Divine harmony: what is the golden ratio in simple words. Secrets of the universe in numbers. Shkrudnev Fedor Dmitrievich Golden ratio of a triangle

Golden ratio - harmonic proportion

During the period of development of architecture, when the physical and mechanical characteristics of building materials were poorly studied, there were no proven methods for calculating building structures - empirical experience and strict adherence to the harmonic proportions of the “golden section” prevailed.

In mathematics, proportion (lat. proportio) is the equality of two ratios: a: b = c: d.

A straight line segment AB can be divided into two parts in the following ways:
into two equal parts – AB: AC = AB: BC;
into two unequal parts in any respect (such parts do not form proportions);
thus, when AB: AC = AC: BC.

The latter is the golden division or division of a segment in extreme and average ratio.

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

a: b = b: c or c: b = b: a.

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler.

From point B a perpendicular equal to half AB is restored. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is laid, ending with point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the golden proportion.

Segments of the golden proportion are expressed by the infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If segment AB is taken to be 100 parts, then the larger part of the segment is 62, and the smaller part is 38 parts.

The properties of the golden ratio are described by the equation:

x2 – x – 1 = 0.

Solution to this equation:

The properties of the golden ratio have created a romantic aura of mystery and almost mystical worship around this number.

Second golden ratio

The Bulgarian magazine "Fatherland" (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives another ratio of 44: 56.

The division is carried out as follows. Segment AB is divided in proportion to the golden ratio. From point C, a perpendicular CD is restored. The radius AB is point D, which is connected by a line to point A. Right angle ACD is divided in half. A line is drawn from point C to the intersection with line AD. Point E divides segment AD in the ratio 56:44.

The figure shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Golden Triangle

To find segments of the golden proportion of the ascending and descending series, you can use the pentagram.

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

We draw straight AB. From point A we lay down on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the resulting points d and d1 with straight lines to point A. We lay off the segment dd1 on line Ad1, obtaining point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to construct a “golden” rectangle.

Rice. 5. Construction of a regular pentagon and pentagram

Rice. 6. Construction of the golden triangle

History of the golden ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, 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. 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 Khesira, 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 the construction of dynamic rectangles.

Plato(427...347 BC) also knew about the golden division. His dialogue " Timaeus"is dedicated to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division.

The façade of the ancient Greek temple of the Parthenon features 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.

Rice. 7. Dynamic rectangles

Rice. 8. Antique golden ratio compass

In the ancient literature that has come down to us, the golden division was first mentioned in “ Beginnings» Euclid. In the 2nd book of the “Principles” the geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd 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 have a lot of empirical experience, but little knowledge. He conceived and began to write a book on geometry, but at that time a monk’s book appeared Luca Pacioli, 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, he was working on the same problems Albrecht Durer. 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. Johann 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).

This proportion is found in architecture, and also occurs when constructing compositions of images of an elongated horizontal format.

The figure shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Golden Triangle

To find segments of the golden proportion of the ascending and descending series, you can use pentagram.

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O- center of the circle, A- a point on a circle and E- the middle of the segment OA. Perpendicular to radius OA, restored at the point ABOUT, intersects the circle at the point D. Using a compass, plot a segment on the diameter C.E. = ED. The side length of a regular pentagon inscribed in a circle is DC. Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

We carry out a direct AB. From point A we plot on it three times a segment O of an arbitrary size, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside the segments ABOUT. Received points d And d1 connect with straight lines to a point A. Line segment dd1 put on line Ad1, getting a point WITH. She split the line Ad1 in proportion to the golden ratio. Lines Ad1 And dd1 used to construct a “golden” rectangle.

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 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 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.

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 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.

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 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.

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 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.

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 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 go deeper into the following 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.

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 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.

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.

Golden ratio

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.

Golden ratio - harmonic proportion

In mathematics proportion(lat. proportio) call the equality of two relations: a : b = c : d.

Straight segment AB can be divided into two parts in the following ways:

into two equal parts - AB : AC = AB : Sun;
into two unequal parts in any respect (such parts do not form proportions);
thus, when AB : AC = AC : Sun.

The latter is the golden division or division of a segment in extreme and average ratio.

a : b = b : c or With : b = b : A.

Rice. 1.Geometric image of the golden ratio

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler.

