Golden ratio in human anatomy. Golden ratio - divine measure of beauty, Fibonacci numbers

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's 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 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 the golden orthogonal quadrilateral 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 a spiral because visually we can easily look at it.”

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, while the outer surface is completely covered with roughness and irregularities. The mollusk was in a shell and for this the inner surface of the shell had to be perfectly smooth. External corners-bends of the shell increase its strength, hardness and thus increase its strength. The perfection and amazing intelligence of the structure of the shell (snail) is amazing. The spiral idea of shells is a perfect geometric form and is amazing in its honed 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 sea shells, which grow and expand in proportion, 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 it is larger than the previous one, and the partitions of the room left behind 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.

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 Installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units.”

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 in a sequence by the number in front of it in the series, the result will always be a value that fluctuates around irrational value 1.61803398875... and every other time it exceeds, That

reaching him.
(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
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
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 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 1:1.61
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,
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
Each finger of our hand consists of three phalanges.
The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the golden number. Just bring your palm closer to you now and look carefully at the index finger, and you will immediately find in it the formula of the golden section (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.
However, the proportions of clothing lose all meaning if they are not linked to the person. Therefore, the ratio of costume details is determined by the characteristics of the figure, its own proportions. In the human body, there are also mathematical relationships between its individual parts. If we take the height of the head as a module, i.e. a conventional unit, then (according to Vitruvius, a Roman architect and engineer of the 1st century BC, the author of the treatise “Ten Books on Architecture”) eight modules will fit in the proportional figure of an adult : from crown to chin; from chin to chest level; from chest to waist; from the waist to the groin line; from the groin line to the middle of the thigh; from mid-thigh to knee; from the knee to the middle of the shin; from shin to floor. A simplified proportion speaks of the equality of the four parts of the figure: from the top of the head to the chest line (along the armpits); from chest to hips; from the hips to the middle of the knee; from knee to floor.
The finished dress is sewn to fit an ideal, standard figure, which not everyone can boast of in real life. However, a person can choose clothes in such a way as to look harmonious.
Proportions play a huge role in clothing.
Proportions in clothing are the ratio of the parts of the costume in size to each other and in comparison with the human figure. The comparative length, width, volume of the bodice and skirt, sleeves, collar, headdress, and details affect the visual perception of the figure in a suit and the mental assessment of its proportionality. The most beautiful, perfect, “correct” ratios look like those that are close to the natural proportions of the human figure. It is known that the height of the head “fits” the height about 8 times, and the waist line divides the figure in a ratio of approximately 3:5.
The most proportional human figure is considered to be the one in which these proportions are also repeated (the ratio of individual parts). The same goes for the suit.
In a costume, you can use both natural proportions and deliberately violated ones. It is impossible to analyze the different options in detail here, since this requires a serious study of the laws of composition. We must remember that natural proportions, as a rule, are “advantageous” for any figure; at the same time, shortcomings of the build can be “corrected” by slightly moving, “looking” for one or another line during fitting (for example, you can slightly raise or lower the waist, narrow or widen the shoulders, change the length of the dress, sleeves, the size of the collar, pockets, belt).
The creation of clothing in many ways has something in common with architecture - both of these arts are intended for direct contact with a person, based on his natural proportions; finally, the suit, together with the person, is almost constantly surrounded by buildings and interior spaces. And the buildings, in turn, are located in natural nature, in the urban architectural environment. Therefore, in different eras, architecture and costume reflect the artistic style of their time; and the folk costume, as it were, absorbs and preserves for centuries all the best, perfect, “eternal”.
The weight of the suit, its apparent “heaviness” or “lightness” depends on various reasons. The more “piled up” of lines, details, decorations, the more massive the figure; but when there is “nothing superfluous,” even a naturally monumental figure will be freer, as if lighter. With physically equal volumes, materials that are dense, dark, embossed, and rough seem more massive than light, light, transparent, smooth, and shiny materials. At the same time, light colors “increase” volume, “reducing” heaviness, dark ones - vice versa. Hence the practical conclusion: overweight people should not be afraid of light-colored materials, but it is better to place them in the upper part of the figure, near the face.

