/ Forens.Ru - 2008.
bibliographic description:
Golden ratio in human anatomy / Forens.Ru - 2008.
Latest additions to the library
Aspects of molecular genetic research of human hair depending on their morphological characteristics. II. Features of genotyping / Aleksandrova V.Yu., Bogatyreva E.A., Kuklev M.Yu., Lapenkov M.I., Plakhina N.V. // Forensic-medical examination. - M., 2019. - No. 2. — P. 22-25.
Possibility of determining the distance of a shot from a 12-gauge hunting weapon based on signs of clothing damage and the corresponding mathematical models / Suvorov A.S., Belavin A.V., Makarov I.Yu., Stragis V.B., Raizberg S.A. , Gyulmamedova N.D. // Forensic-medical examination. - M., 2019. - No. 2. — P. 19-21.
Complex forensic examination of images of a person’s external appearance / Rossinskaya E.R., Zinin A.M. // Forensic-medical examination. - M., 2019. - No. 2. — P. 15-18.
Structure of fatal mechanical injury in Russia (based on materials from 2003-2017) / Kovalev A.V., Makarov I.Yu., Samokhodskaya O.V., Kuprina T.A. // Forensic-medical examination. - M., 2019. - No. 2. - pp. 11-14.
Methodological approaches to the forensic medical examination of the health status of children in cases of neglect of their needs / Kovalev A.V., Kemeneva Yu.V. // Forensic-medical examination. - M., 2019. - No. 2. — P. 4-10.
Geometry is an exact and quite complex science, which at the same time is a kind of art. Lines, planes, proportions - all this helps to create many truly beautiful things. And oddly enough, this is based on geometry in its most varied forms. In this article we will look at one very unusual thing, which is directly related to this. The golden ratio is exactly the geometric approach that will be discussed.
The shape of an object and its perception
People most often rely on the shape of an object in order to recognize it among millions of others. It is by its shape that we determine what kind of thing lies in front of us or stands in the distance. We first recognize people by the shape of their body and face. Therefore, we can confidently say that the shape itself, its size and appearance is one of the most important things in human perception.
For people, the form of anything is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and the feeling of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special ratio were used, which is called the golden ratio.
The concept of the golden ratio
So, golden ratio is the golden proportion, 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 human eye.
From the ancient history of the golden ratio
The golden ratio is often used in many different areas of life today. But the history of this concept goes back to ancient times, when sciences such as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the 6th century BC. But even before that, knowledge about such a ratio was used in practice in Ancient Egypt and Babylon. A clear indication of this are the pyramids, for the construction of which exactly this golden proportion was used.
New period
The Renaissance brought new breath to harmonic division, especially thanks to Leonardo da Vinci. This ratio has increasingly begun to be used both in geometry and in art. Scientists and artists began to study the golden ratio more deeply and create books that examine this issue.
One of the most important historical works related to the golden ratio is a book by Luca Pancholi called The Divine Proportion. Historians suspect that the illustrations of this book were done by Leonardo before Vinci himself.
golden ratio
Mathematics gives a very clear definition of proportion, which says that it is the equality of two ratios. Mathematically, this can be expressed by the following equality: a: b = c: d, where a, b, c, d are some specific values.
If we consider the proportion of a segment divided into two parts, we can encounter only a few situations:
- The segment is divided into two absolutely even parts, which means AB:AC = AB:BC, if AB is the exact beginning and end of the segment, and C is the point that divides the segment into two equal parts.
- The segment is divided into two unequal parts, which can be in very different proportions to each other, which means that here they are completely disproportionate.
- The segment is divided so that AB:AC = AC:BC.
As for the golden ratio, this is a proportional division of a segment into unequal parts, when the entire segment relates to the larger part, just as the larger part itself relates to the smaller one. There is another formulation: the smaller segment is related to the larger one, just as the larger one is to the entire segment. In mathematical terms, it looks like this: a:b = b:c or c:b = b:a. This is exactly what the golden ratio formula looks like.
Golden ratio in nature
The golden ratio, examples of which we will now consider, refers to incredible phenomena in nature. This is very beautiful examples the fact that mathematics is not just numbers and formulas, but a science that has more than a real reflection in nature and our life in general.
