The Golden Ratio and the Golden Ring of Russia In the book of Erich von Däniken (see) I read that sacred places in Ancient Greece are connected to each other by the proportion of the Golden Ratio. I quote the personally verified data that is given in this book (see Fig. 55 and 56): 1. Line

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.

Definition.
The most comprehensive definition of the golden ratio states that the smaller part relates to the larger, as the larger part relates to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship in the forms of space and time operates.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “Asymmetrical Symmetry”, calling it in a broad sense a universal rule that reflects the structure and order of our world order.

Story.
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the son, the large segment the father, and the whole the holy spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden ratio : “It is arranged in such a way that the two Younger Members of This Infinite Proportion in the Sum give the Third Member, and any two Last Members, If Added, Give the Next Member, Moreover, the same Proportion is Preserved to Infinity.” Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations

Fibonacci numbers are a harmonic division, a measure of beauty. The golden ratio in nature, man, art, architecture, sculpture, design, mathematics, music https://psihologiyaotnoshenij.com/stati/zolotoe-sechenie-kak-eto-rabotaet

Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio; most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in the golden division.

Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it a second life in 1855. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his “Mathematical Aesthetics” caused a lot of criticism.

Nature.
Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio and in the shape of an egg, if conditional line pass through its widest part.

The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting.
Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted the attraction of nature to spiral forms, calling the spiral the “Curve of Life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.

Human.
Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.

In Leonardo da Vinci's diary there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo's "Vitruvian Man", created his own scale of "harmonic proportions", which influenced the aesthetics of 20th century architecture.

Adolf Zeising, exploring the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but the main indicator of the golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

The art of spatial forms.
The artist Vasily Surikov said, “that in a Composition there is an Immutable Law, When in a Picture You Can’t Remove or Add Anything, You Can’t Even Put an Extra Point, This is Real Mathematics.” For a long time, artists followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting is no longer complete without a solution geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

Art critic F. v. Kovalev, having examined in detail Nikolai Ge’s painting “Alexander Sergeevich Pushkin in the Village of Mikhailovskoye,” notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair, or the poet himself, is strictly inscribed in golden proportions.

Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the great pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling in the viewer.

Word, sound and film.
Forms are temporary? The Go arts, in their own way, demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin’s work corresponds to the Fibonacci series - 5, 8, 13, 21, 34.

The rule of the golden section also applies in individual works of the Russian classic. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853: 535 = 1, 6) - this is the point of the golden ratio.

Soviet musicologist E. K. Rosenov notes the amazing accuracy of the relationships of the golden section in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point.
Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the golden ratio rule, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

Golden ratio examples. How to get the golden ratio

So, the golden ratio is the golden ratio, which is also a harmonic division. To explain this more clearly, let's look at some features of the form. Namely: a form is something whole, and the whole, in turn, always consists of some parts. These parts most likely have different characteristics, at least different sizes. Well, such dimensions are always in a certain relationship, both among themselves and in relation to the whole.

This means, in other words, we can say that the golden ratio is a ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.

There is much more meaning to a spiral tattoo than it seems at first glance. Such a simple pattern is built according to the so-called golden ratio principle, which is found everywhere in nature. Moreover, this principle has been known since ancient times, which is confirmed by its presence at the base of the Egyptian pyramids.

Symbolism of spiral tattoos

In Ta-moko tattoos or in the same Celtic patterns, spirals are found very often, and this is not surprising. The absence of right angles in this figure symbolizes the connection with nature, which does not like right angles and always tries to smooth them out. A spiral tattoo means unity with nature; as a rule, calm, reasonable people make such a tattoo.

But that's just general meaning, it is not uncommon for people to try to find out the meaning of a spiral tattoo by actually confusing it with other tattoos. Often a tattoo of a spiral shell misleads people, it is Lately very popular. One has a completely different meaning, it suits closed people, loners, who have usually suffered some kind of shock and do not want to share about it, but in his honor they make such a tattoo.

A wave tattoo, which symbolizes love of the sea, or a black sun tattoo, the meaning of which we wrote in detail, is very similar to a spiral.

Often a spiral tattoo is made as a talisman, since it is a symbol of the cyclical nature of life; it conveys the energy of the world and existence. The spiral image can be applied to the shoulders, forearms, chest and back. The tattoo is more suitable for women, since another meaning of the tattoo is the feminine principle.

It is believed that Pythagoras was the first to introduce the concept of the golden ratio. The works of Euclid have survived to this day (he used the golden ratio to build regular pentagons, which is why such a pentagon is called “golden”), and the number of the golden ratio is named after the ancient Greek architect Phidias. That is, this is our number “phi” (denoted by the Greek letter φ), and it is equal to 1.6180339887498948482... Naturally, this value is rounded: φ = 1.618 or φ = 1.62, and in percentage terms the golden ratio looks like 62% and 38%.

