The golden ratio in human anatomy. Fibonacci Golden Ratio

A person distinguishes objects around him by form. Interest in the shape of any object can be dictated by vital necessity, or it can be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, 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 relation to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.
Let's find out what is common between the ancient Egyptian pyramids, the painting by Leonardo da Vinci "Mona Lisa", a sunflower, a snail, pine cone and the fingers of a man?
The answer to this question is hidden in amazing numbers, which were discovered by the Italian mathematician of the Middle Ages Leonardo of Pisa, better known by the name of Fibonacci (born c. 1170 - died after 1228. After his discovery, these numbers began to be called by the name of the famous mathematician. Fibonacci numbers is that each number in this sequence is obtained from the sum of the two previous numbers.
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 the Fibonacci sequence. This is in honor of the 13th century Italian mathematician Fibonacci.
In Fibonacci numbers, there is one very interesting feature... When dividing any number in the sequence by the number in front of it in the row, the result will always be a value fluctuating around irrational value 1.61803398875 ... and once again then rising, then

reaching it.
(Note: an irrational number, i.e. a number whose decimal representation is infinite and not periodic)
Moreover, after the 13th in the sequence, this division result becomes constant indefinitely. It was this constant number of divisions in the Middle Ages that was called Divine proportion, and nowadays it is referred to as the golden ratio, golden mean or 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, the 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 treats its larger part as much as a large one to a smaller one. (If we denote the whole as C, most of A, less than B, then the rule of the golden ratio acts as the ratio C: A = A: B.) The 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 the creation of their creations, Greek masters (Phidias, Miron, Praxitel and others) used this principle of the golden ratio. Center of the golden proportion of the structure human body was located exactly in the place of the navel.
CANONS
Canon - a system of ideal proportions of the human body - was developed by the ancient Greek sculptor Polycletus in the 5th century BC. The sculptor set out to accurately determine the proportions of the human body, in accordance with his ideas about the ideal. Here are the results of his calculations: head - 1/7 of the total height, face and hand - 1/10, foot -1/6. However, even to his contemporaries, the figures of Polycletus seemed too massive, “square”. Nevertheless, the canons became the norm for antiquity and, with some changes, for the artists of the Renaissance and Classicism. Practically the canon of Polykleitos was embodied by him in the statue of Dorifor ("The Spearman"). The statue of the youth is full of confidence; the balance of body parts personifies the power of physical strength. Broad shoulders almost equal to the height of the body, half the height of the body falls on the pubic fusion, the height of the head fits eight times along the height of the body, and the center of the "golden ratio" falls on the level of the navel.
For thousands of years, people have been trying to find mathematical patterns in the proportions of the human body. For a long time, individual parts of the human body served as the basis for all measurements, were natural units of length. So, the ancient Egyptians had three units of length: an elbow (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 fathoms and elbows. In addition, an inch was used - the length of the joint thumb, span - the distance between the apart thumb and forefinger (their shocks), 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 ratio. 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 main book of all modern architects, E. Neufert's reference book "Building Design" contains the basic calculations of the parameters of the human body, containing the golden proportion.
Proportions different parts our body is a number that is 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 perfectly folded. 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 the body parts of men and women differ significantly, but the ratios of these parts correspond in most cases to the ratios of the same whole numbers.
The first example of the golden ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.
In addition, there are several more basic golden proportions of our body:
distance from fingertips to wrist and from wrist to elbow is 1: 1.618
distance from shoulder level to crown of head and head size is 1: 1.618
the distance from the navel point to the crown of the head and from the shoulder level to the crown of the head is 1: 1.618
the distance of the navel point to the knees and from the knees to the feet is 1: 1.618
the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618
the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.618
the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1: 1.61
The golden ratio in human facial features as a criterion for perfect beauty.
In the structure of human facial features, there are also many examples that approach the value of the golden ratio formula. However, do not rush immediately after the ruler to measure the faces of all people. Because exact correspondences to the golden ratio, 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 a person's face is the ideal of beauty for the human eye.