For living organisms, one of the main tasks in life is growth. This desire to take one’s place in space, in fact, occurs in several forms - growing upward, almost horizontally spreading on the ground, or twisting in a spiral on some kind of support. And as incredible as it may be, many plants grow according to the golden ratio.
Another one almost incredible fact- these are the relationships in the body of lizards. Their body looks quite pleasing to the human eye and this is possible due to the same golden ratio. To be more precise, the length of their tail relates to the length of the entire body as 62:38.
Interesting facts about the rules of the golden ratio
The golden ratio is a truly incredible concept, which means that throughout history we can come across many truly interesting facts about this proportion. We present you some of them:
Golden ratio in the human body
In this section it is necessary to mention a very significant person, namely S. Zeizinga. This is a German researcher who has done a tremendous amount of work in the field of studying the golden ratio. He published a work entitled Aesthetic Studies. In his work, he presented the golden ratio as an absolute concept that is universal for all phenomena both in nature and in art. Here we can recall the golden ratio of the pyramid along with the harmonious proportion of the human body and so on.
It was Zeising who was able to prove that the golden ratio, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experiment. According to Zeising's research, the main indicator of the golden ratio is the division of the body by the navel point. Thus, the male body with an average ratio of 13:8 is slightly closer to the golden ratio than the female body, where the golden ratio is 8:5. The golden ratio can also be observed in other parts of the body, such as the hand.
About the construction of the golden ratio
In fact, constructing the golden ratio is a simple matter. As we see, even ancient people coped with this quite easily. What can we say about modern knowledge and technologies of mankind. In this article we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we will confidently declare that it is, in fact, possible. Moreover, this can be done in more than one way.
Since this is a fairly simple geometry, the golden ratio is quite simple to construct even at school. Therefore, information about this can be easily found in specialized books. By studying the golden ratio, 6th graders are fully able to understand the principles of its construction, which means that even children are smart enough to master such a task.
Golden ratio in mathematics
The first acquaintance with the golden ratio in practice begins with a simple division of a straight line segment in the same proportions. Most often this is done using a ruler, compass and, of course, a pencil.
Segments of the golden proportion are expressed as an infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... In order to make these calculations more practical, very often they use not exact, but approximate values, namely - 0 .62 and .38. If the segment AB is taken as 100 parts, then its larger part will be equal to 62, and the smaller part will be equal to 38 parts, respectively.
The main property of the golden ratio can be expressed by the equation: x 2 -x-1=0. When solving, we get the following roots: x 1.2 =. Although mathematics is an exact and rigorous science, like its section - geometry, it is precisely properties such as the laws of the golden section that cast mystery on this topic.
Harmony in art through the golden ratio
In order to summarize, let’s briefly consider what has already been discussed.
Basically, many pieces of art fall under the rule of the golden ratio, where a ratio close to 3/8 and 5/8 is observed. This is the rough formula of the golden ratio. The article has already mentioned a lot about examples of using the section, but we will look at it again through the prism of the ancient and contemporary art. So, the most vivid examples from ancient times:
As for the probably conscious use of proportion, starting from the time of Leonardo da Vinci, it came into use in almost all areas of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.
This harmony is striking in its scale...
Hello, friends!
Have you heard anything about Divine Harmony or the Golden Ratio? Have you ever thought about why something seems ideal and beautiful to us, but something repels us?
If not, then you have successfully come to this article, because in it we will discuss the golden ratio, find out what it is, what it looks like in nature and in humans. Let's talk about its principles, find out what the Fibonacci series is and much more, including the concept of the golden rectangle and the golden spiral.
Yes, the article has a lot of images, formulas, after all, the golden ratio is also mathematics. But everything is described in fairly simple language, clearly. And at the end of the article, you will find out why everyone loves cats so much =)
What is the golden ratio?
To put it simply, the golden ratio is a certain rule of proportion that creates harmony?. That is, if we do not violate the rules of these proportions, then we get a very harmonious composition.
The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole.
