Golden ratio in the human body. Human proportions and the golden ratio

Interesting information on the structure of the human body and its relationship with the golden ratio.

For reference, Wikipedia article:

Golden ratio (golden ratio, division in extreme and mean ratio, harmonic division) - the ratio of two quantities a and b, b > a, when b/a = (a+b)/b is true. The number equal to the ratio b/a is usually denoted by a capital Greek letter in honor of the ancient Greek sculptor and architect Phidias.

The Golden Ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with, and in the very structure of the human body, the rule of the golden ratio is also present.

The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time. The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “asymmetrical symmetry”, calling it in a broad sense a universal rule reflecting the structure and order of our world order.

There is a lot of information and illustrations about the human body and the golden ratio.

Well, the famous drawing by Leonardo da Vinci is about the same thing: how the human body and the golden ratio relate.

What proportions in a person’s face tend to the “golden ratio”? First of all, in people with beautiful faces there is: 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 “golden ratio” proportion. 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 because it’s 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 of human existence, from ways to improve health to 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. By comparing the lengths of the phalanges of the fingers and the hand as a whole, as well as the distances between individual parts of the face, one can find the “golden”

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.

When we look at a beautiful landscape, we are embraced by everything around us. 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. The architectural structures of other Egypt and Babylon were the first to begin 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, solving problems of 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 structures are: 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 the paintings of famous artists are painted according to the cross-section: the versatile Leonardo da Vinci and the inimitable Michelangelo, such relatives in writing as Shishkin and Surikov, the ideal of the purest art - the Spaniard Raphael, and the Italian Botticelli, who gave the ideal of female beauty, 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.

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Golden ratio - dividing 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,

C:B = B:A.

Segments golden ratio are related to each other using the infinite irrational fraction 0.618..., if C take as one 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.

For example, a rectangle with an aspect ratio of 0.618 and 0.382 is a golden rectangle. If you cut a square from it, you will again be left with a golden rectangle. This process can be continued indefinitely.

Another familiar example is the five-pointed star, in which each of the five lines divides the other at the golden ratio point, and the ends of the star are golden triangles.

Golden ratio and the human body

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 foot 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 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

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

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).

Middle finger/little finger ratio = 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 (the 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.

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