What is unique about this proportion (and believe me, it exists)? Let's first try to figure it out using an example of a segment. So, we take a segment and divide it into unequal parts in such a way that its smaller part relates to the larger one, as the larger part relates to the whole. I understand, it’s not very clear yet what’s what, I’ll try to illustrate it more clearly using the example of segments:

So, we take a segment and divide it into two others, so that the smaller segment a relates to the larger segment b, just as the segment b relates to the whole, that is, the entire line (a + b). Mathematically it looks like this:

This rule works indefinitely; you can divide segments as long as you like. And, see how simple it is. The main thing is to understand once and that’s it.

But now let’s look at a more complex example, which comes across very often, since the golden ratio is also represented in the form of a golden rectangle (the aspect ratio of which is φ = 1.62). This is a very interesting rectangle: if we “cut off” a square from it, we will again get a golden rectangle. And so on endlessly. See:

But mathematics would not be mathematics if it did not have formulas. So, friends, now it will “hurt” a little. I hid the solution to the golden ratio under a spoiler; there are a lot of formulas, but I don’t want to leave the article without them.

The principle of the golden ratio. Successful creation or the rule of the golden ratio

Capturing the moment - this is precisely the moment of creation of an artist or photographer. In addition to inspiration, the master must follow strictly defined rules, which include: contrast, placement, balance, the rule of thirds and many others. But the rule of the golden section, also known as the rule of thirds, is still recognized as a priority.

Just something complicated

If we present the basis of the golden ratio rule in a simplified form, then in fact it is the division of the reproduced moment into nine equal parts(three vertically by three horizontally). For the first time, Leonardo da Vinci specifically introduced it, arranging all his compositions in this peculiar grid. It was he who practically confirmed that the key elements of the image should be concentrated at the points of intersection of vertical and horizontal lines.

The rule of the golden ratio in photography is subject to certain correction. In addition to the nine-segment grid, it is recommended to use so-called triangles. The principle of their construction is based on the rule of thirds. To do this, a diagonal is drawn from the extreme upper point to the lower one, and from the opposite upper point a ray is drawn, dividing the already existing diagonal at one of the internal intersection points of the grid. The key element of the composition should be displayed in the average size of the resulting triangles. It’s worth making a remark here: the given diagram for constructing triangles reflects only their principle, and, therefore, it makes sense to experiment with the given instructions.

How to use grid and triangles?

The golden ratio rule in photography operates according to certain standards depending on what is depicted in it.

Horizon factor. According to the rule of thirds, it should be placed along horizontal lines. Moreover, if the captured object is above the horizon, then the factor passes through the bottom line, and vice versa.

Location of the main object. The classic arrangement is one in which the central element is located at one of the intersection points. If the photographer selects two objects, then they should be diagonal or at parallel points.

Using triangles. The golden section rule in the case under consideration deviates from the canons, but only slightly. The object does not have to be located at the intersection point, but is as close as possible to it in the middle triangle.

Direction. This principle of shooting is used in dynamic photography and consists in the fact that two-thirds of the image space should remain in front of the moving object. This will provide the effect of moving forward and indicating the target. Otherwise, the photo may remain misunderstood.

Correction of the golden ratio rule

Despite the fact that the rule of thirds in the existing theory of composition is considered classic, more and more photographers are inclined to abandon it. Their motivation is simple: analysis of paintings by famous artists shows that the rule of the golden ratio does not hold true. One can argue with this statement.

Let's consider the well-known Mona Lisa, which opponents of using the rule of thirds cite as an example (forgetting that da Vinci himself was at the origins of its practical use). Their argument is that the master did not consider it necessary to arrange the key elements of the picture at the points of intersection, as required by the classical image. But they overlook the factor of horizontal lines, according to which the head and torso of the person depicted are positioned in such a way that the silhouette as a whole does not “hurt the eye.” Besides, in this work The spiral is used to a greater extent, which is mostly forgotten by photography theorists. And so it is possible to refute statements regarding almost every creation cited as an example.

The golden ratio rule can be used or abandoned if you want to emphasize the disharmony of the composition. However, it is impossible to say that it is not key in the formation of an art object.

Golden ratio in architecture. How to get the golden ratio

The golden ratio is most easily thought of as the ratio of two parts of the same object of different lengths separated by a point.

