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Human proportions and the golden ratio. The golden ratio in man What are the proportions of the human body? Let's measure the length of parts of the body and head of an average statistical man

Human proportions and the golden ratio. The golden ratio in man What are the proportions of the human body? Let's measure the length of parts of the body and head of an average statistical man

11.04.2019

From open spaces for educational purposes)

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

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 c. 1170 - died after 1228), Italian mathematician . Traveling in the East, he got acquainted with the achievements of Arabic mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called the name of the famous mathematician. The amazing essence of the Fibonacci sequence is that that each number in this sequence is obtained from the sum of the previous two 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 in 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 oscillates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes not reaching it. (Note an irrational number, i.e. a number whose decimal representation is infinite and not 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 division in the Middle Ages that was called the Divine Proportion, and now today it is referred to as the golden section, 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, also created according to the principle of the golden section. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.

The most important book of all modern architects, the reference book by E. Neufert "Building Design" contains the basic calculations of the parameters of the human body, which include the golden ratio.

The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built. The principle of calculating the golden measure on the human body can be depicted as a diagram:

M/m=1.618

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

In addition, 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 the level of the shoulder to the crown of the head and the size of the head is 1:1.618;

* the distance from the point of the navel to the crown of the head and from the level of the shoulder 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 section formula. However, do not immediately rush after the ruler to measure the faces of all people. Because exact correspondences to the golden section, according to scientists and people of art, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden ratio in the face of a person is the ideal of beauty for the human eye.

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

On the human face, there are other embodiments of the golden section rule. Here are some 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 center point of the junction of the lips;

* Mouth width / nose width;

* Width of the nose / distance between the nostrils;

* Distance between pupils / distance between eyebrows.

Human hand.

It is enough just to bring your palm closer to you now and carefully look at your index finger, and you will immediately find the golden section formula 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 and gives the number of the golden section (with the exception of the thumb);

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

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 found that the golden section also exists in the structure of the human lungs.

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

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

The structure of the golden orthogonal quadrangle and spiral.

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

In geometry, a rectangle with this ratio of sides came to be called a golden rectangle. Its long sides are related to the 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 off a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again get a smaller golden rectangle. This process can be continued ad infinitum. As we keep cutting off the squares, we'll get 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 cut off vertical. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also a golden triangle.

English designer and esthetician William Charlton stated that people find spiral shapes pleasing to the eye and have been using them for millennia, explaining this as follows:

"We like the look of a spiral because visually we can easily see it."

In nature.

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

* Botanists have established 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 section is manifested;

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

All springs in a coil have the same shape. Mathematicians have found that even with the 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 a spiral.

The structure of sea shells.

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

"The inner surface of the shells is impeccably smooth, and the outer surface is all covered with roughness, irregularities. The mollusk was in the shell and for this the inner surface of the shell had to be perfectly smooth. The outer corners-bends of the shell increase its strength, hardness and thus increase its strength. Perfection and the striking rationality of the structure of the shell (snail) delights. The spiral idea of \u200b\u200bshells is a perfect geometric form and amazing in its polished beauty. "

In most snails that have shells, the shell grows in 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 shell for themselves ..

But then how could these unintelligent beings 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 the presence of reason and knowledge. But neither primitive mollusks nor unconscious nature, 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 coincidence of some natural circumstances is at least absurd. It is clear that this project is a conscious creation.

Biologist Sir D'Arkey Thompson calls this type of sea shell growth "gnome growth form".

Sir Thompson makes this comment:

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

The nautilus, measuring a few centimeters in diameter, is the most striking example of the gnome-like growth. S. Morrison describes this process of nautilus growth, which even the human mind seems rather difficult to plan:

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

Here are just some types of spiral shells that have a logarithmic growth shape 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 form of growth is found in the animal world not only in molluscs. 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.

The golden ratio in the human ear.

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

Horns and tusks of animals developing in the form of a spiral.

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

The golden section in the structure of microworlds.

Geometric shapes are not limited to just a triangle, square, five- or hexagon. If we combine these figures in various ways with each other, then we will get new three-dimensional geometric shapes. 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 one can name a tetrahedron (a regular four-sided figure), an octahedron, a dodecahedron, an icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easy to transform, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden section.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. . For example, many viruses have a three-dimensional geometric shape 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. In each corner of the icosahedron are 12 units of protein cells in the form of a pentagonal prism, and spike-like structures extend from these corners.