The human body and the golden ratio...
The golden ratio (golden ratio, division in extreme and average ratio) is the ratio of two quantities, equal to the ratio of their sum to the larger of the given quantities. The approximate value of the golden ratio is 1.6180339887.
All human bones are kept in proportion to the golden ratio.

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.
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.
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 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
Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.

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
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
Interpupillary distance/eyebrow distance
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.
Also worth noting is the fact that for most people, the distance between the ends of their outstretched arms is equal to their height.
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.
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 = 73 43'.
Blood pressure changes as the heart works. It reaches its greatest value in the left ventricle of the heart at the moment of its compression (systole). In the arteries, during the systole of the ventricles of the heart, blood pressure reaches a maximum value equal to 115-125 mmHg in a young, healthy person. At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of maximum (systolic) to minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.
If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic pressure is 0.618, that is, their ratio corresponds to the golden proportion. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle - the law of the golden proportion.
In the Universe, all galaxies known to mankind and all the bodies in them exist in the form of a spiral, corresponding to the formula of the golden ratio.

The ideal figure - what is it like? This question is difficult to answer, since the definition of this concept constantly changes depending on preferences and era. However, the most important indicator of success, attractiveness and charm at all times has been and remains proportionality.

Ideal parameters in different centuries

Any generation, people, or person can have their own opinion about what the ideal body proportions of a man and a woman are. In Paleolithic times, as is known, a female figure with more than exaggerated forms was considered beautiful - this is evidenced by archaeological finds.

The ideal proportions of the female body in antiquity included small breasts, slender legs, and wide hips. For the Middle Ages, the canons of beauty were an undefined waist and hips, but at the same time a rounded belly. Curvy figures were at the height of fashion during the Renaissance. And this continued until the era of classicism.

Only the twentieth century brought changes to the idea of what the ideal proportions of the human body should be. Now it is fashionable for a girl to have a flat stomach and slender legs, and for a man to have a muscular figure.

Canons of Polykleitos

The ancient Greek sculptor Polykleitos developed a system of ideal proportions back in the fifth century BC. The sculptor set a goal to accurately determine the proportions of a man’s body in accordance with his ideas about the ideal.

The results of his calculations are as follows: the head should be 1/7 of the total height, the hand and face - 1/10, the foot - 1/6.

However, to Polykleitos’s contemporaries such figures seemed too massive and “square.” These canons, however, became the norm for antiquity, as well as for Renaissance and Classicist artists (with some modifications). In practice, Polykleitos embodied the developed ideal proportions of the human body in the statue “Spearman”. The sculpture of a young man personifies confidence, the balance of body parts demonstrates the power of physical strength.

"Vitruvian Man" by da Vinci

The great Italian artist and sculptor in 1490 created the famous drawing called “Vitruvian Man”. It depicts the figure of a man in two positions, which are superimposed on one another:

With legs and arms spread apart. This position is inscribed in a circle.
With legs brought together and arms spread apart. This position is inscribed in a square.

According to da Vinci's logic, only the ideal proportions of the human body make it possible to fit figures in the indicated positions into a circle and a square.

Vitruvius' theory of proportionation

The ideal body proportions embodied in da Vinci’s drawing were taken as the basis for his theory of proportionation by another Roman scientist and architect Marcus Vitruvius Pollio. Later the theory became widespread in architecture and fine arts. According to it, an ideally proportional body is characterized by the following ratios:

the arm span is equal to the height of a person;
the distance from the chin to the hairline is 1/10 of a person’s height;
from the top of the head to the nipples and from the tips of the fingers to the elbow - 1/4 of the height;
from the top of the head to the chin and from the armpit to the elbow - 1/8 of the height;
maximum shoulder width - 1/4 height;
arm length - 2/5 of the person’s height;
the length of the ears, the distance from the nose to the chin, from the eyebrows to the line - 1/3 of the length of the face.