Simply put, how many lengths of a small segment will fit inside a large one, or the ratio of the largest part to the entire length of a linear object. In the first case, the golden ratio is 0.63, in the second case the aspect ratio is 1.618034.

In practice, the golden ratio is just a proportion, the ratio of segments of a certain length, sides of a rectangle or other geometric shapes, related or conjugate dimensional characteristics of real objects.

Initially, the golden proportions were derived empirically using geometric constructions. There are several ways to construct or derive harmonic proportion:

• Classical splitting of one of the sides right triangle and the construction of perpendiculars and secant arcs. To do this, from one end of the segment it is necessary to restore a perpendicular with a height of ½ of its length and construct a right triangle, as in the diagram.
  If we plot the height of the perpendicular on the hypotenuse, then with a radius equal to the remaining segment, the base is cut into two segments with lengths proportional to the golden ratio;
• Using the method of constructing the pentagram of Dürer, the brilliant German graphic artist and geometer. Today we know Dürer's method of the golden section as a method of constructing a star or pentagram inscribed in a circle in which there are at least four segments of harmonious proportion;
• In architecture and construction, the golden ratio is often used in an improved form. In this case, the division of a right triangle is used not along the leg, but along the hypotenuse, as a diagram.

For your information! Unlike the classic golden ratio, the architectural version implies an aspect ratio of 44:56.

If the standard version of the golden ratio for living beings, paintings, graphics, sculptures and ancient buildings was calculated as 37:63, then the golden ratio in architecture from the end of the 17th century began to be increasingly used as 44:56. Most experts consider the change in favor of more “square” proportions to be the spread of high-rise construction.

Many people dream of an ideal appearance, but not everyone has a clear idea of ​​what proportions can be considered harmonious. The formula for the golden ratio of the face is inextricably linked with the number 1.618 and other ratios. Thus, the proportions of beauty can be described as follows:

• the ratio of the height and width of the face should be 1.618;
• if you divide the length of the mouth and the width of the wings of the nose, you get 1.618;
• when dividing the distances between the pupils and eyebrows, again, the result is 1.618;
• the length of the eyes should match the distance between them, as well as the width of the nose;
• the areas of the face from the hairline to the eyebrows, from the bridge of the nose to the tip of the nose, and the lower part to the chin should be equal;
• If you draw vertical lines from the pupils to the corners of the lips, you will get three sections of equal width.

You need to understand that in nature the coincidence of all parameters is quite rare. But there's nothing wrong with that. This does not mean at all that faces that do not correspond to ideal proportions can be called ugly or unpretty. On the contrary, it is “defects” that sometimes give a face an unforgettable charm.

The golden ratio in the composition of drawings in paint.net
Mathematically, the “Golden Ratio” can be described as follows: the ratio of the whole to its larger part must be equal to the ratio of the larger part to the smaller. Let us illustrate with the example of a segment.

In our case, the entire segment B is divided into two parts - larger A and smaller B. Then, if B / A is equal to A / B, the division of the segment will be carried out according to the principle called the “Golden Section”.
Not exactly accurate, but close to the Golden Ratio, for example a ratio of 2/3 or 5/8. Numbers in such ratios are often called “golden”.
Why do we need this information for drawing in paint.net? The Golden Ratio is important for composition. It is believed that objects containing the “golden ratio” are perceived by people as the most harmonious. It was in similar ratios that famous artists chose the sizes of hosts for their paintings.
Let's consider a simplified version of constructing the “Golden Ratio” for the composition of a drawing, or the “Rule of Thirds”. The rule of thirds is that we mentally divide the frame into three parts horizontally and vertically, and at the intersection points of imaginary lines, we place the key and important details of our drawing or photo collage.

The principle of the "golden ratio" can be applied when cropping an image. So, for example, a frame formed according to the “golden ratio” rule from a large photograph may look like this.

Golden ratio in music. Golden section method in musical works

The “golden ratio” is rather a mathematical concept and its study is a task of science. This is the division of a certain quantity into two parts in such a ratio that the larger part will be related to the smaller one as the whole is to the larger one. This ratio turns out to be equal to the transcendental number Ф = 1.6180339... s amazing properties.

The golden section method is a search for function values ​​on a given interval. This method is based on the principle of dividing a segment in the so-called golden ratio. It is most widely used for searching for extreme values ​​when solving problems related to optimization. In addition to mathematics, the golden section method is used in the most different areas, ranging from architecture, art and ending with astronomy. For example, the famous Soviet director Sergei Eisenstein used it in his film “Battleship Potemkin,” and Leonardo da Vinci used it when he wrote the famous “La Gioconda.”