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

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

“Dr. Kaspar and I have shown that for a spherical shell of a virus, the most optimal shape is icosahedron-type symmetry. and a detailed explanation scheme, while unconscious viruses themselves construct such a complex shell of elastic, flexible protein cell units."

INTRODUCTION

The great creations of Greek sculptors: Phidias, Polyktetus, Myron, Praxiteles have long been considered the standards of the beauty of the human body, 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 ratio. 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 revived with renewed vigor.

Do modern people correspond to those 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 : the study of the golden section, as the ideal proportion of the structure of the human body.

Tasks:

to study the literature on the topic of research work;

define the golden section, 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 the proportions of people;

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

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

Object of study: Human.

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

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

Chapter 1 The Golden Ratio

Pythagoras showed that a segment of unit length 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 - The division of the segment in the extreme and average ratio

By the property of proportion .. x 2=1-x,

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

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

Such a division (point C) Pythagoras calledgolden division , or golden ratio , Euclid - dividing in extreme and average ratio , and Leonardo da Vinci - the now generally accepted term"golden section" .

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

The value of the golden section is usually denoted by the letter F. This is 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 accurate value of F can be seen in Appendix A.

Since the solution of 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

Consider the geometric construction of the golden section (Figure 1.2) using a right-angled triangle DAB, in which the sides AB andAUhave the following lengths: AB = 1, AU= 1/2. Let's draw an arc from the center of the circle C through point A until it intersects with the segment CB, we get the pointD. Then we pass through the pointDan arc with the center of the circle B to the intersection with the segment AB. We got the desired point E, dividing the segment AB in the golden ratio.

Figure 1.2 - Geometric construction of the golden section

Even Pythagoras and the Pythagoreans used the golden ratio to build some regular polyhedra - a tetrahedron, a cube, an octahedron, a dodecahedron, an icosahedron.

Euclid in the 3rd century BC e. uses, following the Pythagoreans, the golden ratio in his "Principles" 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 section, i.e. AB / CB =CB/ D.B. = D.B./ CD .

Figure 1.3 - Pentagram

Arithmetically, the segments of the golden ratio are expressed as an infinite irrational fraction. AC=0.618…, CB=0.382…. In practice, rounding is used: 0.62 and 0.38. If the segment AB is taken as 100 parts (Figure 1.4), then the larger part of the segment is 62, and the smaller one 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 ratio is 62, and the smaller one is 38 parts. These three quantities allow us to construct a series of segments of the golden ratio. 100, 62, 38, 24, 14, 10 - this is a series of values of the golden ratio, expressed arithmetically.

Figure 1.4 - Lines of the golden section 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 ratios between other elements of the picture.

Leonardo da Vinci and Albrecht Dürer 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 in the manufacture of his famous violins.

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

The doctrine of the golden section has been 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 section in architecture and art. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and decorations from the tomb of Tutankhamun indicate that the 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 school of Pythagoras and, in particular, to the questions of the golden division.

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

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

In the era of the Italian Renaissance, a new wave of passion for the golden ratio arises. The golden ratio is elevated to the rank of the main aesthetic principle. Leonardo da Vinci calls her "Sectioautea", from where the term "golden section" or "golden number" comes from. Luca Pacioli in 1509 writes the first essay on the golden ratio, entitled "DedivinaProportioned", which means "On the divine proportion." Johannes Kepler, who was the first to mention the meaning 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 section). 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, for almost two centuries, the golden ratio was forgotten. In the middle of the XIX century. the German scientist Zeising makes an attempt to formulate the universal law of proportionality and, at the same time, rediscovers the golden section. In his Aesthetic Investigations (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 relations 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 XIX 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 section.

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

The origins of the mathematical theory of biological populations go back to the "rabbit problem", which is associated with the emergence of Fibonacci numbers. Patterns described by Fibonacci numbers and the golden ratio are found in many phenomena of the physical and biological world ("magic" cores in physics, brain rhythms, etc.).

The Soviet mathematician Yu. V. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Academician GV Tsereteli discovers the golden ratio in Shota Rustaveli's poem "The Knight in the Panther's Skin". There are elegant methods for solving problems in search theory and programming theory based on Fibonacci numbers and the golden ratio.

In recent decades, Fibonacci numbers and the golden ratio have unexpectedly shown themselves to be the basis of digital technology.