Rice. 2.Dividing a straight line segment using the golden ratio. B.C. = 1/2 AB; CD = B.C.

From point IN a perpendicular equal to half is restored AB. Received point WITH connected by a line to a point A. A segment is plotted on the resulting line Sun ending with a dot D. Line segment AD transferred to direct AB. The resulting point E divides a segment AB in the golden ratio ratio.

Segments of the golden ratio are expressed as an infinite irrational fraction A.E.= 0.618..., if AB take as one BE= 0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then the larger part of the segment is equal to 62, and the smaller part is 38 parts.

The properties of the golden ratio are described by the equation:

x 2 - x - 1 = 0.

Solution to this equation:

The properties of the golden ratio have created a romantic aura of mystery and almost mystical worship around this number.

Second golden ratio

The Bulgarian magazine “Fatherland” (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash “On the second golden section”, which follows from the main section and gives a different ratio 44:56.

This proportion is found in architecture, and also occurs when constructing compositions of images of an elongated horizontal format.

Rice. 3.Construction of the second golden ratio

The division is carried out as follows. Line segment AB divided according to the golden ratio. From point WITH the perpendicular is restored CD. Radius AB there is a point D, which is connected by a line to a point A. Right angle ACD is divided in half. From point WITH a line is drawn until it intersects with the line AD. Dot E divides a segment AD in the ratio 56:44.

Rice. 4.Dividing a rectangle with the line of the second golden ratio

The figure shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Golden Triangle

To find segments of the golden proportion of the ascending and descending series, you can use pentagram.

Rice. 5.Construction of a regular pentagon and pentagram

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O- center of the circle, A- a point on a circle and E- the middle of the segment OA. Perpendicular to radius OA, restored at the point ABOUT, intersects the circle at the point D. Using a compass, plot a segment on the diameter C.E. = ED. The side length of a regular pentagon inscribed in a circle is DC. Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

Rice. 6.Building a golden
triangle

We carry out a direct AB. From point A lay a segment on it three times ABOUT arbitrary value, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside the segments ABOUT. Received points d And d 1 connect with straight lines to a point A. Line segment dd 1 put on line Ad 1 , getting a point WITH. She split the line Ad 1 in proportion to the golden ratio. Lines Ad 1 and dd 1 used to construct a “golden” rectangle.

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 at 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.

Rice. 7.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.

Rice. 8.Antique golden ratio compass

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,” a 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 gold division we got to know each other 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 painter Piero della Francesca, who wrote two books, one of which was entitled "On Perspective in Painting". He is considered the creator of descriptive geometry.

Luke 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 - God 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. That's why 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 remains 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, set aside the segment m, put the segment next to it M. Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending series

Rice. 9.Construction of a scale of golden proportion segments

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 Studies”. 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 to be “mathematical aesthetics.”

Rice. 10.Golden proportions in parts of the human body

Rice. eleven.Golden proportions in the human figure

Zeisingdid a tremendous job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the 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, for which the average 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. The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

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 the 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 edition does not mention a single work of painting.

At the end of the 19th - beginning of the 20th centuries. Many purely formalistic theories appeared about the use of the golden ratio in works of art and architecture.With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

Fibonacci series

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers:

Months

etc.

Pairs of rabbits

etc.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, 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. This relationship is denoted by the symbol F. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden proportion, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to the whole.

Fibonacci also dealt with the practical needs of trade: what is the smallest number of weights that can be used to weigh a product? Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...

Generalized golden ratio

The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden division.

Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this field is the discovery of generalized Fibonacci numbers and generalized golden ratios.

The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of weights discovered by him 1, 2, 4, 8, 16... at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 = 2 + 2..., in the second - this is the sum of the two previous numbers 2 = 1 + 1, 3 = 2 + 1, 5 = 3 + 2.... Is it possible to find a general mathematical formula from which we obtain “ binary series and Fibonacci series? Or maybe this formula will give us new numerical sets that have some new unique properties?

Indeed, let us set the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Let's consider the number series, S+ 1 of the first terms of which are units, and each of the subsequent ones is equal to the sum of two terms of the previous one and separated from the previous one by S steps. If n We denote the th term of this series by φ S (n), then we obtain the general formula φ S (n) = φ S (n- 1) + φ S (n - S - 1).