The golden ratio method is also used in music. It turned out that this golden proportion occurs very often in musical works. At the beginning of the 20th century, at a meeting of the Moscow Music Circle, a message was made containing information about the application of the golden ratio in music. The message was listened to with great interest by members of the musical circle, composers S. Rachmaninov, S. Taneyev, R. Gliere and others. Report by musicologist E.K. Rosenov “The Law of the Golden Ratio in Music and Poetry” marked the beginning of research into mathematical patterns associated with the golden ratio in music. He analyzed the musical works of Mozart, Bach, Beethoven, Wagner, Chopin, Glinka and other composers and showed that this “divine proportion” was present in their works.

The climax of many musical works is not located in the center, but is slightly shifted towards the end of the work in a ratio of 62:38 - this is the point of the golden proportion. Doctor of Art History, Professor L. Mazel noticed, while studying the eight-bar melodies of Chopin, Beethoven, Scriabin, that in many works of these composers the climax, as a rule, falls on the weak beat of the fifth, that is, at the point of the golden ratio - 5/8. L. Mazel believed that almost every composer who adheres to the harmonic style can find a similar musical structure: five bars of ascent and three bars of descent. This suggests that the golden section method was actively used by composers, either consciously or unconsciously. Probably, this structural arrangement of climactic moments gives a musical work a harmonic sound and emotional coloring.

A serious study of musical works for the manifestation of the golden proportion in them was undertaken by composer and musicologist L. Sabaneev. He studied about two thousand works of different composers and came to the conclusion that in approximately 75% of cases the golden ratio was present in a musical work at least once. The most a large number of works in which the golden proportion occurs, he noted in such composers as Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%) , Schubert (91%). He studied Chopin's etudes most closely and came to the conclusion that the golden ratio was determined in 24 out of 27 etudes. Only in three of Chopin's etudes was the golden ratio not found. Sometimes the structure of a musical work included both symmetry and the golden ratio. For example, many of Beethoven's works are divided into symmetrical parts, and in each of them the golden ratio appears.

So, we can say that the presence of the golden ratio in a piece of music is one of the criteria for the harmony of a musical composition.

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's Mona Lisa, a sunflower, a snail, a pine cone and human fingers have in common?

The answer to this question is hidden in the amazing numbers that have been discovered Italian medieval mathematician Leonardo of Pisa, better known by the name Fibonacci (born about 1170 - died after 1228), Italian mathematician . Traveling around the East, he became acquainted with the achievements of Arab mathematics; contributed to their transfer to the West.

After his discovery, these numbers began to be called after the famous mathematician. The amazing essence of the Fibonacci number sequence is that that each number in this sequence is obtained from the sum of the two previous numbers.

So, the numbers forming the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

are called “Fibonacci numbers”, and the sequence itself is called the Fibonacci sequence.

In Fibonacci numbers there is one very interesting feature. When dividing any number from the sequence by the number in front of it in the series, the result will always be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. (Approx. irrational number, i.e. a number whose decimal representation is infinite and non-periodic)

Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series... Exactly this constant number division in the Middle Ages was called the Divine proportion, and now in our days it is referred to as the golden section, golden average or golden proportion . In algebra, this number is denoted by the Greek letter phi (Ф)

So, Golden ratio = 1:1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio

Artists, scientists, fashion designers, designers make their calculations, drawings or sketches based on the ratio of the golden ratio. They use measurements from the human body, which was also created according to the principle of the golden ratio. Before creating their masterpieces, Leonardo Da Vinci and Le Corbusier took the parameters of the human body, created according to the law of the Golden Proportion.

The most main book For all modern architects, E. Neufert’s reference book “Building Design” contains basic calculations of the parameters of the human torso, which contain the golden proportion.

Proportions various parts our body is a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned. The principle of calculating the gold measure on the human body can be depicted in the form of a diagram:

M/m=1.618

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

In addition to this, there are several more basic golden proportions of our body:

* the distance from the fingertips to the wrist to the elbow is 1:1.618;

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

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

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

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

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

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

In the structure of human facial features there are also many examples that are close in value to the golden ratio formula. However, do not immediately rush for a ruler to measure the faces of all people. Because exact correspondences to the golden ratio, according to scientists and artists, artists and sculptors, exist only in people with perfect beauty. Actually, the exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.

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

There are other embodiments of the golden ratio rule on the human face. Here are a few of these relationships:

*Face height/face width;

* Central point of connection of the lips to the base of the nose / length of the nose;

* Face height / distance from the tip of the chin to the central point where the lips meet;

*Mouth width/nose width;

* Nose width / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand

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

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

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

* A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence:

The golden ratio in the structure of the human lungs

American physicist B.D. West and Dr. A.L. Goldberger, during physical and anatomical studies, established that the golden ratio also exists in the structure of the human lungs.