In the second half of the 20th century, representatives of almost all sciences and arts turn to Fibonacci numbers and the golden ratio (mathematics, physics, chemistry, botany, biology, psychology, poetry, architecture, painting, music), 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 the ancient Greek statues that have survived to this day. Zeising even identified parts of the human body that, in his opinion, correspond most closely to the golden ratio. If you divide the human body according to the rule of the golden section, then the line will pass in the navel. 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 section (Figure 2.1).

Figure 2.1 - The golden ratio in the structure of the human body

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

The height of the face (to the roots of the hair) is related to the vertical distance between the arches of the eyebrows and the bottom of the chin, as the distance between the bottom of the nose and the bottom of the chin is related 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 for 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 ratio.

Leonardo applied scientific knowledge of the proportions of the human body to Pacioli and Vitruvius's theories of beauty. In the drawing by Leonardo "The Vitruvian Man" a male figure is 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 genitalia is 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.

"Vitruvian Man" represents the approximate proportions of the body of an ordinary adult, which since the time of ancient Greece has been used as an artistic canon for depicting a person. The proportions are formulated as follows:

Human height \u003d arm span (the distance between the fingertips of arms spread apart) \u003d 8 palms \u003d 6 feet \u003d 8 faces \u003d 1.618 multiplied by 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 "Dorifor" ("Spear-bearer"), sculpted by Poliktetom (Figure 2.3).

Figure 2.3 - The statue of "Dorifor" by the Greek sculptor Polyktetos

The figure of a young man expresses the unity of the beautiful and the valiant, underlying the Greek principles of art. 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 body of the athlete 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 of its individual members are connected by a mathematically rigorous system of proportional relationships, among which the golden ratio occupies the most important place. Having measured thousands of human bodies, he found that the golden ratio is the average value characteristic of all well-developed bodies. The average proportion of the male body is close to 13/8 = 1.625, and the female is close 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 head and body of a person on different stages of development

Belgian mathematician L. QueteletXIXcentury established that a person is ideal only when calculating the arithmetic mean. In 1871 his studies of the proportions of the bodies of the inhabitants of Europe fully confirmed the ideal proportions.

Chapter 3 The golden section in the proportions of the human body. Study

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

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

In the human body, the navel is the point of dividing the height of the body in the golden ratio. Therefore, we measured the height of people (a), navel height ( b) and the distance from the head to the navel (c). Then, in the Microsoft Office Excel 2007 program, the ratios of these quantities were found (a/ b, b/ c) for each person separately,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 chose 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 section in people of different ages.

Class

Number of persons

The resulting arithmetic mean

attitude

The number of people with the golden ratio

1,701

1,652

1,640

1,622

teachers

1,630

11th grade and teachers

1,626

Visually, these data can be presented in the form of diagrams (appendices C and D).

Based on the results of the study, the following can be doneconclusions:

Therefore, the golden ratio in the proportions of the human body is the average value, to which the proportions of the body of an adult are approaching. Only in some people the proportions of the body 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 section has been widely used in mathematics, physics, chemistry, painting, aesthetics, biology, music, and technology.

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

To achieve the goal, we studied the literature on the topic of research work, got acquainted with the golden ratio, with 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 section, tested the following hypothesis: the proportions of each human body correspond to the golden section.

To test the hypothesis, we measured the height of people and some parts of the body 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 separatelycmiddle valueie for a group of people of the same age, compared the obtained ratios with the value of the golden ratio and chose people with the golden ratio.

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

with age, the proportions of the body 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 USED SOURCES

Vasyutinskiy, N.A. Golden proportion / N.A. Vasyutinskiy - M.: Mol. guard, 1990. - 238 p.

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

Lukashevich, I.G. Mathematics in nature / I.G. Lukashevich. -Minsk: Belarus. assoc. "Competition", 2013. - 48s.

World of mathematics: in 40 volumes. T.1: Fernando Corbalan. Golden section. Mathematical language of beauty / Translation from English. - M.: De Agostini, 2014. - 160s.

Stakhov, A.P. Codes of the golden ratio / A.P. Stakhov. - M.: "Radio and communication", 1984. - 152s.

Timerding, G.E. Golden Section / G.E. Timerding; ed. G.M. Fikhtengolts; per. from German. - Petrograd: Scientific book publishing, 1924. - 86p.