For example, if we add up the width of the two front upper teeth and divide this amount by the height of the teeth, then, having received the Golden Ratio number, it can be argued that the structure of these teeth is ideal.
On human face there are other incarnations of the rule of the golden ratio. Here are some of these relationships:
Face height / face width,
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 number of gold.It is enough just to bring your palm closer to you now and carefully look at forefinger and you will immediately find the Golden Ratio formula in it (excluding the thumb).
In addition, the ratio between middle finger and little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, a total of 10, but with the exception of two biphalangeal 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 proportions in clothes.
The most important means of creating a harmonious image are proportions (for artists and architects, they are of paramount importance). Harmonious proportions are based on certain mathematical relationships. This is the only means by which it is possible to "measure" beauty. The golden ratio is the most famous example harmonious proportion. Using the principle of the golden ratio, you can create the most perfect proportions in the composition of the costume and establish an organic connection between the whole and its parts.
However, the proportions of clothing lose all meaning if they are not linked to a person. Therefore, the ratio of the details of the costume is determined by the characteristics of the figure, its own proportions. In the human body, there are also mathematical relationships between its individual parts. If taken as a module, i.e. conventional unit, the height of the head, then (according to Vitruvius, the Roman architect and engineer of the 1st century BC, the author of the treatise "Ten Books on Architecture"), eight modules will fit in the proportional figure of an adult: from the crown to the chin; from chin to chest level; from chest to waist; from the waist to the groin line; from the groin line to mid-thigh; from mid-thigh to knee; from the knee to the middle of the lower leg; from shin to floor. The simplified proportion speaks of 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 hips to mid-knee; from knee to floor.
The finished dress is sewn on an ideal, standard folded figure, which in real life not everyone boasts. However, a person can choose clothes in such a way as to look harmonious.
Proportions play a huge role in clothing.
The proportions in clothes are the proportions of the parts of the suit in size among themselves 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" are those ratios that are close to the natural proportions of the human figure. It is known that the height of the head "fits" in height about 8 times, and the waist line divides the figure in a ratio of about 3: 5.
The most proportional figure of a person is considered to be the one in which these proportions are also repeated (the ratio of individual parts). The same goes for the costume.
Both natural proportions and deliberately disturbed ones can be used in a suit. It is impossible to make out in detail here different variants, since for this you need to seriously study the laws of composition. It must be remembered that natural proportions are generally "beneficial" for any figure; at the same time, the flaws in addition can be "corrected" by slightly moving, "looking" during the fitting of this or that line (for example, you can slightly overestimate or underestimate the waist, narrow or widen the shoulders, change the length of the dress, sleeves, the size of the collar, pockets, belt).
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, proceed from his natural proportions; finally, the suit, together with the person, is almost constantly surrounded by buildings and interior spaces. And buildings, in turn, are in natural nature, in an urban architectural environment. Therefore, in different eras architecture and costume reflect art style its time; a folk costume as if it absorbs and preserves for centuries all the best, perfect, "eternal".
The weight of the suit, its apparent "heaviness" or "lightness" depends on different reasons... The more "heaped up" lines, details, ornaments, the more massive the figure; but when there is "nothing superfluous", even a naturally monumental figure will be freer, as if lighter. With physically equal volumes, materials are dense, dark, embossed, rough seem more massive than light, light, transparent, smooth, shiny. At the same time, light tones "increase" the volume, "decreasing" the severity, dark tones - on the contrary. Hence the practical conclusion: overweight people you should not be afraid of light-colored materials, but it is better to place them in the upper part of the figure, near the face.

The golden ratio is the division of a segment into unequal parts, while the entire segment (A) refers to the greater part (B), like this most of(B) refers 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.

Human bones are sustained in a 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 the person's feet and the navel point = 1, then the person's height = 1.618.

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

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

The distance from the navel point to the knees and from the knees to the feet is 1: 1.618.

The distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618.

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

Other proportional ratios:

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; width of the nose / distance between the nostrils; distance between pupils / distance between eyebrows.