But besides this, the golden ratio is mathematics: it has a specific formula and a specific number. Many mathematicians, in general, consider it the formula of divine harmony, and call it “asymmetrical symmetry”.
The golden ratio has reached our contemporaries since the times Ancient Greece However, there is an opinion that the Greeks themselves had already spotted the golden ratio among the Egyptians. Because many works of art Ancient Egypt clearly constructed according to the canons of this proportion.
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 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 take a closer look 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.
Fibonacci series and golden ratio
We continue to create and observe the magic of mathematics and the golden ratio. In the Middle Ages there was such a comrade - Fibonacci (or Fibonacci, they spell it differently everywhere). He loved mathematics and problems, he also had an interesting problem with the reproduction of rabbits =) But that’s not the point. He opened number sequence, the numbers in it are called “Fibonacci numbers”.
The sequence itself looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233... and so on ad infinitum.
In other words, the Fibonacci sequence is a sequence of numbers where each subsequent number is equal to the sum of the previous two.
What does the golden ratio have to do with it? You'll see now.
Fibonacci Spiral
To see and feel the whole connection between the Fibonacci number series and the golden ratio, you need to look at the formulas again.
In other words, from the 9th term of the Fibonacci sequence we begin to obtain the values of the golden ratio. And if we visualize this whole picture, we will see how the Fibonacci sequence creates rectangles closer and closer to the golden rectangle. This is the connection.
Now let's talk about the Fibonacci spiral, it is also called the “golden spiral”.
The golden spiral is a logarithmic spiral whose growth coefficient is equal to φ4, where φ is the golden ratio.
In general, from a mathematical point of view, the golden ratio is perfect proportion. But this is just the beginning of her miracles. Almost the entire world is subject to the principles of the golden ratio; nature itself created this proportion. Even esotericists see numerical power in it. But we will definitely not talk about this in this article, so in order not to miss anything, you can subscribe to site updates.
Golden ratio in nature, man, art
Before we begin, I would like to clarify a number of inaccuracies. Firstly, the very definition of the golden ratio in this context is not entirely correct. The fact is that the very concept of “section” is a geometric term, always denoting a plane, but not a sequence of Fibonacci numbers.
And, secondly, the number series and the ratio of one to the other, of course, have been turned into a kind of stencil that can be applied to everything that seems suspicious, and one can be very happy when there are coincidences, but still, common sense should not be lost.
However, “everything was mixed up in our kingdom” and one became synonymous with the other. So, in general, the meaning is not lost from this. Now let's get down to business.
You will be surprised, but the golden ratio, or rather the proportions as close as possible to it, can be seen almost everywhere, even in the mirror. Don't believe me? Let's start with this.
You know, when I was learning to draw, they explained to us how easier it is to build a person’s face, his body, and so on. Everything must be calculated relative to something else.
Everything, absolutely everything is proportional: bones, our fingers, palms, distances on the face, the distance of outstretched arms in relation to the body, and so on. But even that's not all internal structure of our body, even it, is equal or almost equal to the golden section formula. Here are the distances and proportions:
from shoulders to crown to head size = 1:1.618
from the navel to the crown to the segment from the shoulders to the crown = 1:1.618
from navel to knees and from knees to feet = 1:1.618
from the chin to the extreme point of the upper lip and from it to the nose = 1:1.618
Isn't this amazing!? Harmony in pure form, both inside and outside. And that is why, on some subconscious level, some people do not seem beautiful to us, even if they have a strong, toned body, velvety skin, beautiful hair, eyes and stuff and everything else. But, all the same, the slightest violation of the proportions of the body, and the appearance already slightly “hurts the eyes.”
In short, the more beautiful a person seems to us, the closer his proportions are to ideal. And this, by the way, can be attributed not only to the human body.
Golden ratio in nature and its phenomena
A classic example of the golden ratio in nature is the shell of the mollusk Nautilus pompilius and the ammonite. But this is not all, there are many more examples:
in the curls of the human ear we can see golden spiral;
its same (or close to it) in the spirals along which galaxies twist;
and in the DNA molecule;
According to the Fibonacci series, the center of a sunflower is arranged, cones grow, the middle of flowers, a pineapple and many other fruits.