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

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

APPENDIX A

THE SIGNIFICANCE OF THE GOLDEN RATIO

Figure A.1 - More accurate value of Ф

APPENDIX B

CORRESPONDENCE OF THE PROPORTIONS OF THE HUMAN BODY TO THE GOLDEN SECTION

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)

Belly 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 Daniel

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 Xenia

155

1,632

1,583

1,652

Larchenko Evgeniya

158

1,681

1,469

Listvyagov Sergei

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

Bulai 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 CALCULATION OF BODY PROPORTIONS IN PEOPLE OF DIFFERENT AGES

Figure B.1 - The results of calculating the body proportions of students in grade 1

Figure B.2 - The results of calculating body proportions for students in grade 5

Figure B.3 - The results of calculating the body proportions of students in grade 9

Figure B.4 - The results of calculating body proportions for students in grade 11

Figure B.5 - The 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 the average proportions of the body of people of different ages with the value of the golden section

APPENDIX E

STAGES OF WORK ON THE RESEARCH

a B C)

Figure E.1 - Literature study

a B C)

d) e)

Figure D.2 - Taking measurements of students and teachers

Figure D.3 - Input and processing of received data

A person distinguishes objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of a sense 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 section 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 is common between the ancient Egyptian pyramids, the painting by Leonardo da Vinci "Mona Lisa", a sunflower, a snail, a pine cone and human fingers?
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 c. 1170 - died after 1228. After his discovery, these numbers began to be called the name of 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 previous two 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 in Fibonacci numbers. When dividing any number in the sequence by the number preceding it in the series, the result will always be a value that fluctuates around irrational value 1.61803398875... That

