5th century BC. The sculptor set out to accurately determine the proportions of the human body, in accordance with his ideas about the ideal. Here are the results of his calculations: head - 1/7 of the total height, face and hand - 1/10, foot -1/6. However, to contemporaries the figures of Polykleitos seemed too massive and “square”. Nevertheless, the canons became the norm for antiquity and, with some changes, for artists of the Renaissance and classicism. Almost the canon of Polykleitos was embodied by him in the statue of Doryphoros (“Spear-bearer”). The statue of the youth is full of confidence; the balance of body parts represents the power of physical strength. The broad shoulders are almost equal to the height of the body, half the height of the body is at the pubic fusion, the height of the head is eight times the height of the body, and the center of the “golden proportion” is at the level of the navel.

For thousands of years, people have been trying to find mathematical patterns in the proportions of the human body. For a long time, individual parts of the human body served as the basis for all measurements and were natural units of length. Thus, the ancient Egyptians had three units of length: a cubit (466 mm), equal to seven palms (66.5 mm), a palm, in turn, equal to four fingers. The measure of length in Greece and Rome was the foot.
The main measures of length in Russia were the sazhen and the cubit. In addition, an inch was used - the length of the joint of the thumb, a span - the distance between the spread thumb and index fingers (their heads), a palm - the width of the hand.

The human body and the golden ratio

The most important book of all modern architects, E. Neufert's reference book "Building Design", contains basic calculations of the parameters of the human torso, which contain the golden proportion.

It is characteristic that the sizes of the body parts of men and women differ significantly, but the ratios of these parts correspond in most cases to the ratios of the same integers.

The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.

In addition to this, there are several more basic golden proportions of our body:
the distance from fingertips to wrist and from wrist to elbow is 1:1.618
the distance from shoulder level to the top of the head and the size of the head is 1:1.618
the distance from the navel point to the crown of the head and from shoulder level to the crown of the head is 1:1.618
the distance of the navel point to the knees and from the knees to the feet is 1: 1.618
the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1.618
the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1.618

The golden ratio in human facial features as a criterion of perfect beauty.

There are other embodiments of the golden ratio rule on the human face. Here are a few of these relationships:
Face height / face width,
The central point where the lips connect to the base of the nose/length of the nose.
Face height / distance from the tip of the chin to the center point of the lips
Mouth width/nose width,
Nose width / distance between nostrils,
Distance between pupils / distance between eyebrows.

Human hand

The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb).

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

A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence.

Proportions in clothing.

The most important means of creating a harmonious image are proportions (for artists and architects they are of paramount importance). Harmonious proportions are based on certain mathematical relationships. This is the only means by which one can “measure” beauty. The golden ratio is the most famous example of harmonious proportion. Using the principle of the golden ratio, you can create the most perfect proportions in the composition of a costume and establish an organic connection between the whole and its parts.

Municipal educational institution

Secondary school No. 4, Rostov

Research Job

Golden ratio in human body proportions

Completed: Roshchina Natalia,

and Vyatkina Maria, 10th grade students

Supervisor: Gorokhova Galina Viktorovna,

Mathematic teacher

Rostov, 2014

Introduction........................................................ ...................................................3- 4

Chapter 1. Theoretical foundations…………………………………….4-10

Chapter 2. Practical research and data analysis…………….10-11

Conclusion……………………………………………………11

Literature…………………………………………………………….12

Introduction

“I like nothing but beauty

in beauty there is nothing but forms,

in forms - nothing but proportions,

in proportions there is nothing but number.”

Aurelius Augustine

For a long time, people have strived to surround themselves with beautiful things.

The household items of the ancient inhabitants already show man's desire for beauty. On

At a separate stage of his development, a person began to wonder: why this or that

an object is beautiful and what is the basis of beauty? Already in ancient Greece

The study of the essence of beauty, the beautiful, has formed into a separate science of aesthetics.

At the same time, the idea was born that the basis of beauty is harmony -

proportionality of parts and the whole, merging of various components of an object into one whole.

A person distinguishes objects around him by color, taste, smell, shape. Interest in the shape of an object may be caused by a vital necessity, or perhaps by the beauty of the form.