It is obvious that when S= 0 from this formula we get a “binary” series, with S= 1 - Fibonacci series, with S= 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

Overall golden S-proportion is the positive root of the golden equation S-sections x S+1 - x S - 1 = 0.

It is easy to show that when S= 0, the segment is divided in half, and when S= 1 - the familiar classical golden ratio.

Relations between neighbors S- Fibonacci numbers coincide with absolute mathematical accuracy in the limit with gold S-proportions! Mathematicians in such cases say that gold S-sections are numerical invariants S-Fibonacci numbers.

Facts confirming the existence of gold S-sections in nature, cites the Belarusian scientist E.M. Soroko in the book “Structural Harmony of Systems” (Minsk, “Science and Technology”, 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermal stable, hard, wear-resistant, resistant to oxidation, etc.) only if the specific gravities of the original components are related to each other by one of gold S-proportions. This allowed the author to put forward hypotheses e about the fact that they are golden S-sections are numerical invariants of self-organizing systems. Once confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

Using golden codes S-proportions can be expressed by any real number as a sum of powers of gold S-proportions with integer coefficients.

The fundamental difference between this method of encoding numbers is that the bases of the new codes, which are golden S-proportions, with S> 0 turn out to be irrational numbers. Thus, new number systems with irrational bases seem to put the historically established hierarchy of relations between rational and irrational numbers “from head to foot.” The fact is that natural numbers were first “discovered”; then their ratios are rational numbers. And only later - after the discovery of incommensurable segments by the Pythagoreans - irrational numbers were born. For example, in decimal, quinary, binary and other classical positional number systems, natural numbers were chosen as a kind of fundamental principle - 10, 5, 2 - from which, according to certain rules, all other natural numbers, as well as rational and irrational numbers, were constructed.

A kind of alternative to the existing methods of notation is a new, irrational system, as a fundamental principle, the beginning of which is an irrational number (which, recall, is the root of the golden ratio equation); other real numbers are already expressed through it.

In such a number system, any natural number can always be represented as finite - and not infinite, as previously thought! - the sum of the degrees of any of the gold S-proportions. This is one of the reasons why “irrational” arithmetic, having amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and “Fibonacci” arithmetic.

Principles of formation in nature

Everything that took on some form was formed, grew, strived to take a place in space and preserve itself. This desire is realized mainly in two options - growing upward or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly shorter than the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The idea of the golden ratio will be incomplete without talking about the spiral.

Rice. 12.Archimedes spiral

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and came up with an equation for the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

Goethe also emphasized the tendency of nature towards spirality. The helical and spiral arrangement of leaves on tree branches was noticed a long time ago. The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that the Fibonacci series manifests itself in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, and pine cones, and therefore, the law of the golden ratio manifests itself. The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the “curve of life.”

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there.

Rice. 13. Chicory

The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

Rice. 14.Viviparous lizard

At first glance, the lizard has proportions that are pleasant to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38.

In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.

Nature has carried out division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Rice. 15. bird egg

The great Goethe, a poet, naturalist and artist (he drew and painted in watercolors), dreamed of creating a unified doctrine of the form, formation and transformation of organic bodies. It was he who introduced the term morphology into scientific use.

Pierre Curie at the beginning of this century formulated a number of profound ideas about symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.

The laws of “golden” symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms. These patterns, as indicated above, exist in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.

Golden ratio and symmetry

The golden ratio cannot be considered on its own, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulf (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.

The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern ideas, the golden division is asymmetrical symmetry. The science of symmetry includes such concepts as static And dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments and equal values. Dynamic symmetry is characterized by an increase in segments or their decrease, and it is expressed in the values of the golden section of an increasing or decreasing series.