reaching him.
(Note an irrational number, i.e. a number whose decimal representation is infinite and not 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 division in the Middle Ages that was called the Divine Proportion, and now today it is referred to as the golden section, the golden mean 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 is 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- the ratio of proportions in which the whole is related to its larger part as the larger is to the smaller. (If we denote the whole as C, the larger part of A, the smaller B, then the golden section rule acts as the ratio C: A = A: B.) Author of the golden rule- Pythagoras - considered perfect such 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 the weight and volume of the body 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 ratio. The center of the golden ratio of the structure of the human body was located exactly at the navel.
CANONS
Canon - a system of ideal proportions of the human body - was developed by the ancient Greek sculptor Polykleitos 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, even to his contemporaries, the figures of Polikleitos seemed too massive, “square”. Nevertheless, the canons became the norm for antiquity and, with some changes, for artists of the Renaissance and classicism. In practice, the canon of Polykleitos was embodied by him in the statue of Doryphorus (“Spear-bearer”). The statue of a young man is full of confidence; the balance of the parts of the body personifies the power of physical strength. 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 center of the “golden proportion” falls at the level of the navel.
For millennia, 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 of all measurements, were natural units of length. So, the ancient Egyptians had three units of length: a cubit (466 mm), equal to seven palms (66.5 mm), a palm, in turn, was equal to four fingers. The measure of length in Greece and Rome was the foot.
The main measures of length in Russia were sazhen and cubit. In addition, an inch was used - the length of the joint of the thumb, a span - the distance between the spread thumb and forefinger (their mop), 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, also created according to the principle of the golden section. Leonardo Da Vinci and Le Corbusier, before creating their masterpieces, took the parameters of the human body, created according to the law of the Golden Ratio.
The most important book of all modern architects, the reference book by E. Neufert "Building Design" contains the basic calculations of the parameters of the human body, which include the golden ratio.
The proportions of the various parts of our body make up a number very close to the golden ratio. If these proportions coincide with the formula of the golden ratio, then the appearance or body of a person is considered to be ideally built. The principle of calculating the golden measure on the human body can be depicted as a diagram
M/m=1.618
It is characteristic that the sizes of 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 section in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.
In addition, there are several more basic golden proportions of our body:
the distance from the fingertips to the wrist and from the wrist to the elbow is 1:1.618
the distance from the level of the shoulder to the crown of the head and the size of the head is 1:1.618
the distance from the point of the navel to the crown of the head and from the level of the shoulder 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 top of the head 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 section formula. However, do not immediately rush after the ruler to measure the faces of all people. Because exact correspondences to the golden section, according to scientists and people of art, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden ratio in the face of a person is the ideal of beauty for the human eye.
For example, if we sum up the width of the two upper front teeth and divide this sum by the height of the teeth, then, having obtained the golden ratio, we can say that the structure of these teeth is ideal.
On the human face, there are other embodiments of the golden section rule. Here are some of these relationships:
Face height / face width,
The center point of the junction of the lips 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 junction 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. It is enough just to bring your palm closer to you now and carefully look at the index finger, and you will immediately find the formula of the golden section in it. section (except for the thumb).
In addition, the ratio between the middle finger and the little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 clothes.
Proportions are the most important means of creating a harmonious image (for artists and architects they are of paramount importance). Harmonious proportions are based on certain mathematical relationships. This is the only means by which it is possible to "measure" beauty. The golden ratio is the most famous example of a harmonious proportion. Using the principle of the golden section, it is possible to create the most perfect proportions in the costume composition 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 the details of the costume is determined by the features of the figure, its own proportions. In the human body, too, there are mathematical relationships between its individual parts. If we take the height of the head as a module, that is, a conventional unit, then (according to Vitruvius, a Roman architect and engineer of the 1st century BC, 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 the middle of the thigh to the knee; from the knee to the middle of the lower leg; from ankle to floor. A simplified proportion indicates the equality of the four parts of the figure: from the crown 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 on an ideal, standard folded 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 ratios of the parts of the costume in size to each other and in comparison with the figure of a person. The comparative length, width, volume of the bodice and skirt, sleeves, collar, headdress, details affect the visual perception of the figure in the suit, 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" in growth about 8 times, and the waistline divides the figure in a ratio of approximately 3:5.
The most proportional human figure is the one in which these proportions are also repeated (the ratio of individual parts). The same goes for the costume.
In a suit, you can use both natural proportions and deliberately violated ones. It is impossible to analyze in detail the different options here, since for this you need to seriously study the laws of composition. It must be remembered that natural proportions are, as a rule, "favorable" for any figure; at the same time, the shortcomings of the addition can be "corrected" by slightly moving, "search" during the fitting of one or another line (for example, you can slightly raise or lower the waist, narrow or widen the shoulders, change the length of the dress, sleeve, collar size, pockets, belts).
The creation of clothes in many ways seems to have something in common with architecture - both of these arts are intended for direct contact with a person, they proceed from his natural proportions; finally, the suit, along with the person, is almost constantly surrounded by buildings and interior spaces. And the buildings, in turn, are in the natural environment, 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 mass of the costume, its apparent "heaviness" or "lightness" depends on various reasons. The more "piled up" lines, details, decorations, the more massive the figure; but when there is "nothing superfluous", even a naturally monumental figure will be freer, as it were, easier. With physically equal volumes, dense, dark, embossed, rough materials seem more massive than light, light, transparent, smooth, shiny ones. At the same time, light tones "increase" the volume, "reducing" heaviness, dark ones - vice versa. Hence the practical conclusion: overweight people should not be afraid of light materials, but it is better to place them in the upper part of the figure, near the face.

From the history

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

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

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 c. 1170 - died after 1228. After his discovery, these numbers began to be called the name of 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 previous two 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 in 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 oscillates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes not reaching it.
(Note an irrational number, i.e. a number whose decimal representation is infinite and not 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 division in the Middle Ages that was called the Divine Proportion, and now today it is referred to as the golden section, the golden mean 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 is 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- the ratio of proportions in which the whole is related to its larger part as the larger is to the smaller. (If we denote the whole as C, the larger part of A, the smaller B, then the golden section rule acts as the ratio C: A = A: B.) Author of the golden rule- Pythagoras - considered perfect such 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 the weight and volume of the body 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 ratio. The center of the golden ratio of the structure of the human body was located exactly at the navel.

CANONS
Canon - a system of ideal proportions of the human body - was developed by the ancient Greek sculptor Polykleitos 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, even to his contemporaries, the figures of Polikleitos seemed too massive, “square”. Nevertheless, the canons became the norm for antiquity and, with some changes, for artists of the Renaissance and classicism. In practice, the canon of Polykleitos was embodied by him in the statue of Doryphorus (“Spear-bearer”). The statue of a young man is full of confidence; the balance of the parts of the body personifies the power of physical strength. 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 center of the “golden proportion” falls at the level of the navel.