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

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

Structure of the golden orthogonal quadrilateral and spiral

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

In geometry, a rectangle with this aspect ratio came to be called the golden rectangle. Its long sides are in relation to its short sides in a ratio of 1.168:1.

The golden rectangle also has many amazing properties. The Golden Rectangle has many unusual properties. By cutting a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again obtain a golden rectangle of smaller dimensions. This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located along a logarithmic spiral, having important V mathematical models natural objects (for example, snail shells).

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

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

“We like the look of a spiral because visually we can easily look at it.”

In nature

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

* Botanists have found that in the arrangement of leaves on a branch, sunflower seeds or pine cones, the Fibonacci series is clearly manifested, and therefore the law of the golden ratio is manifested;

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

All springs in the spiral have the same shape. Mathematicians have found that even with an increase in the size of the springs, the shape of the spiral remains unchanged. There is no other form in mathematics that has the same unique properties as the spiral.

The structure of sea shells

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

“The inner surface of the shells is impeccably smooth, while the outer surface is completely covered with roughness and irregularities. The mollusk was in a shell and for this the inner surface of the shell had to be perfectly smooth. External corners-bends of the shell increase its strength, hardness and thus increase its strength. The perfection and amazing intelligence of the structure of the shell (snail) is amazing. The spiral idea of ​​shells is a perfect geometric form and is amazing in its honed beauty."

In most snails that have shells, the shell grows in the shape of a logarithmic spiral. However, there is no doubt that these unreasonable creatures not only have no idea about the logarithmic spiral, but do not even have the simplest mathematical knowledge to create a spiral-shaped shell for themselves.

But then how were these unreasonable creatures able to determine and choose for themselves the ideal form of growth and existence in the form of a spiral shell? Could these living beings, whom scientists world calls primitive life forms, calculate that the logarithmic shape of a shell would be ideal for their existence?

Of course not, because such a plan cannot be realized without intelligence and knowledge. But neither primitive mollusks nor unconscious nature possess such intelligence, which, however, some scientists call the creator of life on earth (?!)

Trying to explain the origin of such even the most primitive form of life by a random combination of certain natural circumstances is absurd, to say the least. It is clear that this project is a conscious creation.

Biologist Sir D'arky Thompson calls this type of growth of sea shells "growth form of dwarves."

Sir Thompson makes this comment:

“There is no simpler system than the growth of sea shells, which grow and expand in proportion, maintaining the same shape. The most amazing thing is that the shell grows, but never changes shape.”

The Nautilus, measuring several centimeters in diameter, is the most striking example of the gnome growth habit. S. Morrison describes this process of nautilus growth as follows, which seems quite difficult to plan even with the human mind:

“Inside the nautilus shell there are many compartments-rooms with partitions made of mother-of-pearl, and the shell itself inside is a spiral expanding from the center. As the nautilus grows, another room grows in the front part of the shell, but this time it is larger than the previous one, and the partitions of the room left behind are covered with a layer of mother-of-pearl. Thus, the spiral expands proportionally all the time.”

Here are just some types of spiral shells with a logarithmic growth pattern in accordance with their scientific names:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa, Solarium Trochleare.

All discovered fossil remains of shells also had a developed spiral shape.

However, the logarithmic growth form is found in the animal world not only in mollusks. The horns of antelopes, wild goats, rams and other similar animals also develop in the form of a spiral according to the laws of the golden ratio.

Golden ratio in the human ear

In the human inner ear there is an organ called Cochlea (“Snail”), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also shaped like a snail, containing a stable logarithmic spiral shape = 73º 43'.

Animal horns and tusks developing in a spiral shape

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

Golden ratio in the structure of microcosms

Geometric shapes are not limited to just a triangle, square, pentagon or hexagon. If we connect these figures with each other in different ways, we get new three-dimensional geometric figures. Examples of this are figures such as a cube or a pyramid. However, besides them, there are also other three-dimensional figures that we have not encountered in everyday life, and whose names we hear, perhaps for the first time. Among such three-dimensional figures are the tetrahedron (regular four-sided figure), octahedron, dodecahedron, icosahedron, etc. The dodecahedron consists of 13 pentagons, the icosahedron of 20 triangles. Mathematicians note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden ratio.

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

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

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

“Dr. Kaspar and I showed that for the spherical shell of the virus, the most optimal shape is symmetry such as the icosahedron shape. This order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. 14 Installation of such cubes requires an extremely accurate and detailed explanatory diagram. Whereas unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units.”