The exact presence of the golden ratio in the human face is the ideal of beauty for the human eye.

The Golden Ratio 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 (excluding the thumb). Ratio middle finger/ little finger = golden ratio.

A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, only 10, but with the exception of two biphalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio (numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence).

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

The human body and the golden ratio ...
Golden ratio (golden ratio, division in extreme and average ratio) - the ratio of two values, equal to the ratio of their sum to the larger of these values. The approximate value of the golden ratio is 1.6180339887.
All human bones are sustained in the proportion of 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 perfectly folded.
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 1.618.
Distance from shoulder level to crown of head and head size is 1: 1.618
The distance from the navel point to the crown of the head and from the shoulder level to the crown of the head is 1: 1.618
The distance of the navel point to the knees and from the knees to the feet is 1: 1.618
The distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1: 1.618
Actually, the exact presence of the golden ratio in a person's face is the ideal of beauty for the human eye.

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
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
Pupil distance / eyebrow distance
It is enough just to bring your palm closer to yourself now and carefully look at the index finger, and you will immediately find the formula of the golden ratio in it.
Each finger of our hand consists of three phalanges. The sum of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (excluding the thumb).
In addition, the ratio between middle finger and little finger is also equal to the golden ratio.
A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, a total of 10, but with the exception of two biphalangeal 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.
It should also be noted that for most people, the distance between the ends of their apart arms is equal to height.
The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi are made up of two main airways, one of which (left) is longer and the other (right) is shorter.
It was found that this asymmetry continues in the branches of the bronchi, in all the smaller airways.
Moreover, the ratio of the length of the short and long bronchi also makes up the golden ratio and is equal to 1: 1.618.
In the inner ear of a person there is an organ called Cochlea ("snail"), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also created in the form of a snail, containing a stable logarithmic spiral shape = 73 43 '.
Blood pressure changes as the heart works. It reaches its greatest value in the left ventricle of the heart at the time of its compression (systole). In the arteries during the systole of the ventricles of the heart, the blood pressure reaches a maximum value equal to 115-125 mm Hg in a young, healthy person... At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of maximum (systolic) to minimum (diastolic) pressure is 1.6 on average, that is, close to the golden ratio.
If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic pressure is 0.618, that is, their ratio corresponds to the golden ratio. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle - the law of the golden ratio.
In the Universe, all galaxies known to mankind and all bodies in them exist in the form of a spiral, corresponding to the golden ratio formula.

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

Ideal parameters in different ages

Any generation, people, person can have their own opinion about what are perfect proportions bodies of a man and a woman. In Paleolithic times, as you know, was considered beautiful female figure with more than hypertrophied forms - this is evidenced by archaeological finds.

Perfect proportions female body in the period of antiquity, they assumed a small chest, slender legs, wide hips. For the Middle Ages, the canons of beauty were unexpressed waist and hips, but at the same time a rounded belly. At the height of fashion during the Renaissance, there were curvaceous forms. And so it continued until the era of classicism.

Only the twentieth century has made a change in the idea of ​​what the ideal proportions of the human body should be. Now it's fashionable for a girl to have slim stomach and slender legs, and the man has a muscular figure.

The canons of Polycletus

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

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

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

Da Vinci's Vitruvian Man

Great italian artist and the sculptor in 1490 created famous drawing called "Vitruvian Man". He depicts the figure of a man in two positions, which are superimposed on one another:

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

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

Vitruvius' proportioning theory

The ideal body proportions embodied in da Vinci's drawing were taken as the basis for his theory of proportioning by another Roman scientist and architect, Mark Vitruvius Pollio. Later the theory became widespread in architecture and fine arts... According to her, for an ideally proportional body, the following ratios are characteristic:

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

Golden ratio concept

Vitruvius' theory of proportioning arose much later than the theory of the golden ratio. It is believed that objects that contain the golden ratio are the most harmonious. Egyptian pyramid of Cheops, Parthenon in Athens, cathedral Notre dame de paris, paintings by Leonardo da Vinci " The last supper"," Mona Lisa ", Botticelli's work" Venus ", Raphael's painting" School of Athens "were created on this principle.