The concept of the golden ratio

Vitruvius' theory of proportionation arose much later than the theory of the golden section. It is believed that objects that contain the golden ratio are the most harmonious. The Egyptian pyramid of Cheops, the Parthenon in Athens, Notre Dame Cathedral, Leonardo da Vinci's paintings "The Last Supper", "Mona Lisa", Botticelli's work "Venus", Raphael's painting "The School of Athens" were created according to this principle.

The concept of the golden ratio was first given by the ancient Greek philosopher Pythagoras. He may have borrowed this knowledge from the Babylonians and Egyptians. This concept is then used in Euclidean Elements.

The term golden ratio was introduced into use by Leonardo da Vinci. After him, many artists consciously applied this principle in their paintings.

Rule of golden symmetry

From a mathematical point of view, the golden ratio consists of proportionally dividing a segment into unequal parts, while the entire segment is related to the larger part as the larger part itself is to the smaller one, that is, the smaller segment is related to the larger one as the larger one is to the whole.

If the whole is designated as C, the larger part as A, and the smaller part as B, the golden ratio rule will take the form of the ratio C: A = A: B. The basic geometric figures are based precisely on this ideal proportion.

The rule in question subsequently became the academic canon. It is used in the genetic structures of organisms, the structure of chemical compounds, space and planetary systems. Such patterns exist in the structure of the human body in general and individual organs in particular, as well as in the biorhythms and functioning of visual perception and the brain.

Zeising's Aesthetic Studies

In 1855, the German professor Zeising published his work, in which, based on the results obtained from measuring about two thousand bodies, he concluded that the division of a figure by the navel point is the most important indicator of the golden ratio. The ideal proportions of a man's body fluctuate within the average ratio of 13: 8 = 1.625 and come closer to the golden ratio than the proportions of a woman's figure, where the average value is expressed in the ratio 8: 5 = 1.6.

Such indicators are also calculated for other parts of the body: shoulder and forearm, fingers and hand, and so on.

90-60-90 - the ideal of beauty?

In society, the ideal proportions of the human body are revised approximately every fifteen years. During this period of time, due to acceleration, ideas about beauty undergo significant changes.

Therefore, the ideal proportions of a female body are not the notorious 90-60-90. These indicators are not suitable for everyone. After all, every girl has her own body type, which is inherited.

Ideal female body proportions

In our country, many now take as the ideal the physique standards drawn up by Dr. A.K. Anokhin back in the late nineteenth century. According to them, the proportions of a woman’s body are ideal if for every 1 cm of a woman’s height there is:

0.18-0.2 cm neck circumference;
0.18-0.2 cm shoulder circumference;
0.21-0.23 cm calf circumference;
0.32-0.36 cm hip circumference;
0.5-0.55 cm chest circumference (not bust);
0.35-0.40 cm waist circumference;
0.54-0.62 cm pelvic circumference.

Multiply your height (in centimeters) by the numbers given above. Then take the corresponding measurements of the body parts. Based on the results, it will become clear to what extent you meet the standards.

Male body proportions

Many varieties have a modern idea of the ideal male figure. In fact, it is impossible to name ideal body proportions for all men at the same time. There are subjective opinions, and there is reality, which is created by statistics and science. And objective evidence suggests that the ideal physique of a man has remained unchanged for thousands of years. From a female point of view, the V-shaped torso is considered the most attractive, which has ensured its owner success in society throughout the centuries.

Currently, ideal body proportions can be calculated in different ways: using the McCallum formula, Brock's method or the Wilks coefficient. McCallum, for example, talks about the need to have the same length of the torso and legs. And the size of the chest, in his opinion, should exceed the size of the pelvis (approximately 10 to 9). The chest and waist should be in proportions of 4 to 3, and the arms spread to the sides should be the height of the man. These same parameters were once incorporated into the “Vitruvian Man” phenomenon.

