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 =)
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 of Ancient Greece, however, there is an opinion that the Greeks themselves had already spied the golden ratio among the Egyptians. Because many works of art of Ancient Egypt are clearly built 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 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 the 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 many times. 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.
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 discovered a 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.
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 φ4, where φ is the golden ratio.
In general, from a mathematical point of view, the golden ratio is an ideal 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.
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 this is not all, the internal structure of our body, even this, 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 its purest form, both inside and outside. And that is why, at some subconscious level, some people do not seem beautiful to us, even if they have a strong, toned body, velvety skin, beautiful hair, eyes, etc., 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.
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 a 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.
One can read the highest beauty and harmony in absolutely everything living and non-living.
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.
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 the musical works of Mozart, Chopin, Schubert, Bach and others.
In painting (this is clearly visible): all the most famous paintings by famous artists are 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.
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.
The rule of the “golden ratio” in painting, photography, mathematics, architecture, artThe one-third rule, or the golden ratio. This rule was derived by Leonardo Da Vinci and is one of the most important. The most important element of the image is located at a distance of approximately 1/3 of the height or width of the frame from its border. Divide the frame into nine equal squares. The points of intersection of the lines are the “golden ratio”.
Photo by Andrey Popov
Another diagram confirming the “golden ratio” is shown below. Let's draw a diagonal of the photo, then from the free corner we lower a line to this diagonal at a right angle. This way our photo will be divided into three right triangles. The diagram can be rotated any way you want, but the most important parts of the plot should be located in these triangles. 
Here is a drawing illustrating two “golden ratio” schemes at once. 
A person distinguishes objects around him by their shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the shape. The form, the construction of which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a feeling of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points, and they are located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.
This discovery was called the “golden ratio” of the painting by artists of that time. Therefore, in order to draw attention to the main element of the photograph, it is necessary to combine this element with one of the visual centers.
The properties of the golden ratio have created a romantic aura of mystery and almost mystical worship around this number.
History of the golden ratio
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Hesira, depicted on a relief of a wooden board from a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded. The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for constructing dynamic rectangles. Plato (427...347 BC) also knew about the golden division. His dialogue “Timaeus” is devoted to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division. The facade of the ancient Greek temple of the Parthenon contains golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world. The Pompeian compass (museum in Naples) also contains the proportions of the golden division. In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. In the 2nd book of “Principles” the geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (II century BC), Pappus (III century AD), and others. In medieval Europe, with the golden division We met through Arabic translations of Euclid’s Elements. The translator J. Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates.
During the Renaissance, interest in the golden division increased among scientists and artists due to its use in both geometry and art, especially in architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry.
Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “The Divine Proportion” was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity - God the son, God the father and God the holy spirit (it was implied that the small segment is the personification of God the son, the larger segment is the god of the father, and the entire segment - God of the Holy Spirit).
Leonardo da Vinci also paid a lot of attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in the golden division. Therefore, he gave this division the name golden ratio. So it still remains as the most popular.
At the same time, in the north of Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches the introduction to the first version of the treatise on proportions. Dürer writes. “It is necessary that someone who knows how to do something should teach it to others who need it. This is what I set out to do.”
Judging by one of Dürer's letters, he met with Luca Pacioli while in Italy. Albrecht Durer develops in detail the theory of proportions of the human body. Dürer assigned an important place in his system of relationships to the golden section. A person’s height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is well known.
Great astronomer of the 16th century. Johannes Kepler called the golden ratio one of the treasures of geometry. He was the first to draw attention to the importance of the golden proportion for botany (plant growth and their structure).
Kepler called the golden proportion self-continuing. “It is structured in such a way,” he wrote, “that the two lowest terms of this never-ending proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion is maintained until infinity."
The construction of a series of segments of the golden proportion can be done both in the direction of increase (increasing series) and in the direction of decrease (descending series).
If on a straight line of arbitrary length, put aside segment m, put aside segment M next to it.
In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Research”. What happened to Zeising was exactly what should inevitably happen to a researcher who considers a phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions “mathematical aesthetics.”
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied. Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the numbers expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction or the other. His next book was titled “The Golden Division as a Basic Morphological Law in Nature and Art.” In 1876, a small book, almost a brochure, was published in Russia outlining this work of Zeising. The author took refuge under the initials Yu.F.V. This publication does not mention a single work of painting.
Golden proportions in parts of the human body
"Golden Ratio" has long been synonymous with the word “harmony”. Collocation "golden ratio" It simply has a magical effect. If you are carrying out some kind of artistic commission (it doesn’t matter if it’s a painting, sculpture or design), the phrase “the work was done in full accordance with the rules golden ratio“can be an excellent argument in your favor - the customer most likely will not be able to check, but it sounds solid and convincing. At the same time, few understand what is hidden under these words. Meanwhile, figure out what it is golden ratio and how it works is quite simple.
The golden ratio is a division of a segment into 2 proportional parts, in which the whole is to the larger part as the larger is to the smaller . Mathematically, this formula looks like this: With : b = b : a or a : b = b : c.