For millennia, 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 of all measurements, were natural units of length. So, the ancient Egyptians had three units of length: a cubit (466 mm), equal to seven palms (66.5 mm), a palm, in turn, was equal to four fingers. The measure of length in Greece and Rome was the foot.
The main measures of length in Russia were sazhen and cubit. In addition, an inch was used - the length of the joint of the thumb, a span - the distance between the spread thumb and forefinger (their mop), palm - the width of the hand.

The human body and the golden ratio

It is characteristic that the sizes of 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 section in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between the human foot and the navel point as a unit of measurement, then the height of a person is equivalent to the number 1.618.

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

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

On the human face, there are other embodiments of the golden section rule. Here are some of these relationships:
Face height / face width,
The center point of the junction of the lips 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 junction of the lips
Mouth width / nose width,
Nose width / distance between nostrils,
Distance between pupils / distance between eyebrows.

Human hand

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

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

A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). Each hand has 5 fingers, that is, 10 in total, but with the exception of two two-phalangeal 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 clothes.

Proportions are the most important means of creating a harmonious image (for artists and architects they are of paramount importance). Harmonious proportions are based on certain mathematical relationships. This is the only means by which it is possible to "measure" beauty. The golden ratio is the most famous example of a harmonious proportion. Using the principle of the golden section, it is possible to create the most perfect proportions in the costume composition 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 the details of the costume is determined by the features of the figure, its own proportions. In the human body, too, there are mathematical relationships between its individual parts. If we take the height of the head as a module, that is, a conventional unit, then (according to Vitruvius, a Roman architect and engineer of the 1st century BC, 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 the middle of the thigh to the knee; from the knee to the middle of the lower leg; from ankle to floor. A simplified proportion indicates the equality of the four parts of the figure: from the crown 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 on an ideal, standard folded 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 ratios of the parts of the costume in size to each other and in comparison with the figure of a person. The comparative length, width, volume of the bodice and skirt, sleeves, collar, headdress, details affect the visual perception of the figure in the suit, 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" in growth about 8 times, and the waistline divides the figure in a ratio of approximately 3:5.

The most proportional human figure is the one in which these proportions are also repeated (the ratio of individual parts). The same goes for the costume.
In a suit, you can use both natural proportions and deliberately violated ones. It is impossible to analyze in detail the different options here, since for this you need to seriously study the laws of composition. It must be remembered that natural proportions are, as a rule, "favorable" for any figure; at the same time, the shortcomings of the addition can be "corrected" by slightly moving, "search" during the fitting of one or another line (for example, you can slightly raise or lower the waist, narrow or widen the shoulders, change the length of the dress, sleeve, collar size, pockets, belts).

The creation of clothes in many ways seems to have something in common with architecture - both of these arts are intended for direct contact with a person, they proceed from his natural proportions; finally, the suit, along with the person, is almost constantly surrounded by buildings and interior spaces. And the buildings, in turn, are in the natural environment, 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 mass of the costume, its apparent "heaviness" or "lightness" depends on various reasons. The more "piled up" lines, details, decorations, the more massive the figure; but when there is "nothing superfluous", even a naturally monumental figure will be freer, as it were, easier. With physically equal volumes, dense, dark, embossed, rough materials seem more massive than light, light, transparent, smooth, shiny ones. At the same time, light tones "increase" the volume, "reducing" heaviness, dark ones - vice versa. Hence the practical conclusion: overweight people should not be afraid of light materials, but it is better to place them in the upper part of the figure, near the face (blouse, headdress, even a coat or raincoat with strict vertical lines).

Theoretical foundations of color combinations

When choosing the color range of the collection, it is important for the designer to take into account the rules for the compatibility of the colors used. And although, as they say, rules exist in order to break them, every self-respecting designer should know the theoretical foundations of the interaction of colors.

So, there are chromatic and achromatic colors.

Achromatic- white, gray and black. White is the brightest achromatic color, black is the darkest.

chromatic circle is a color chart based on the interaction of three primary colors: red, yellow and blue. They are defined as primary because they cannot be separated into other colors. If we mix the primary colors together, we get the rest of the colors, which we define as secondary.

All intermediate colors in the color wheel, including the main color that forms them, are related (moreover, the main colors located nearby are not related). In the color wheel, there are four groups of related colors: yellow-red, yellow-green, blue-green, blue-red. The harmonies of related colors are based on the presence of impurities in them of the same primary colors. Related chromatic combinations represent a restrained, calm range of colors, and the introduction of impurities of black and white colors enhances their emotional expressiveness.