For a man, the ideal height is 180-185 centimeters. Weight is hardly worth citing as a standard; it is more important to link it with body proportions and height. After all, even with optimal weight, a loose figure will not bring success to its owner.

INTRODUCTION

The great creations of Greek sculptors: Phidias, Polyctetus, Myron, Praxiteles have long been considered the standards of beauty of the human body and examples of harmonious physique. Is it possible to express the beauty of a person using formulas and equations? Mathematics gives an affirmative answer. In creating their creations, Greek masters used the principle of the golden proportion. The golden ratio has been a measure of harmony in nature and in works of art for many centuries. It was studied by people of antiquity and the Renaissance. B XIIn the 10th and 20th centuries, interest in the golden ratio was revived with renewed vigor.

Do modern people correspond to the ideal proportions of the structure of the human body that have come down to us since ancient times? We will try to answer this question in the research work “The Golden Ratio in the Proportions of the Human Body.”

Goal of the work : study of the golden ratio as the ideal proportion of the structure of the human body.

Tasks:

study the literature on the topic of the research work;

define the golden ratio, get acquainted with its construction, application and history;

learn mathematical patterns in the proportions of the human body;

learn to find the golden ratio in human proportions;

determine the correspondence of the proportions of the human body to the golden ratio.

Hypothesis : The proportions of every human body correspond to the golden ratio.

Object of study: Human.

Subject of study : Golden ratio in the proportions of the human body.

Research methods : measuring the height and parts of the human body, processing the results obtained using mathematical methods using Microsoft Office Excel 2007, comparative analysis of the obtained measurements with the value of the golden ratio.

Chapter 1 Golden Ratio

Pythagoras showed that a segment of unit length is AB (Figure 1.1). can be divided into two parts so that the ratio of the larger part (AC = x) to the smaller one (CB = 1-x) will be equal to the ratio of the entire segment (AB = 1) to the larger part (AC = x):

Figure 1.1 – Division of a segment in extreme and average ratio

By the property of proportion.. x 2 = 1's,

x 2 + x-1 = 0. (1)

The positive root of this equation is, so the ratios in the given proportion are equal: =≈1.61803 each.

Pythagoras called this division (point C)golden division , or golden ratio , Euclid – division in extreme and average ratio , and Leonardo da Vinci – the now generally accepted term"golden ratio" .

Zolo that section - it's so proportionale division of a segment into unequal parts, within which the entire segment is related to the larger part, as the larger part is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.

The value of the golden ratio is usually denoted by the letter F. This was done in honor of Phidias, the creator of immortal sculptural works.

Ф=1.618033988749894. This is the value of the golden ratio with 15 decimal places. A more precise value of Ф can be seen in Appendix A.

Since the solution to equation (1) is the ratio between the lengths of the parts of the segment, it does not depend on the length of the segment itself. In other words, the value of the golden ratio does not depend on the original length.

1.2 Construction and application of the golden ratio

Let's consider the geometric construction of the golden section (Figure 1.2) using a right triangle ACB, in which sides AB andAChave the following lengths: AB = 1, AC= 1/2. Let's draw an arc from the center of circle C through point A until it intersects with segment CB, we get a pointD. Then we draw through the pointDarc with the center of circle B until it intersects with segment AB. We obtained the desired point E, dividing the segment AB in the golden ratio.

Figure 1.2 – Geometric construction of the golden section

Pythagoras and the Pythagoreans also used the golden ratio to construct some regular polyhedra - tetrahedron, cube, octahedron, dodecahedron, icosahedron.

Euclid in the 3rd century. BC e. Following the Pythagoreans, he uses the golden proportion in his “Elements” to construct regular (golden) pentagons, the diagonals of which form a pentagram.

In the pentagram in Figure 1.3, the intersection points of the diagonals divide them in the golden ratio, i.e. AB/CB =C.B./ D.B. = D.B./ CD .