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It was known even in ancient Egypt Golden ratio, Leonardo da Vinci and Euclid studied its properties.A person’s visual perception is designed in such a way that he distinguishes by shape all the objects that surround him. His interest in an object or its form is sometimes dictated by necessity, or this interest could be caused by the beauty of the object. If at the very basis of form construction, a combination is used golden ratio and the laws of symmetry, then this is the best combination for visual perception by a person who feels harmony and beauty. The whole whole consists of parts, large and small, and these parts of different sizes have a certain relationship, both to each other and to the whole. And the highest manifestation of functional and structural perfection in nature, science, art, architecture and technology is the Principle golden ratio. Concept of golden ratio introduced into scientific use by the ancient Greek mathematician and philosopher (VI century BC) Pythagoras. But the very knowledge of golden ratio he borrowed from the ancient Egyptians. The proportions of all temple buildings, the Cheops pyramid, bas-reliefs, household items and decorations from tombs show that the ratio golden ratio was actively used by ancient masters long before Pythagoras. As an example: the bas-relief from the temple of Seti I in Abydos and the bas-relief of Ramses used the principle golden ratio in the proportions of the figures. The architect Le Corbusier found this out. On a wooden board recovered from the tomb of the Architect Khesir, there is a relief drawing on which the architect himself is visible, holding measuring instruments in his hands, which are depicted in a position fixing the principles golden ratio. Knew about the principles golden ratio and Plato (427...347 BC). The dialogue "Timaeus" is proof of this, since it is devoted to questions golden division, aesthetic and mathematical views of the Pythagorean school. Principles Golden ratio used by ancient Greek architects in the facade of the Parthenon Temple. The compasses that ancient architects and sculptors of the ancient world used in their work were discovered during excavations of the Parthenon Temple.

Parthenon, Acropolis, Athens In Pompeii (museum in Naples) proportions golden division also available.In ancient literature that has come down to us, the principle golden ratio mentioned for the first time in Euclid's Elements. In the book "Beginnings" in the second part the geometric principle is given golden ratio. The followers of Euclid were Pappus (III century AD), Hypsicles (II century BC), and others. To medieval Europe with the principle golden ratio We met through translations from Arabic of Euclid's Elements. Principles golden ratio were known only to a narrow circle of initiates, they were jealously guarded and kept in strict confidence. The era of renaissance and interest in the principles has arrived golden ratio increases among scientists and artists, since this principle is applicable in science, architecture, and art. And Leonardo Da Vinci began to use these principles in his works, even moreover, he began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, who preceded him and published the book “Divine Proportion”, after which Leonardo left his work unfinished. According to historians of science and contemporaries, Luca Pacioli was a real luminary, a brilliant Italian mathematician who lived in the period between Galileo and Fibonacci. As a student of the artist Piero della Francesca, Luca Pacioli wrote two books, “On Perspective in Painting,” the title of one of them. He is considered by many to be the creator of descriptive geometry. Luca Pacioli, at the invitation of the Duke of Moro, came to Milan in 1496 and lectured there on mathematics. Leonardo da Vinci worked at the Moro court at this time. Luca Pacioli's book The Divine Proportion, published in Venice in 1509, became an enthusiastic hymn. golden ratio, with beautifully executed illustrations, there is every reason to believe that the illustrations were done by Leonardo da Vinci himself. Monk Luca Pacioli, as one of the virtues golden ratio highlighted its “divine essence.” Understanding the scientific and artistic value of the golden ratio, Leonardo da Vinci devoted a lot of time to studying it. Carrying out a section of a stereometric body consisting of pentagons, he obtained rectangles with aspect ratios in accordance with golden ratio. And he gave it the name “ golden ratio" Which still holds up to this day. Albrecht Dürer, also studying golden ratio in Europe, meets with the monk Luca Pacioli. Johannes Kepler, the greatest astronomer of his time, was the first to draw attention to the meaning golden ratio for botany calling it the treasure of geometry. He called the golden proportion self-continuing. “It is structured this way,” he said, “the sum of the two junior terms of an infinite proportion gives the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum.”

Golden Triangle:: Golden Ratio and Golden Ratio:: Golden Rectangle:: Golden Spiral

Golden Triangle

To find the segments of the golden proportion of the descending and ascending rows, we will use a pentagram.

Rice. 5. Construction of a regular pentagon and pentagram

In order to build a pentagram, you need to draw a regular pentagon according to the construction method developed by the German painter and graphic artist Albrecht Durer. If O is the center of the circle, A is a point on the circle and E is the midpoint of the segment OA. The perpendicular to the radius OA, restored at point O, intersects with the circle at point D. Using a compass, mark a segment on the diameter CE = ED. Then the side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and get five points to draw a regular pentagon. Then, through one corner, we connect the corners of the pentagon with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio. We draw straight AB. From point A we lay down on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the resulting points d and d1 with straight lines to point A. We lay off the segment dd1 on line Ad1, obtaining point C. She divided line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to construct a “golden” rectangle.