Colors that are located in adjacent quarters on the color wheel are called related-contrasting. Combinations of related-contrasting colors are the most common and richest type of color harmonies in terms of color possibilities. Not all combinations of this kind are equally harmonious. Artistic practice shows that related-contrasting colors are in harmony with each other if the number of unifying main color and the number of contrasting main colors in them are the same. The simplest harmonious combination of related-contrasting colors is significantly enriched when achromatic or colors from their shadow series are added to them.

The interval of colors through one color sector is called the average. The combination of colors in the middle intervals often produces an unpleasant impression, for example, green with blue, red with purple.

Contrasting (complementary colors) are located in opposite quarters of the color wheel. The eye immediately notices this combination, so it is used where it is necessary to attract attention.

To two harmonious related-contrasting colors, a third can be added - the main color, their related, weakened saturation. The colors will be pairwise related-contrasting and pairwise complementary. Such combinations are highly harmonious and coloristically rich.

To two harmonious related colors, one contrasting one can be added. So, harmony is formed if the related greenish-yellow and leaf-green colors are supplemented with red-blue, i.e. additional intermediate of the first two.

Successful color combinations

According to French designers, a combination of the following colors is always appropriate: light brown with black, gray with red, gray with pink, gray with white, gray with blue, mustard with black, red with pale blue.

Laws of costume composition

When creating clothes, it is important to endow it with not only functional, but also aesthetic content - beauty, harmony, proportionality of the parts of the whole. A beautiful costume has a composition characteristic of it, that is, a correctly harmonized ratio of all its elements, parts and details.

What is a "suit"? This strange, from the point of view of a number of people, question, in fact, has a far from unambiguous answer. In the everyday sense, this is clothing. From the point of view of the image maker, a suit is a means of forming an image. An artist can give another definition: a costume is a plastic art that has its own composition.

* Straight lines. Cause a feeling of peace, stillness. * Soft, smooth, wavy lines give the impression of movement. * Vertical lines. They create the appearance of lengthening the figure, enhance the dynamics of forms. * Horizontal lines. Visually expand the figure, reduce growth, give the figure greater stability, stability. * Diagonal lines. They enhance the dynamics of the form of clothing, visually expand the figure or can narrow it from smaller to larger.

It often happens that the suit seems to be sewn, that is, the clothes fit well and the color matches the person’s face, but still something is not right. It can be assumed that in this case, during the creation of the costume, its composition was violated.

The composition of a costume is the combination of all its elements into one whole, expressing a certain idea, thought, image. The elements of a costume are all its components: shape, material and its properties, color, constructive and decorative lines.
First of all, a person in a suit perceives:

general dress code,
color and component elements of the form,
details and details.

Giving the composition certain properties depends on the use of certain composition means, which include:

proportions;
rhythm;
symmetry-asymmetry;
nuance and contrast;
rhythm;
color solution.

The use of these means allows the creator of the costume to express his idea, fill the costume with artistic content and, thus, influence the thoughts and feelings of the audience.

Let's take a closer look at composition tools.

The first law of composition. Integrity or the presence of a whole.

The main property of composition is wholeness.
Composition is such a composition and arrangement of parts of the whole, when:

nothing can be taken away without damage to the whole;
nothing can be changed;
nothing can be attached.

An important principle of giving the costume integrity is the consistency of all elements of the costume according to three principles - contrast, nuance or similarity.

Contrast is a pronounced opposition, opposition, which can be carried out in the form, color, volume and texture of the material.

Nuance is a kind of transitional value from contrast to similarity. The nuance is expressed by a subtle change in the form of the elements of the costume, their texture and color scheme.

Similarity - repetition in the costume of an element that occurs in various variations

The second law of composition. The law of proportions.

The most important means of creating a harmonious image are proportions. The law of proportions determines the ratio of the parts of the whole to each other and to the whole.
Proportions appear in the form of various mathematical relations - simple and irrational. The most harmonious irrational proportion is considered to be the "golden section", when the smaller part is related to the larger one in the same way that the larger part is related to the whole. According to this canon, the head of a person is 1/8 of the length of the body, and the waistline divides it as 5/8.