Figure 1.3 - Pentagram

Arithmetically, segments of the golden proportion are expressed as an infinite irrational fraction. AC=0.618…, CB=0.382…. In practice, rounding is used: 0.62 and 0.38. If segment AB is taken to be 100 parts (Figure 1.4), then the larger part of the segment is 62, and the smaller part is 38 parts.

This method of constructing the golden ratio is used by artists. If the height or width of the picture is divided into 100 parts, then the larger segment of the golden proportion is 62, and the smaller is 38 parts. These three quantities allow us to construct a series of segments of the golden proportion. 100, 62, 38, 24, 14, 10 are a series of golden proportion values expressed arithmetically.

Figure 1.4 - Golden ratio lines and diagonals in the picture

The proportions of the golden section were often used by artists not only when drawing the horizon line, but also in the relationships between other elements of the picture.

Leonardo da Vinci and Albrecht Durer found the golden ratio in the proportions of the human body. The ancient Greek sculptor Phidias used it not only in his statues, but also in the design of the Parthenon Temple. Stradivari used this ratio when making his famous violins.

A form organized using the proportions of the golden section evokes the impression of beauty, pleasantness, consistency, proportionality, harmony.

The doctrine of the golden ratio is widely used in mathematics, physics, chemistry, painting, aesthetics, biology, music, and technology.

1.3 History of the golden ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, the ancient Greek philosopher and mathematician (VIV. BC.). However, long before the birth of Pythagoras, the ancient Egyptians and Babylonians used the principles of the golden ratio in architecture and art. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

Plato (427...347 BC) also knew about the golden division. His dialogue “Timaeus” is devoted to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division.

Ancient sculptors and architects widely used the number 1.62 or numerical ratios close to it in their artistic works. For example, the facade of the ancient Greek temple of the Parthenon contains golden proportions.

In the ancient literature that has come down to us, the golden proportion is first mentioned in Euclid’s Elements (325...265 BC) in the second book, and in the sixth book the definition and construction of the division of a segment in extreme and average ratio is given.

During the Italian Renaissance, a new wave of fascination with the golden ratio emerged. The golden proportion is elevated to the rank of the main aesthetic principle. Leonardo da Vinci calls her "Sectioautea", which is where the term “golden ratio” or “golden number” comes from. Luca Pacioli in 1509 wrote the first essay on the golden proportion, entitled “DedivinaProportioned", which means "On divine proportion." Johannes Kepler, who was the first to mention the significance of this proportion in botany, speaks of it as "a priceless treasure, as one of the two treasures of geometry" and calls it "Sectiodivina"(divine ratio). The Dutch composer Jacob Obrecht (1430-1505) makes extensive use of the golden ratio in his musical compositions, which are likened to "a cathedral created by a brilliant architect."

After the Renaissance, the golden ratio was forgotten for almost two centuries. In the middle of the 19th century. The German scientist Zeising makes an attempt to formulate the universal law of proportionality and at the same time rediscovers the golden ratio. In his “Aesthetic Studies” (1855), he shows that this law is manifested in the proportions of the human body (Figure 1.5) and in the body of those animals whose forms are distinguished by grace. In the body of ancient statues and well-built people, the navel is the point of dividing the height of the body in the golden ratio.

Figure 1.5 – Numerical relationships in the human body (according to Zeising)

Zeising finds proportional relationships close to the golden ratio in some temples (in particular, in the Parthenon), in the configurations of minerals, plants, and in the sound chords of music.

At the end of the 19th century. German psychologist Fechner conducts a series of psychological experiments to determine the aesthetic impression of rectangles with different aspect ratios. The experiments turned out to be extremely favorable for the golden ratio.

In the 20th century interest in the golden ratio is being revived with renewed vigor. In the first half of the century, composer L. Sabaneev formulated the general law of rhythmic balance and at the same time substantiated the golden ratio as a certain norm of creativity, a norm of the aesthetic design of a musical work. G. E. Timerding, M. Ghika, G. D. Grimm write about the meaning of the golden section in nature and art.