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

Leonardo da Vinci introduced the term “golden section” into everyday life. After him, many artists consciously applied this principle in their paintings.

Golden symmetry rule

From a mathematical point of view, the golden ratio consists in proportionally dividing a segment into unequal parts, while the entire segment refers to the greater part in the same way as the largest part to the smaller one, that is, the smaller segment refers to the larger one, as the larger segment refers to everything.

If we designate the whole as C, most of it - A, and the smaller part - B, the rule of the golden ratio will look like the ratio C: A = A: B. The main geometric shapes are based on this ideal proportion.

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

Zeising's "Aesthetic Research"

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

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

90-60-90 - the ideal of beauty?

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

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

Ideal proportions of the female body

In our country, now many people take for the ideal the standards of physique drawn up by Dr. A.K. Anokhin at the end of the nineteenth century. According to them, the proportions of the female body are ideal if 1 cm of a woman's height accounts for:

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

Multiply your height (in centimeters) by the numbers above. Then take the appropriate measurements of the body parts. Based on the results, it will become clear how you comply with the regulations.

Male body proportions

Many varieties have modern performance about the ideal male figure. In fact, the ideal body proportions for all men cannot be named at the same time. There are subjective opinions, and there is a reality that is created by statistics and science. And objective evidence suggests that the ideal physique of a man has remained unchanged for millennia. From a female point of view, the most attractive is the V-shaped torso, which has provided its owner with success in society throughout the ages.

Currently, you can calculate the ideal body proportions different ways: Using McCallum's formula, Brock's method, or Wilkes coefficient. McCallum, for example, talks about the need to have the same length of the torso and legs. And the size chest, in his opinion, should exceed the size of the pelvis (approximately 10 to 9). The chest and waist should be in a ratio of 4 to 3, and the arms, spread apart, should be the height of a man. The same parameters at one time were laid in the phenomenon of the "Vitruvian Man".

For a man perfect height 180-185 centimeters are considered. Weight as a reference is hardly worth citing, it is more important to link it with body proportions and height. Indeed, even with an optimal weight, a loose figure will not bring success to its owner.

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 ratio. 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, E. Neufert's reference book "Building Design" contains the basic calculations of the parameters of the human body, containing the golden proportion.

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 perfectly folded. 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 ratio in the structure of the human body:
If we take the navel point as the center of the human body, and the distance between a person's feet and the navel point as a unit of measurement, then a person's height is equivalent to 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;

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

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

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

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

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

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

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

In the structure of human facial features, there are also many examples that approach the value of the golden ratio formula. However, do not rush immediately after the ruler to measure the faces of all people. Because exact correspondences to the golden ratio, 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 a person's face is the ideal of beauty for the human eye.

For example, if we add up the width of the two front upper teeth and divide this amount by the height of the teeth, then, having received the Golden Ratio number, it can be argued that the structure of these teeth is ideal.

On the human face, there are other incarnations of the rule of the golden ratio. 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;

* Width of the mouth / width of the nose;

* Width of nose / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand.

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

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

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

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (excluding the thumb). Each hand has 5 fingers, that is, a total of 10, but with the exception of two biphalangeal 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 proportion 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 ratio also exists in the structure of the human lungs.

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

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

The structure of the golden orthogonal quadrilateral and spiral.

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

In geometry, a rectangle with this aspect ratio has come to be called the golden rectangle. Its long sides compare to short sides in a 1.168: 1 ratio.

The golden rectangle also has many amazing properties. The golden rectangle has many unusual properties. Cutting off a square from the gold rectangle, the side of which is equal to the smaller side of the rectangle, we again get a smaller gold rectangle. This process can be continued indefinitely. As we continue to cut out the squares, we will get smaller and smaller golden rectangles. Moreover, they will be located along a logarithmic spiral, which has essential v mathematical models natural objects (for example, snail shells).

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

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

"We like the look of the spiral, because visually we can see it with ease."