Friends, there are so many examples that I’ll just leave the video here (it’s just below) so as not to overload the article with text. Because if you dig into this topic, you can delve into such a jungle: even the ancient Greeks proved that the Universe and, in general, all space is planned according to the principle of the golden ratio.
You will be surprised, but these rules can be found even in sound. See:
The highest point of sound that causes pain and discomfort in our ears is 130 decibels.
We divide the proportion 130 by the golden ratio number φ = 1.62 and we get 80 decibels - the sound of a human scream.
We continue to divide proportionally and get, let’s say, the normal volume of human speech: 80 / φ = 50 decibels.
Well, the last sound that we get thanks to the formula is a pleasant whispering sound = 2.618.
Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. I haven’t tested it, and I don’t know how true this theory is, but you must agree, it sounds impressive.
You can read in absolutely everything living and non-living supreme beauty and harmony.
The main thing is not to get carried away with this, because if we want to see something in something, we will see it, even if it is not there. For example, I paid attention to the design of the PS4 and saw the golden ratio there =) However, this console is so cool that I wouldn’t be surprised if the designer really did something clever there.
Golden ratio in art
This is also a very large and extensive topic that is worth considering separately. Here I will just note a few basic points. The most remarkable thing is that many works of art and architectural masterpieces of antiquity (and not only) were made according to the principles of the golden ratio.
Egyptian and Mayan pyramids, Notre Dame de Paris, Greek Parthenon and so on.
IN musical works Mozart, Chopin, Schubert, Bach and others.
In painting (this is clearly visible there): all the most famous paintings famous artists made taking into account the rules of the golden ratio.
These principles can be found in Pushkin’s poems and in the bust of the beautiful Nefertiti.
Even now, the rules of the golden ratio are used, for example, in photography. Well, and of course, in all other arts, including cinematography and design.
Golden Fibonacci cats
And finally, about cats! Have you ever wondered why everyone loves cats so much? They've taken over the Internet! Cats are everywhere and it's wonderful =)
And the whole point is that cats are perfect! Don't believe me? Now I’ll prove it to you mathematically!
Do you see? The secret is revealed! Cats are ideal from the point of view of mathematics, nature and the Universe =)
*I'm kidding, of course. No, cats are really ideal) But no one has measured them mathematically, probably.
That's basically it, friends! We'll see you in the next articles. Good luck to you!
P.S. Images taken from medium.com.
What do they have in common? Egyptian pyramids, the Mona Lisa painting by Leonardo da Vinci and the Twitter and Pepsi logos?
Let’s not delay the answer - they were all created using the golden ratio rule. The golden ratio is the ratio of two quantities a and b, which are not equal to each other. This proportion is often found in nature; the golden ratio rule is also actively used in fine arts and design - compositions created using “divine proportions” are well balanced and, as they say, pleasing to the eye. But what exactly is the golden ratio and can it be used in modern disciplines, for example, in web design? Let's figure it out.
A LITTLE MATH
Let's say we have a certain segment AB, divided in two by point C. The ratio of the lengths of the segments is: AC/BC = BC/AB. That is, a segment is divided into unequal parts in such a way that the larger part of the segment makes up the same share in the whole, undivided segment as the smaller segment makes up in the larger one.
This unequal division is called the golden ratio. The golden ratio is designated by the symbol φ. The value of φ is 1.618 or 1.62. In general, to put it very simply, this is the division of a segment or any other value in the ratio of 62% and 38%.
“Divine proportion” has been known to people since ancient times; this rule was used in the construction of the Egyptian pyramids and the Parthenon; the golden ratio can be found in paintings Sistine Chapel and in Van Gogh's paintings. The golden ratio is still widely used today - examples that are constantly before our eyes are the Twitter and Pepsi logos.
The human brain is designed in such a way that it considers as beautiful those images or objects in which an unequal proportion of parts can be detected. When we say about someone that “he is well-proportioned,” we unknowingly mean the golden ratio.
The golden ratio can be applied to various geometric shapes. If we take a square and multiply one side by 1.618, we get a rectangle.