The “rabbit problem”, with which the emergence of Fibonacci numbers is associated, has its origins in the mathematical theory of biological populations. The patterns described by Fibonacci numbers and the golden ratio are found in many phenomena of the physical and biological world ("magic" nuclei in physics, brain rhythms, etc.).

Soviet mathematician Yu. V. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Academician G.V. Tsereteli discovers the golden ratio in Shota Rustaveli’s poem “The Knight in the Skin of a Tiger.” Elegant methods for solving problems in search theory and programming theory emerge, based on Fibonacci numbers and the golden ratio.

In recent decades, Fibonacci numbers and the golden ratio have unexpectedly emerged as the foundation of digital technology.

In the second half of the 20th century, representatives of almost all sciences and arts (mathematics, physics, chemistry, botany, biology, psychology, poetry, architecture, painting, music) turned to Fibonacci numbers and the golden ratio, because the golden ratio is the key to understanding the secrets of perfection in nature and art.

Chapter 2 Ideal proportions of the human body

For thousands of years people have been trying to find mathematical patterns in the proportions of the human body, especially a well-built, harmonious person.

The ancient Greeks, who considered the golden ratio to be a manifestation of harmony in nature, created statues of people in compliance with the rule of the golden ratio. INXIXcentury, Professor Zeising confirmed this by measuring ancient Greek statues that have survived to this day. Zeising even identified parts of the human body that, in his opinion, most closely correspond to the golden ratio. If you divide the human body according to the rule of the golden section, the line will pass in the navel area. Shoulder length refers to the total length of the arm also according to the golden ratio. The ratio of parts of the face, the length of the phalanges of the fingers and many other parts of the body fall under the rule of the golden ratio (Figure 2.1).

Figure 2.1 – Golden ratio in the structure of the human body

The golden proportion occupies a leading place in the artistic canons of Leonardo da Vinci and Durer. In accordance with these canons, the golden proportion corresponds to the division of the body into two unequal parts by the waist line.

The height of the face (to the roots of the hair) refers to the vertical distance between the arches of the eyebrows and the bottom of the chin, just as the distance between the bottom of the nose and the bottom of the chin refers to the distance between the corners of the lips and the bottom of the chin, this ratio is equal to the golden ratio.

Human fingers consist of three phalanges: main, middle and nail. The length of the main phalanges of all fingers, except the thumb, is equal to the sum of the lengths of the other two phalanges, and the lengths of all phalanges of each finger are related to each other according to the rule of the golden proportion.

Leonardo applied scientific knowledge of the proportions of the human body to Pacioli's and Vitruvius' theories of beauty. In Leonardo's drawing "Vitruvian Man" there is a male figure inscribed in a circle and a square (Figure 2.2).

Figure 2.2 – “Vitruvian Man” by Leonardo da Vinci

A square and a circle have different centers. The human genitals are the center of the square, and the navel is the center of the circle. The ideal proportions of the human body in such an image correspond to the ratio between the side of a square and the radius of a circle: the golden ratio.

The "Vitruvian Man" represents the approximate body proportions of a normal adult human, which have been used as an artistic canon for the depiction of humans since ancient Greece. The proportions are formulated as follows:

Human height = arm span (distance between the fingertips of arms spread out to the sides) = 8 palms = 6 feet = 8 faces = 1.618 times the height of the navel (distance from the navel to the ground).

One of the highest achievements of classical Greek art can be the statue “Doriphoros” (“Spearman”), sculpted by Polyctetus (Figure 2.3).

Figure 2.3 – Statue “Doriphoros” by the Greek sculptor Polyktetus

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, half the height of the body falls on the pubic fusion, 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.

In the middle of the 19th century, the German scientist Zeising found that the entire human body as a whole and each individual member of it are connected by a mathematically strict system of proportional relationships, among which the golden ratio occupies the most important place. Having measured thousands of human bodies, he established that the golden proportion is an average statistical value characteristic of all well-developed bodies. The average proportion of the male body is close to 13/8 = 1.625, and the female one - to 8/5 = 1.60; in a newborn the proportion is 2, by the age of 13 it is 1.6, and by the age of 21 it is equal to the male (Figure 2.4).

