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. In Fibonacci numbers there is one very interesting feature. 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 referred to as golden ratio, golden mean or golden ratio . 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 main book For 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.
Proportions various parts our body is 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 top line of the eyebrows and from the top 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.
On human face There are other incarnations of the golden ratio rule. 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 forefinger, 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 (except thumb);
* In addition, the ratio between the middle finger and little finger is also equal to the golden ratio;
* A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:
The golden ratio in the structure of the human lungs.
American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.
The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter.
* It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
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 along a logarithmic spiral, having 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 shell (snail) structure is amazing. The spiral idea of shells is a perfect geometric shape and amazing in its sharp 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 beings, whom scientists world calls primitive life forms, calculate that the logarithmic shape of a shell 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, to say the least, absurd. 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 growth seashells, which grow and expand proportionately, maintaining the same shape. The most amazing thing is that the shell 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 remain 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 that have 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 you connect these figures in various ways among themselves, 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 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 is 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 diagram-explanation. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units."
The golden ratio is the division of a segment into unequal parts, with the entire segment (A) related to the larger part (B), as this larger part (B) is related to the smaller part (C), or A: B = B: C, or C:B = B:A.
Segments golden ratio relate to each other through the infinite irrational number Ф = 0.618... If C taken as one, then A= 0.382. The numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence, on which the basic geometric figures are built.
Human bones are kept in proportion close to the golden ratio. And the closer the proportions are to the golden ratio formula, the more ideal a person’s appearance looks.
If the distance between a person's feet and the navel point = 1, then the person's height = 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 top of the head and from shoulder level to the top of the head is 1:1.618.
The distance from 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.
Other proportional relationships:
Face height/face width; center point of connection of lips to base of nose/nose length; 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.
The exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.
The formula of the golden ratio is visible when looking at the index finger. Each finger of the hand consists of three phalanges. The sum of the first two phalanges of the finger in relation to the entire length of the finger = the golden ratio (excluding the thumb). Ratio middle finger/ little finger = 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 double-phalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio (numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence).
Already in the Middle Ages, measures of parts of the human body were used as standards. When building cathedrals in France, a device was used consisting of 5 rods, which represented the lengths of a palm, a large and small span, a foot and an elbow. All these lengths were multiples of a smaller unit of length called line and was equal to 1/12 inch, i.e. about 2.5 mm. If we convert these numbers to the metric system, we can see that the quantities lines are numbers from the Fibonacci series. The ratio of each to the previous one is equal to F, which is even more surprising, because these units correspond to arbitrary parts of the human body.
What proportions in a person’s face tend to the “golden ratio”? First of all, people with beautiful faces observed: An ideal proportion between the distances from the medial corner of the eye to the wing of the nose and from the wing of the nose to the chin. This relationship is called "dynamic symmetry" or "dynamic equilibrium". The ratio of the height of the upper and lower lip will be 1.618
The height of the supralabial fold (the distance between the upper lip and the lower border of the nose) and the height of the lips will be in the ratio 62: 38. The width of one nostril in total with the width of the bridge of the nose relates to the width of the other nostril in the proportion of the “golden ratio”. The width of the mouth also refers to the width between the outer edges of the eyes, and the distance between the outer corners of the eyes refers to the width of the forehead at the level of the eyebrow line, like all proportions of the “golden ratio”.
The distance between the line of closure of the lips to the wings of the nose refers to the distance from the line of closure of the lips to the lower point of the chin, as 38: 62: And to the distance from the wings of the nose to the pupil - as 38: 62 = 0 The distance between the line of the upper part of the forehead to the line of the pupils and the distance between the line of the pupils and the line of closure of the lips has the proportion of the “golden ratio”
The navel divides the height of a person in golden ratio. The base of the neck divides the distance from the crown to the navel in golden ratio. For most people, the top of the ear shares the height of the head with the neck in a golden ratio. By dividing the segment between the crown and the Adam's apple in relation to the golden ratio, we get a point lying on the line of the eyebrows. The lowest point of the ear divides in golden ratio the distance from the top of the ear to the base of the neck. The chin divides the distance from the bottom of the ear to the base of the neck in golden ratio.
The span of a person’s outstretched arms is approximately equal to his height, as a result of which the human figure fits into a square and a circle. “Pentagonal” or “five-ray” symmetry, so characteristic of the world of plants and animals, is manifested in the structure of human bodies. And the human body can be considered as five-rayed, where the rays are the head, two arms and two legs. The human body can be inscribed in a pentagram. So called the pose of a person with arms spread 180° and legs spread 90°.
The most basic principle of harmonizing a costume according to this principle is the ratio of parts 3:5, or 5:3. That is, we do not divide the shape of the suit in half. If the skirt is long, then the jacket or jacket should be short. If the skirt is short - accordingly. Any detail can be built according to the principle of the golden ratio. The bodice and yoke can be in a ratio of 3:5. The dress and the length of the legs remaining after the dress are like 5:3.
It’s hard to take your eyes off the beauty, it’s so attractive, maybe the reason is it – golden and divine. It should be noted that a person is able to intuitively feel the proportions of the section. While working on a painting, embroidery or costume, without knowing it, he puts Him into his creations.
A beautiful smile is not only about snow-white, healthy, even teeth, but also about their correct relationship and location. And here we are again faced with the pattern of the “golden ratio”
Surprisingly, in dentistry you can trace the proportions of the “golden section”.
The “Golden Section” method for prolonging active longevity is a path of self-knowledge and self-improvement. This is a special system of exercises and knowledge that combines many elements human existence, ranging from ways to promote health and ending with interpersonal relationships.