Beauty and harmony have always been the most important categories of knowledge, to a certain extent, even its goal, because ultimately the artist seeks truth in beauty, and the scientist seeks beauty in truth. The study of beauty has always been part of the study

the harmony of nature, the basic laws of its organization. Household items

Antiquity already shows man's desire for beauty. At a separate stage of his

development, people began to wonder: why this or that object is

beautiful and what is the basis of beauty? We also wanted to find the answer to this

We learned that the law of the golden ratio is widely used in fine arts, architecture, music, and even poetry. We were surprised that the perfectly proportioned human body is also entirely built on the principle of the golden division. Ancient sculptors were well aware of the use of the golden division to dismember the human body and knew how to use it; ancient statues are the best proof of this. You can check this peculiar law on any ancient statue. Modern researchers come to the conclusion that the Egyptians, back in the era of the ancient kingdom, developed a system of “harmonic proportioning” of the image, which is based on the principle of the golden division.

If the height of a well-built figure is divided into extreme and average ratios, then the dividing line will be exactly at the height of the waist, or, more precisely, the navel. If each of the resulting parts is in turn divided in extreme and average ratios, then the dividing line will again pass at very specific (anatomical) points: at the height of the so-called Adam's apple and the patella. But that's not all. Each individual part of the body - head, hand, etc. is also divided into natural parts according to the law of the golden division. In a word, the division of the external forms of a correctly folded human body is subject to the principle of the golden division down to the smallest parts.

We were also struck by the fact that the male figure satisfies this proportion especially well, and artists have long known that, contrary to general opinion, men are more beautifully built than women.

The last statement seemed more than controversial to us. We decided to study the structure of the modern human body.

Goal of the work: explore the principle of the “golden ratio” using the example of the human body.

Object of study: 8th grade students.

Tasks:

Get acquainted with the concept of the “golden ratio” and its use in life;

Consider the use of the “golden section” in human anatomy;

Find out from your classmates whether the concept of “beauty” corresponds to the rules of the golden ratio

Hypothesis: if a person’s body is built according to the “golden ratio” principle, then such a person can be considered beautiful.

Research methods: 1) analysis of information on this topic,

2) conducting a survey among classmates,

3) mathematical calculations of proportional relationships.

4) comparison of the obtained data.

Chapter 1. Theoretical foundations

History of the "golden ratio"

In the ancient literature that has come down to us, the first mention of the “golden ratio” is found in the works of Euclid “Principles” (about 300 BC). The “golden ratio” was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid applied it when creating his geometry.

The proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the “golden section” ratios when creating them.

What is the “golden ratio” or, in other words, the “golden ratio”? The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the largest part is related to the smaller one, and this is approximately equal to 1.62, that is, c:d = d:c

Golden ratio in mathematics.

The division of a segment in the mean and extreme ratio is called golden ratio. Another name has become established in history - “ golden ratio."

Let, C AB, and produces, as they say, the “golden ratio” of the segment

AC: AB =CB: AC (1)

Golden ratio is called such a division of a segment in which the larger part is related to the whole as the smaller part is to the larger one.

If the length of segment AB is denoted by A, and the length of AC is through X, That (Oh)- the length of the segment CB, and proportion (1) will take the form:

(2)

In a proportion, as is known, the product of the extreme terms is equal to the product of the middle terms and we rewrite proportion (2) in the form:

x 2 = a (a – x).

We get a quadratic equation:

x 2 + ax – a 2 = 0

The length of the segment is expressed as a positive number, therefore, from two roots, you should choose the positive one

X=
or X =

Number
denoted by the letter in honor of the ancient Greek sculptor Phidias (born at the beginning of the 5th century BC), in whose works this number appears many times. Number
approximately equal to 0.61803398...

Thus, the parts of the “golden ratio” make up approximately 62% and 38% of the entire segment.

Golden figures.

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler.

Dividing a line segment using the golden ratio

From point B a perpendicular equal to half AB is restored. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is laid, ending with point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the golden proportion.

Segments of the golden proportion are expressed by the infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If segment AB is taken to be 100 parts, then the larger part of the segment is 62, and the smaller part is 38 parts.

Dividing a rectangle with the line of the second golden ratio

The figure shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Construction of a regular pentagon and pentagram

The pentagram served as a symbol of the Pythagorean Union. The Pythagoreans believed it was possible to achieve purification of the spirit with the help of mathematics. According to their theory, the world order is based on numbers. The world, they believed, consists of opposites, and harmony leads opposites to unity. Harmony lies in numerical relationships. The Pythagoreans attributed various properties to numbers. Thus, they called even numbers feminine, odd numbers (except 1) – masculine. The number 5 - as the sum of the first female number (2) and the first male number (3) - was considered a symbol of love. Hence such attention to the pentagram, which has 5 angles. The five-pointed star - the pentagram - is very beautiful, it’s not for nothing that many countries place it on their flags and coats of arms! Her beauty, it turns out, has a mathematical basis.