Now, if we superimpose a square on this rectangle, we can see the golden ratio line:
If we continue to use this proportion and break the rectangle into smaller parts, we get this picture:
It is not yet clear where this fragmentation of geometric figures will lead us. A little more and everything will become clear. If we draw a smooth line equal to a quarter of a circle in each of the squares of the diagram, then we will get a Golden Spiral.
This is an unusual spiral. It is also sometimes called the Fibonacci spiral, in honor of the scientist who studied the sequence in which each number is early to the sum of the two previous ones. The point is that this mathematical relationship, which we visually perceive as a spiral, is found literally everywhere - sunflowers, sea shells, spiral galaxies and typhoons - there is a golden spiral everywhere.
HOW CAN YOU USE THE GOLDEN RATIO IN DESIGN?
So, the theoretical part is over, let's move on to practice. Is it really possible to use the golden ratio in design? Yes, you can. For example, in web design. Considering this rule, you can get correct ratio compositional elements of the layout. As a result, all parts of the design, down to the smallest ones, will be harmoniously combined with each other.
If we take a typical layout with a width of 960 pixels and apply the golden ratio to it, we will get this picture. The ratio between the parts is the already known 1:1.618. The result is a two-column layout, with a harmonious combination of two elements.
Sites with two columns are very common and this is far from accidental. Here, for example, is the National Geographic website. Two columns, golden ratio rule. Good design that is orderly, balanced and respects the requirements of visual hierarchy.
One more example. Design studio Moodley has developed a corporate identity for the Bregenz performing arts festival. When the designers worked on the event poster, they clearly used the golden ratio rule in order to correctly determine the size and location of all elements and, as a result, obtain the ideal composition.
Lemon Graphic, who created the visual identity for Terkaya Wealth Management, also used a 1:1.618 ratio and a golden spiral. Three design elements business card fit perfectly into the scheme, as a result of which all parts fit together very well
Here's another interesting use of the golden spiral. Before us again is the National Geographic website. If you take a closer look at the design, you can see that there is another NG logo on the page, only a smaller one, which is located closer to the center of the spiral.
Of course, this is not accidental - the designers knew very well what they were doing. This is a great place to duplicate a logo, as our eye naturally moves toward the center of the composition when viewing a site. This is how the subconscious works and this must be taken into account when working on design.
GOLDEN CIRCLES
“Divine proportion” can be applied to any geometric shapes, including circles. If we inscribe a circle in squares, the ratio between which is 1:1.618, then we get golden circles.
Here is the Pepsi logo. Everything is clear without words. Both the ratio and the way the smooth arc of the white logo element was achieved.
With the Twitter logo, things are a little more complicated, but here too you can see that its design is based on the use of golden circles. It doesn't follow the "divine proportion" rule a little, but for the most part all of its elements fit into the scheme.
CONCLUSION
As you can see, despite the fact that the golden ratio rule has been known since time immemorial, it is not at all outdated. Therefore, it can be used in design. It is not necessary to try your best to fit into the scheme - design is an imprecise discipline. But if you need to achieve a harmonious combination of elements, then it won’t hurt to try to apply the principles of the golden ratio.
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 is related to the larger one, just as the larger part is related to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.
The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as "asymmetrical symmetry", calling it in a broad sense a universal rule reflecting the structure and order of our world order.
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 in the golden section the divine trinity: 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 proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.
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 in the shape of an egg, if a conditional line is drawn 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. Goethe later noted nature’s attraction 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 " harmonic proportions”, which influenced the aesthetics of 20th century architecture.
Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many antique 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 male body 13:8 is closer to the golden ratio than the proportions of the female body - 8:5.
ART OF SPATIAL FORMS
The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you can neither remove nor add anything, you cannot even add an extra point, this is real mathematics.” For a long time, artists followed this law intuitively, but after Leonardo da Vinci, the creation process painting can no longer do without solving 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 the painting by Nikolai Ge “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: on their list are the Great Pyramids of Giza, the Cathedral Notre Dame of Paris, St. Basil's Cathedral, 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
The forms of temporary art 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 late period Pushkin’s creativity 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. So climax « Queen of Spades" is dramatic scene 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 golden ratio ratios 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 rule of the golden ratio, 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.