Rice. 6. Building gold

triangle

Golden Ratio and Golden Ratio

In mathematics and art, two quantities are in the golden ratio if the ratio between the sum of these quantities and the larger is the same as the ratio between the larger and the smaller. Expressed algebraically: The golden ratio is often denoted by the Greek letter phi (? or?). The figure of the golden ratio illustrates the geometric relationships that define this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

golden rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:? (one-to-fi), that is 1: or approximately 1:1.618. The golden rectangle can only be constructed with a ruler and a compass: 1. Construct a simple square 2. Draw a line from the middle of one side of the area to the opposite corner 3. Use this line as a radius to draw an arc that defines the height of the rectangle 4. Complete the golden rectangle

Golden spiral

In geometry, the golden spiral is a logarithmic spiral whose growth factor b is related to? , golden ratio. In particular, the golden spiral becomes wider (further from its origin) by a factor ? for every quarter turn it makes.

Consecutive points of dividing the golden rectangle into squares lie on logarithmic spiral, which is sometimes known as the golden spiral.

Golden ratio in architecture and art.

Many architects and artists executed their works in accordance with the proportions of the golden section, especially in the form of a golden rectangle, in which the ratio of the larger side to the smaller side has the proportions of the golden section, believing that this ratio would be aesthetically pleasing. [Source: Wikipedia.org ]

Here are some examples:

Parthenon, Acropolis, Athens . This ancient temple fits almost exactly into the golden rectangle.

Vitruvian Man by Leonardo da Vinci you can make many lines of rectangles in this figure. Then, there are three different sets of golden rectangles: Each set is for the head, torso, and legs area. Leonardo Da Vinci's Vitruvian Man drawing is sometimes confused with the Golden Rectangle principles, however, this is not the case. The construction of the Vitruvian Man is based on drawing a circle with a diameter equal to the diagonal of the square, moving it upward so that it touches the base of the square and drawing a final circle between the base of the square and the midpoint between the area of the center of the square and the center of the circle: Detailed explanation about geometric construction >>

Golden ratio in nature.

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden proportion in the arrangement of branches along the stem of a plant and the veins in the leaves. He expanded his research and moved from plants to animals, studying the skeletons of animals and the branches of their veins and nerves, as well as the proportions of chemical compounds and the geometry of crystals, up to the use of the golden ratio in the visual arts. In these phenomena, he saw that the golden ratio was used everywhere as a universal law, Zeising wrote in 1854: The Golden Ratio is a universal law, which contains the basic principle shaping the desire for beauty and completeness in such areas as nature and art, which permeates, as a primary spiritual ideal, all structures, forms and proportions, whether cosmic or physical, organic or inorganic, acoustic or optical, but the principle of the golden ratio finds its most complete realization in the human form.

Examples:

Cutting through the Nautilus shell reveals the golden principle of spiral construction.

Mozart divided his sonatas into two parts, the lengths of which reflect golden ratio, although there is much debate as to whether he did this deliberately. In more modern times, Hungarian composer Béla Bartók and French architect Le Corbusier deliberately incorporated the principle of the golden ratio into their works. Even today golden ratio surrounds us everywhere in artificial objects. Look at almost any Christian cross, the ratio of the vertical part to the horizontal part is the golden proportion. To find the golden rectangle, look in your wallet and you will find credit cards there. Despite this abundant evidence from works of art created over the centuries, there is currently debate among psychologists about whether people actually perceive golden proportions, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 journal article, Professor Christopher Green, of York University in Toronto, discusses a number of experiments over the years that have not shown any preference for the golden rectangle shape, but notes that several others have provided evidence that such a preference does not exist. . But regardless of the science, the golden ratio retains its mystique, in part because it has excellent applications in many unexpected places in nature. Spiral Nautilus shells are surprisingly close to golden ratio, and the ratio of the length of the chest and abdomen in most bees is almost golden ratio. Even a cross-section of the most common forms of human DNA fits perfectly into the golden decagon. Golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to attract the interest of mathematical communities. Dr. Steven Marquardt, a former plastic surgeon, used this mysterious proportion golden ratio, in his work, which had long been responsible for beauty and harmony, to make a mask that he considered the most beautiful form of the human face that could be.