Figure 2.4 - Comparison of the proportions of the human head and body on various stages of development

Belgian mathematician L. Quetelet inXIXcentury, he established that a person is ideal only when calculating the arithmetic mean. In 1871 his studies of the proportions of the bodies of Europeans fully confirmed the ideal proportions.

Chapter 3 Golden ratio in the proportions of the human body. Study

We tested the hypothesis that the proportions of every human body correspond to the golden ratio.

Students of 1st, 5th, 9th and 11th grades and teachers of different ages (from 25 to 53 years old) were involved in the study.

In the human body, the navel is the point of dividing the height of the body in the golden ratio. That's why we measured people's height (a), navel height ( b) and the distance from the head to the navel (c). Then, in Microsoft Office Excel 2007, the ratios of these quantities were found (a/ b, b/ c) for each person individually,cmiddle valueie for a group of people of the same age (a/ b), compared the ratios with the value of the golden ratio (1.618) and selected people with the golden ratio (Appendix B).

We presented the results of the study in the form of a table (Table 3.1).

Table 3.1 – Correspondence of the proportions of the human body to the golden ratio for people of different ages.

Class

Number of persons

The resulting arithmetic mean

attitude

Number of people with the golden ratio

1,701

1,652

1,640

1,622

Teachers

1,630

11th grade and teachers

1,626

These data can be visually presented in the form of diagrams (Appendices C and D).

Based on the results of the study, the following can be done:conclusions:

Consequently, the golden ratio in the proportions of the human body is the average statistical value to which the proportions of the body of an adult person approach. Only some people have body proportions that correspond to the golden ratio.

CONCLUSION

The golden ratio has been a measure of harmony in nature and in works of art for many centuries. The doctrine of the golden ratio is widely used in mathematics, physics, chemistry, painting, aesthetics, biology, music, and technology.

The purpose of the research work was to study the golden ratio as the ideal proportion of the structure of the human body.

To achieve the goal, we studied the literature on the topic of the research work, got acquainted with the golden ratio, its construction, application and history; learned mathematical patterns in the proportions of the human body; learned to find the golden ratio in the proportions of people (Appendix E).

In the practical part, we determined the correspondence of the proportions of the human body to the golden ratio and tested the following hypothesis: the proportions of each human body correspond to the golden ratio.

To test the hypothesis, we measured the height of people and some body parts of students in grades 1, 5, 9, 11 and teachers of different ages. Then, in Microsoft Office Excel 2007, we found the ratios of values for each person individually,cmiddle valueie for a group of people of the same age, compared the resulting ratios with the value of the golden ratio and selected people with the golden ratio.

Based on the results of the study, the following conclusions can be drawn:

With age, a person’s body proportions change;

the proportions of the human body differ even among people of the same age;

in adults, body proportions approach the golden ratio, but rarely correspond to it;

The ideal proportions of the golden ratio do not apply to all people.

LIST OF SOURCES USED

Vasyutinsky, N.A. Golden proportion / N.A. Vasyutinsky - M.: Mol. Guard, 1990. – 238 p.

Kovalev, F.V. Golden section in painting: textbook. allowance / F.V. Kovalev. - K.: Higher school. Head publishing house, 1989.-143 p.

Lukashevich, I.G. Mathematics in nature / I.G. Lukashevich. -Minsk: Belarusian. assoc. “Competition”, 2013. - 48 p.

The world of mathematics: in 40t. T.1: FernandoCorbalan. Golden ratio. Mathematical language of beauty / Translated from English. - M.: De Agostini, 2014. - 160 p.

Stakhov, A.P. Golden ratio codes/A.P. Stakhov. - M.: “Radio and Communication”, 1984. – 152s.

Timerding, G.E. Golden ratio / G.E. Timerding; edited by G.M. Fikhtengolts; lane from German. - Petrograd: Scientific Book Publishing, 1924. – 86 p.