Knowledge of the functioning of all organs provides a powerful incentive to strengthen the body and soul. A set of recommendations compiled individually for each person is a sequential change of tasks as the complexity increases. As a result, the vascular system is improved and optimal conditions are created for biochemical and biophysical processes in the body. The complex is selected in such a way that its implementation leads to changes in all organs and tissues. When performing these exercises, people normalize the functions of the entire body, increase immunity and resistance to stress.
The law of the golden ratio is visible in the quantitative division of the human body, corresponding to the numbers of the Fibonacci series. The morphogenesis of the hand approaches the golden ratio of 1.618, since 8:5 = 1.6. Comparing the lengths of the phalanges of the fingers and the hand as a whole, as well as the distances between in separate parts faces, you can find “golden” ones
Conclusions: Man is the crown of nature’s creation... It has been established that golden relationships can also be found in the proportions of the human body. In addition, man himself is a creator, creating wonderful works of art in which the golden proportion is visible. Man, like other creations of nature, is subject to the universal laws of development. The roots of these laws must be sought deeper - in the structure of cells, chromosomes and genes, and further - in the emergence of life itself on Earth.
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 top line of the eyebrows and from the top 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 respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.
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.”
When we look at beautiful landscape, we cover everything around. Then we pay attention to details. A murmuring river or a majestic tree. We see a green field. We notice how the wind gently hugs him and shakes the grass from side to side. We can feel the aroma of nature and hear the singing of birds... Everything is harmonious, everything is interconnected and gives a feeling of peace, a sense of beauty. Perception proceeds in stages in slightly smaller fractions. Where will you sit on the bench: on the edge, in the middle, or anywhere? Most will answer that it is a little further from the middle. The approximate number for the proportion of the bench from your body to the edge would be 1.62. It’s the same in the cinema, in the library, everywhere. We instinctively create harmony and beauty, which I call the “Golden Ratio” all over the world.
Golden ratio in mathematics
Have you ever wondered whether it is possible to determine the measure of beauty? It turns out that from a mathematical point of view it is possible. Simple Arithmetic gives the concept of absolute harmony, which is reflected in impeccable beauty, thanks to the principle of the Golden Ratio. Architectural structures Egypt and Babylon were the first to comply with this principle. But Pythagoras was the first to formulate the principle. In mathematics, this is a division of a segment slightly more than half, or more precisely 1.628. This ratio is presented as φ =0.618= 5/8. A small segment = 0.382 = 3/8, and the entire segment is taken as one.
A:B=B:C and C:B=B:A 
The principle of the golden ratio was used by great writers, architects, sculptors, musicians, people of art, and Christians who drew pictograms (five-pointed stars, etc.) with its elements in churches, fleeing from evil spirits, and people studying exact sciences, problem solver cybernetics.
Golden ratio in nature and phenomena.
Everything on earth takes shape, grows upward, to the side or in a spiral. Archimedes paid close attention to the latter and composed an equation. According to the Fibonacci series, there is a cone, a shell, a pineapple, a sunflower, a hurricane, a spider’s web, a DNA molecule, an egg, a dragonfly, a lizard... 
Titirius proved that our entire Universe, space, galactic space - everything is planned based on the Golden Principle. One can read the highest beauty in absolutely everything living and non-living. 
Golden ratio in man.
The bones are also designed by nature according to the proportion 5/8. This eliminates people’s reservations about “wide bones.” Most body parts in ratios apply to the equation. If all parts of the body obey the Golden Formula, then the external data will be very attractive and ideally proportioned.
The segment from the shoulders to the top of the head and its size = 1:1 .618
The segment from the navel to the top of the head and from the shoulders to the top of the head = 1:1 .618
The segment from the navel to the knees and from them to the feet = 1:1 .618
The segment from the chin to the extreme point of the upper lip and from it to the nose = 1:1 .618 
All facial distances give a general idea of the ideal proportions that attract the eye.
Fingers, palm, also obey the law. It should also be noted that the length of the spread arms with the torso is equal to the height of a person. Why, all organs, blood, molecules correspond to the Golden Formula. True harmony inside and outside our space.
Parameters from the physical side of surrounding factors.
Sound volume. The highest point of sound, causing an uncomfortable feeling and pain in the auricle = 130 decibels. This number can be divided by the proportion 1.618, then it turns out that the sound of a human scream will be = 80 decibels.
Using the same method, moving further, we get 50 decibels, which is typical for the normal volume of human speech. And the last sound that we get thanks to the formula is a pleasant whisper sound = 2.618.
Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. The simple arithmetic of harmony is embedded in our entire environment.
Golden ratio in art.
In architecture the most famous buildings and facilities: Egyptian pyramids, Mayan pyramids in Mexico, Notre Dame de Paris, Greek Parthenon, Peter's Palace, and others. 
In music: Arensky, Beethoven, Havan, Mozart, Chopin, Schubert, and others.
In painting: almost all paintings famous artists written according to the cross-section: the versatile Leonardo da Vinci and the inimitable Michelangelo, such relatives in the writings of Shishkin and Surikov, the ideal of the purest art - the Spaniard Raphael, and who gave the ideal female beauty- Italian Botticelli, and many, many others.
In poetry: the ordered speech of Alexander Sergeevich Pushkin, especially “Eugene Onegin” and the poem “The Shoemaker”, the poetry of the wonderful Shota Rustaveli and Lermontov, and many other great masters of words.
In sculpture: a statue of Apollo Belvedere, Olympian Zeus, beautiful Athena and graceful Nefertiti, and other sculptures and statues.
Photography uses the “rule of thirds.” The principle is this: the composition is divided into 3 equal parts vertically and horizontally, key points are located either on the lines of intersection (horizon) or at the points of intersection (object). Thus the proportions are 3/8 and 5/8. 
According to the Golden Ratio, there are many tricks that are worth examining in detail. I will describe them in detail in the next one. 