The result of the algebraic solution of this proportion will be the irrational number Ф (Ф in honor of the ancient Greek sculptor Phidias).
I will not give the equation itself so as not to load the text. If desired, it can be easily found on the Internet. I will only say that F will be approximately equal to 1.618. Remember this number, this is a numerical expression golden ratio.
So, golden ratio– this is a rule of proportion, it shows the relationship between parts and the whole.
On any segment you can find a “golden point” - a point that divides this segment into parts perceived as harmonious. Accordingly, you can also divide any object. For example, let's construct a rectangle divided in accordance with the “golden” proportion:
The ratio of the larger side of the resulting rectangle to the smaller one will be approximately 1.6 (note that the smaller rectangle resulting from the construction will also be golden).
In general, in articles explaining the principle golden ratio, there are many similar drawings. This is explained simply: the fact is that finding the “golden point” by conventional measurement is problematic, since the number F, as we remember, is irrational. But such problems are easily solved using geometric methods, using a compass and a ruler.
However, the presence of a compass is not at all necessary to apply the law in practice. There are a number of numbers that are considered to be an arithmetic expression of the golden ratio. This Fibonacci series . This is the row:
0 1 1 2 3 5 8 13 21 34 55 89 144 etc.
It is not necessary to memorize this sequence; it can be easily calculated: each number in the Fibonacci series is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division. So, 21: 34 = 0.617, and 34: 55 = 0.618.
One of the most ancient (and still attractive) symbols, the pentagram is an excellent illustration of the principle golden ratio.
In a regular five-pointed star, each segment is divided by a segment intersecting it in golden ratio(in the above figure, the ratio of the red segment to the green, as well as the green to blue, as well as the blue to violet, are equal). (quote from Wikipedia).
Why does the “golden proportion” seem so harmonious?
The theory golden ratio there are a lot of both supporters and opponents. In general, the idea that beauty can be measured and calculated using a mathematical formula is not attractive to everyone. And, perhaps, this concept would indeed seem far-fetched mathematical aesthetics, if not for the numerous examples of natural shape formation, corresponding golden ratio.
The term itself golden ratio"introduced by Leonardo da Vinci. Being a mathematician, da Vinci also sought a harmonious relationship for the proportions of the human body.
“If we tie a human figure - the most perfect creation of the Universe - with a belt and then measure the distance from the belt to the feet, then this value will relate to the distance from the same belt to the top of the head, just as the entire height of a person relates to the length from the waist to the feet.”
The division of the body by the navel point is the most important indicator golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man. Proportions golden ratio manifest themselves in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Gradually, golden ratio turned into an academic canon, and when a revolt against academicism matured in art, about golden ratio forgotten for a while. However, in the mid-19th century, this concept again became popular thanks to the works of the German researcher Zeising. He made many measurements (about 2000 people), and concluded that golden ratio expresses the average statistical law. Besides people , Zeising explored architectural structures, vases, flora and fauna, poetic meters and musical rhythms. According to his theory, golden ratio is an absolute, a universal rule for any phenomena of nature and art.
The principle of the golden proportion is used in various fields, not only in art, but also in science and technology. Being so universal, it is, of course, subject to many doubts. Often manifestations golden ratio are declared the result of erroneous calculations or a simple coincidence (or even fraud). In any case, any comments from both supporters of the theory and opponents should be treated critically.
You can read about how to apply this principle in practice.
The laws of shape formation in nature and art, visual perception and compositional construction of the image are outlined. The role of the golden ratio is shown. Recommendations are given for the practical application of the golden proportion in creating a holistic harmonious form that most fully expresses the content of a work of art and satisfies the human need for beauty.
About the golden ratio.
The debate about whether science should or should not intrude into the reserved areas of art has been going on for a long time. And this dispute is clearly scholastic in nature. In all eras of prosperity, art has entered into an alliance with science. Artist-thinkers, theorists and teachers who reflected on the problems of educating young people always came to the conclusion that without science art cannot develop and prosper. The artist and teacher N.P. Krymov wrote: “They say: art is not science, not mathematics, that it is creativity, mood, and that nothing in art can be explained - look and admire. In my opinion, this is not so. Art is explicable and very logical, you need and can know about it, it is mathematical... You can prove exactly why a picture is good and why it is bad” 1 V. I. Surikov argued that in composition there is some kind of immutable law, when in a picture You can’t remove or add anything, you can’t even add an extra point, this is real mathematics. Famous French architect and architectural theorist of the 19th century. Viollet-le-Duc believed that a form that cannot be explained will never be beautiful. On the doors of the Sikyon School of Drawing in Ancient Greece it was written: “People who do not know geometry are not allowed here.” Artists should not be afraid of mathematics; it is outside and inside us. Behind the apparent simplicity and randomness of the living perception of the surrounding reality lies mathematics. When we listen to music, our brain does algebra. When we look at something, our brains do geometry.