Urmantsev, Yu.A. Symmetry of nature and the nature of symmetry / Yu.A. Urmantsev. - M., Mysl, 1974. - 229s.

I explore the world: Children's encyclopedia: Mathematics /Auth.-comp. A.P.Savin and others; artist A.V. Kardashuk and others - M.: AST: Astrel, 2002. - 475 p.

APPENDIX A

MEANING OF THE GOLDEN RATIO

Figure A.1 – More accurate value of Ф

APPENDIX B

COMPLIANCE OF THE PROPORTIONS OF THE HUMAN BODY WITH THE GOLDEN RATIO

Table B.1-Results of measuring people and calculating the arithmetic mean values of body proportions for students in grades 1, 5, 9, 11 and teachers

Class

Height(s)

Navel line height (b)

Distance from navel to head (s)

a/b

b/c

Arithmetic mean (a/ b)

Golden ratio

1,618

Andreev Vladislav

130

1,688

1,453

Grabtsevich Daria

125

1,760

1,315

Vavanova Daria

127

1,716

1,396

Zakharenko Rodion

124

1,676

1,480

1 class

Kaporikov Daniil

133

1,684

1,463

1,701

Karsakov Zakhar

120

1,690

1,449

Lazovy Maxim

128

1,707

1,415

Lasotskaya Anna

125

1,645

1,551

Morgunova Maria

116

1,758

1,320

Pavlyushchenko Egor

129

1,675

1,481

Rakovsky Alexander

128

1,707

1,415

Bakhareva Ksenia

146

1,678

1,475

Bytkovsky Maxim

145

1,706

1,417

Zhdanovich Victoria

146

1,698

1,433

5th grade

Klimova Ksenia

155

1,632

1,583

1,652

Larchenko Evgenia

158

1,681

1,469

Listvyagov Sergey

143

1,644

1,554

Mukhina Anastasia

144

1,636

1,571

Paderina Anastasia

151

1,659

1,517

Prochukhanov Denis

151

1,641

1,559

Savkina Anastasia

140

1,609

1,642

Simakovich Alevtina

137

1,631

1,585

Surganova Daria

150

1,630

1,586

Smolyarov Vladislav

142

1,651

1,536

Tikhinsky Alexander

144

1,636

1,571

Averkov Alexey

171

104

1,644

1,552

Continuation of Table B.1

Teachers

Bulay E.I.

teaches.

163

101

1,614

1,629

1,630

Volkova O.V.

teaches.

1,64

1,563

Grinevskaya N.A.

teaches.

1,644

1,554

Grinchenko E.B.

teaches.

1,636

1,571

Kireenko A.S.

teaches.

175

108

1,62 0

1,612

Stukalov D.M.

teaches.

1,634

1,578

11th grade and teachers

Tsedrik N.E.

teaches.

1,646

1,548

Shkorkina N.N.

teaches.

1,602

1,661

1,626

Yatsenko V.N.

teaches.

1,604

1,656

APPENDIX B

RESULTS OF CALCULATING BODY PROPORTIONS IN PEOPLE OF DIFFERENT AGES

Figure B.1 – Results of calculating body proportions for 1st grade students

Figure B.2 – Results of calculating body proportions for 5th grade students

Figure B.3 – Results of calculating body proportions for 9th grade students

Figure B.4 – Results of calculating body proportions for 11th grade students

Figure B.5 – Results of calculating body proportions for teachers

APPENDIX D

COMPARISON OF BODY PROPORTIONS OF PEOPLE OF DIFFERENT AGES

WITH THE VALUE OF THE GOLDEN RATIO

Figure D.1 – Comparison of average body proportions of people of different ages with the value of the golden ratio

APPENDIX E

STAGES OF WORK ON THE RESEARCH

a B C)

Figure D.1 - Study of literature

a B C)

d) e)

Figure D.2 - Taking measurements of students and teachers

Figure D.3 – Input and processing of received data