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 Durer (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, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

Construction of the golden triangle

We draw straight AB. From point A we lay down on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the resulting points d and d1 with straight lines to point A. We lay off the segment dd1 on line Ad1, obtaining point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to construct a “golden” rectangle.

Fibonacci numbers

The series of Fibonacci numbers 1,1,2,3,5,8,13,21,34,55,89, etc. is closely related to the golden ratio. In this series, each subsequent number is the sum of the two previous numbers. Four centuries after Fibonacci's discovery of a series of numbers, I. Kepler established that the ratio of adjacent numbers in the limit tends to the golden proportion F. This property is inherent not only to Fibonacci numbers. Starting with any two numbers and constructing an additive series in which each term is equal to the sum of the previous two (for example, the series 7, 2, 9, 11, 20, ...), we discovered that the ratio of two consecutive terms of such a series also tends to the number  : the further we move from the beginning of the row, the better the approximation will be. If you take a calculator and divide each of them by the previous one, you get: 1:1=1; 2:1=2; 3:2=1.5; 5:3=1.666666; 8:5=1.6; 13:8=1.625; 21:13=1.615384;…

Golden ratio in art.

E Even during the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane.

This discovery was called the “golden ratio” of the painting by artists of that time. Therefore, in order to draw attention to the main element of the photograph, it is necessary to combine this element with one of the visual centers.

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. His personality is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.”

He gained fame as an unsurpassed artist, a great scientist, a genius who anticipated many inventions that were not realized until the 20th century.

There is no doubt that Leonardo da Vinci was a great artist, this was already recognized by his contemporaries, but his personality and activities will remain shrouded in mystery, since he left to his descendants not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “about everyone in the world."

He wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence.

The portrait of Monna Lisa (La Gioconda) has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon. There are many versions about the history of this portrait. Here is one of them.

One day, Leonardo da Vinci received an order from the banker Francesco de le Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint the portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became lively and interesting.

Fairy tale

Once upon a time there lived one poor man, he had four sons: three were smart, and one of them was this and that. And then death came for the father. Before losing his life, he called his children to him and said: “My sons, I will soon die. As soon as you bury me, lock the hut and go to the ends of the world to find happiness for yourself. Let each of you learn something so that you can feed yourself.” The father died, and the sons dispersed around the world, agreeing to return to the clearing of their native grove three years later. The first brother came, who learned to carpenter, cut down a tree and hewed it, made a woman out of it, walked away a little and waited. The second brother returned, saw the wooden woman and, since he was a tailor, dressed her in one minute: like a skilled craftsman, he sewed beautiful silk clothes for her. The third son decorated the woman with gold and precious stones - after all, he was a jeweler. Finally, the fourth brother came. He did not know how to carpenter or sew, he only knew how to listen to what the earth, trees, grass, animals and birds were saying, he knew the movements of the celestial bodies and could also sing wonderful songs. He sang a song that made the brothers hiding behind the bushes cry. With this song he revived the woman, she smiled and sighed. The brothers rushed to her and each shouted the same thing: “You must be my wife.” But the woman replied: “You created me - be my father. You dressed me, and you decorated me - be my brothers.

And you, who breathed my soul into me and taught me to enjoy life, you are the only one I need for the rest of my life.”

Having finished the tale, Leonardo looked at Monna Lisa, her face lit up with light, her eyes shone. Then, as if awakening from a dream, she sighed, ran her hand over her face and without a word went to her place, folded her hands and assumed her usual pose. But the job was done - the artist awakened the indifferent statue; a smile of bliss, slowly disappearing from her face, remained in the corners of her mouth and trembled, giving her face an amazing, mysterious and slightly sly expression, like that of a person who has learned a secret and, carefully keeping it, cannot contain his triumph. Leonardo worked silently, afraid to miss this moment, this ray of sunshine that illuminated his boring model...

The portrait of Monna Lisa is attractive because the composition of the drawing is built on “golden triangles” (more precisely, on triangles that are pieces of a regular star-shaped pentagon).

The Golden Ratio and the Human Body

D Zealous sculptors knew and used the golden ratio as a criterion of harmony, a canon of beauty, the roots of which lie in the proportions of the human body. “The human body is the best beauty on earth,” stated N. Chernyshevsky. The great works of Greek sculptors: Phidias, Polykleitos, Myron, Praxiteles have long been rightfully considered the standards of beauty of the human body, examples of harmonious physique. In creating their creations, Greek masters used the principle of the golden proportion. The center of the golden proportion of the human body structure was located exactly at the navel. And it is no coincidence that the value of the golden proportion is usually denoted by the letter F; this was done in honor of Phidias, the creator of immortal sculptural works.