A suit will look elegant if the following rules of proportions are followed when sewing it:

The principle of the "golden section"(3:5, 5:8, 8:13) - causes the most harmonious perception, recommended for business style. Proportions are based on the length of the skirt. The most suitable length of the skirt is selected and the length of the jacket is calculated according to the “golden section” rule (Figure 1).
Contrasting proportions(1:4, 1:5) - more actively attract the attention of others. It is more expedient to use them for evening suits (Figure 2).
Similar proportions(1:1) - cause a feeling of static, peace, are recommended for everyday and home wear (Figure 3).

Figure 1 - Principle Figure 2 - Contrasting Figure 3 - Similar

"golden section" proportion proportions

Symmetry law.

The third law of composition. Symmetry law

Symmetry has long been considered one of the important conditions for the beauty of form.

A suit is considered symmetrical if it consists of geometrically equal parts and elements arranged in a certain order relative to the vertical axis of symmetry. The symmetrical composition creates the impression of stability, balance, grandeur, significance, solemnity.

Asymmetry in a suit is a lack of symmetry or a deviation from it. Asymmetry indicates a lack of balance, a disturbance of peace. It focuses the viewer's attention more on the dynamism of the composition, revealing its hidden ability to move. If a symmetrical composition is always balanced, then in an asymmetric composition the balance depends on the distribution of large and small values, lines, color spots, and the use of contrasts.

Fig 1 - Symmetry in a suit Fig 2 - Balanced asymmetry

Fig 3 - Unbalanced asymmetry

The fourth law of composition. The law of rhythm

The law of rhythm expresses the nature of the repetition or alternation of parts of the whole. Rhythm always implies movement.

The rhythm can be: active, impetuous, fractional or smooth, calm, slow. Rhythm in a costume can be created by costume elements: divisions - constructive or decorative lines, color - stripes, a cage, accessories - buttons, etc.
According to the method of organization, rhythm in a suit can be:

horizontal - horizontal stripes;
vertical;
spiral;
diagonal;
radial beam.

The latter views give the form a rapid movement (in the figure).

The fifth law of composition. The law of the chief in general

In a tree, the main thing is the trunk, in animals, the spine. In composition, it is the compositional center. The law of the main as a whole shows around what the parts of the whole are united. The compositional center is that object, part of an object or a group of objects that are located in the picture so that they are the first to catch the eye.

The compositional center does not have to be the largest in size, it just needs to attract the attention of the viewer, muffle distracting contrasts and minor details - everything should be subordinated to the main .

Examples of the use of the golden section in women's clothing.

It is difficult to take your eyes off the beauty, it is so attractive, maybe the reason is in it - golden and divine. It should be noted that a person is able to intuitively feel the proportions of the section. Working on a painting, embroidery or costume, without knowing it, he puts Him into his creations. No wonder, because the golden ratio is always before our eyes, in the form of ourselves.

The golden ratio is the division of a segment into unequal parts, while the entire segment (A) is related to the larger part (B), as this larger part (B) is related to the smaller part (C), or A:B=B:C, or C:B=B:A.

Segments golden ratio correlate with each other through an infinite irrational number Ф = 0.618 ... If C take as a unit, then A= 0.382. The numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence, on which the main geometric figures are built.

Human bones are in proportion close to the golden ratio. And the closer the proportions to the formula of the golden section, the more ideal the appearance of a person looks.

If the distance between a person's feet and the navel point = 1, then the person's height = 1.618.

The distance from the level of the shoulder to the crown of the head and the size of the head is 1:1.618.

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

The distance from the navel 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 top of the head is 1:1.618.

Other proportional ratios:

Face height / face width; the 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 of the junction of the lips; mouth width / nose width; width of the nose / distance between the nostrils; distance between pupils / distance between eyebrows.

The exact presence of the golden proportion in the face of a person is the ideal of beauty for the human eye.

The golden section formula 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 (with the exception of the thumb). Middle finger / little finger ratio = golden ratio.

A person has 2 hands, the fingers on each hand consist of 3 phalanges (with the exception of the thumb). There are 5 fingers on each hand, that is, only 10, but with the exception of two two-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).

Already in the Middle Ages, the measurements of parts of the human body were used as standards. When building cathedrals in France, a device was used, consisting of 5 rods, which were the lengths of the palm, large and small span, foot and elbow. All these lengths were multiples of a smaller unit of length, which was called line and was equal to 1/12 inch, i.e. about 2.5 mm. If we translate these figures into the metric system, we can see that the quantities lines are numbers from the Fibonacci series. The ratio of each to the previous is F, which is even more surprising, because these units correspond to arbitrary parts of the human body.