Table of contents
Preface
Introduction
Chapter first
Golden ratio and questions of composition theory
About the golden ratio
Golden ratio - harmonic proportion
Golden ratio and symmetry
History of the golden ratio
Natural scientific foundations of composition theory
Principles of formation in nature
Patterns of visual perception
Objectification of light impressions
Scientific theory of composition
Definition of composition
Search for the laws of composition
What is the scientific theory of composition
Human creativity
Laws, rules, techniques and means of composition
Chapter two
Practical composition
Composition when working from life
Point of view
Distance to object. The size of the image on the seed. Transferring the distance to an object
The imaginary picture and the real picture
Methods for determining viewing angles when working from nature
Techniques for mechanical image acquisition
Techniques of compositional constructions
Analysis of the picture
Composition of still life and interior
Landscape composition
About the portrait. Live performances
The artist's place in front of the painting
Image integrity
Chapter Three
Working on a painting
The golden ratio in the linear construction of a picture
Idea, format, rhythm and golden ratio
Sketch of the painting. Distance calculations and solving the “inverse problem”
The geometric center of the picture and the line of the golden ratio. Harmonization of form
The main line of sight in the picture
Compositional algorithm for linear construction of a picture
Golden ratio and composition of tones
Light and eye
General light tone
The law of three components and the principle of close relationships
Composition of light tones
Golden ratio and color composition
General color tone of the painting
Palette limitation
Color systems and models
Symmetry of color. Contrast and nuance
Color harmony
Construction and development of color. Complete compositional algorithm of the painting
Chapter Four
Scientific and intuitive in the artist’s work
Painting size
Artistic structure of the painting
Beautiful and mysterious
Subject - visual image - artistic image
Artists and scientists. Scientific and everyday terminology
Subject and color
Two poles of painting
Draw with form, write with color
Chapter Five
Explicable and sensual in painting
About color
The language of painting is a special language of art
Color is singular, color is multiple. Psychological assessment of color
What does color depend on?
Good tradition
Theories of painting - scientific basis
A word about the author
Painter's memo
In the world of wise thoughts
List of used and recommended literature.
Download the e-book for free in a convenient format, watch and read:
Download the book The Golden Ratio in Painting, Kovalev F.V., 1989 - fileskachat.com, fast and free download.
Description of the presentation by individual slides:
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Golden ratio in painting Prepared by: Kharlamova Elizaveta Di-1B Teacher Khakimova Odina Rasulovna Department of Education Moscow College of Decorative and Applied Arts named after. Carla Faberge
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Sometimes professional artists, having learned to draw and paint from life, due to their own weak fundamental training, believe that knowledge of the laws of beauty (in particular the law of the golden ratio) interferes with free intuitive creativity. This is a big and deep misconception of many artists who never became true creators. The entire ancient culture passed under the sign of the golden proportion. Knowledge of the laws of the golden section or continuous division, as some researchers of the study of proportions call it, helps the artist create consciously and freely. Using the laws of the golden ratio, you can explore the proportional structure of any work of art, even if it was created on the basis of creative intuition. This aspect of the matter is of no small importance in the study of the classical heritage and in the art historical analysis of works of all types of art.
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A little history In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. And the discovery of proportions belongs to the merits of ancient Eastern mathematics, while ancient tradition connects it with the name of the outstanding mathematician of the 6th century BC. e. Pythagoras and his student Nicomachus. Familiarity with the golden ratio played a significant role in the work of ancient architects and sculptors. It will be interesting to know the rule, clearly visible in ancient Greek statues: when dividing a person’s torso in accordance with the golden ratio, it is easy to find the level of the navel and elbow; when repeatedly dividing two segments in opposite directions, the height of the knee and the lower level of the neck are discovered.
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It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.
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Leonardo da Vinci There is no doubt that Leonardo was a great artist, this was already recognized by his contemporaries, but his personality and activities will remain shrouded in mystery, since he left to his descendants not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “ about everything in the world". He wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence. The term "Golden Ratio" was introduced by Leonardo da Vinci (1452-1519) (brilliant painter, scientist and engineer)
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Mona Lisa (La Gioconda) In this masterpiece, researchers noticed that Leonardo's deep knowledge of the structure of the human body helped him capture this mysterious smile. They emphasized the expressiveness of individual parts of the painting and landscape, the new companion of the portrait, the naturalness of expression, the simplicity of the pose, the beauty of the hands. The artist did something unprecedented: the painting depicts air, it envelops the figure in a transparent haze. There are many versions about the history of this portrait. Here is one of them. One day, Leonardo da Vinci received an order from the banker Francesco de le Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint the portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became lively and interesting.
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Mona Lisa (La Gioconda) The composition of the portrait "La Gioconda" is based, according to Luca Pacioli (a medieval monk), on golden triangles, which are parts of a star pentagon.
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There was an opinion that the composition was successful because of its construction on “golden rectangles”.
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The picture has points that involuntarily attract our attention, the so-called visual centers..
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Slide description:
The golden ratio in the painting by I.I. Shishkin “Pine Grove” In this famous painting by I. Shishkin, the motifs of the golden ratio are clearly visible. A brightly sunlit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio. To the left of the main pine there are many pines - if you wish, you can successfully continue dividing the picture further according to the golden ratio.