Albrecht Durer began developing the theory of proportions of the human body during the Renaissance. Dürer assigned an important place in his system of relationships to the golden section. A person’s height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is well known.

In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Studies”. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art.

Zeising did a tremendous job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man.

The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo in most detail

Z The golden proportion was used by many ancient sculptors. The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

But let's analyze other proportions of the famous statue. One of the highest achievements of classical Greek art can be the statue of “Doriphoros”, sculpted by Polycletus. The figure of the young man expresses the unity of beauty and valor, underlying the Greek principles of art. The broad shoulders are almost equal to the height of the body, the height of the head is eight times the height of the body, and the position of the navel on the athlete’s body corresponds to the golden proportion.

The distance from the sole of the spearman to his knee is j 3, neck height with head - j 4, length of neck to ear - j 5, and the distance from the ear to the top of the head is j 6. Thus, in this statue we see a geometric progression with the denominator j: 1, j, j 2 , j 3 , j 4 , j 5 , j 6 .

Chapter 2: Case Studies and Data Analysis

For the first time we encounter the concept of the “golden ratio” in a 6th grade mathematics course. We were interested in this concept and decided to study it. Before starting work on the topic “Golden Ratio,” we conducted a survey among students from grades 7 to 11 and teachers of our school. It was necessary to answer the question “Do you know what the “golden ratio” or “golden section” is? The survey results are shown in the diagram.

Most teachers know what the “Golden Ratio” and “Golden Ratio” are, and students from grades 7 to 11 have no idea about the “Golden Ratio” and “Golden Ratio”.

In order to check whether the golden ratio holds true in the proportions of the human body, we conducted research among 10th grade students. Each participant had two types of measurements taken: a measurement from the top of the head to the navel, and a measurement from the navel to the floor. Their ratio was compared with the ratio number of the golden ratio.

In order to select students for the study, we conducted a sociological survey “The Most Beautiful Classmate,” in which 56 people participated.

As a result of the survey, we identified 2 boys and 2 girls who, according to classmates, are the most beautiful.

We present the results.

Gryazeva A.

Borisova K.

Kuvinov V.

Semeletko R.

For the second study, we took measurements from the top of the forehead to the eyebrows and from the eyebrows to the bottom of the chin.

We compared the results with the numbers of the golden proportion.

Gryazeva A.

Borisova K.

Kuvinov V.

Semeletko R.

Based on the results of the study, we identified two students who best fit the golden ratio - Borisova K. and Semeletko R.

Conclusion: Our work proves that a person whose body obeys the rule of the “golden proportion” is considered truly beautiful.

Conclusion.

The importance of the golden ratio in modern science is very great. This proportion is used in almost all areas of knowledge. Many famous scientists and geniuses tried to study it: Aristotle, Herodotus, Leonardo da Vinci, but no one completely succeeded.

This paper discusses methods for finding the “Golden Section” and provides examples taken from art and anatomy.

In our work, we wanted to demonstrate the beauty and breadth of the Golden Ratio in real life.

At the beginning of our work, we were interested in the opinion of scientists that the male figure is built better than the female one. As a result of research, we found that in women the approach to the “golden proportion” is more pronounced than in men. Therefore, despite the assertion of ancient scientists, a woman is more beautiful than a man.

Research has proven that the human body obeys the golden ratio rule.

I would like to tell my contemporaries that the beauty of a girl, a woman, is not in the currently accepted volumetric indicators: 90 x 60 x 90, but in the proportionality of body parts proven by the ancients. I hope that my research work will help many to look at themselves differently. Obviously, the golden ratio has some special property; it contains a mystery of nature that has yet to be discovered. The golden ratio is a mathematical concept and its study is the task of science. But it is also a criterion of beauty and harmony, and these are already categories of art. Therefore, we will end our research work with poetry.

“Whatever life teaches us,

But the heart believes in miracles.

There is endless strength

There is also imperishable beauty"

F. Tyutchev

Literature:

Brunov N. Proportion of ancient and medieval architecture, Moscow, publishing house of the All-Union Academy of Architecture, 1936.

Vasyutinsky N. L.

20 is the golden ratio. – M.: Mol. Guard, 1990.

Zverev I.D. ecology in school education: a new anapaest of education. Series “Pedagogy and Psychology”. – M., Knowledge, 1980.

D. Pidow. Geometry and art. – M.: mir, 1989

Magazine "Quantum", 1973, No. 8.