Harmonious proportions. Divine harmony: what is the golden ratio in simple words

The essay was completed by an 8th grade student at Municipal Educational Institution Gymnasium No. 9 Veronica Vyushina

Ekaterinburg

1. Introduction. Golden ratio proportion. F and φ.

"Geometry has two great treasures. The first is the Pythagorean theorem, the second is the division of a segment in extreme and mean ratios"

Johannes Kepler

Regular polygons attracted the attention of ancient Greek scientists long before Archimedes. The Pythagoreans, who chose a pentagram - a five-pointed star - as the emblem of their union, attached great importance to the problem of dividing a circle into equal parts, that is, of constructing a regular inscribed polygon. Albrecht Durer (1471-1527), who became the personification of the Renaissance in Germany, provides a theoretically accurate method for constructing a regular pentagon, borrowed from Ptolemy’s great work “Almagest”.

Dürer's interest in constructing regular polygons reflects their use in the Middle Ages in Arabic and Gothic designs, and after the invention of firearms in the planning of fortresses.

Medieval methods for constructing regular polygons were approximate, but were (or could not help but be) simple: preference was given to construction methods that did not even require changing the opening of the compass. Leonardo da Vinci also wrote a lot about polygons, but it was Dürer, not Leonardo, who passed on medieval methods of construction to his descendants. Dürer, of course, was familiar with Euclid’s “Elements,” but did not present in his “Guide to Measurement” (on constructions using compasses and rulers) the method proposed by Euclid for constructing a regular pentagon, which was theoretically accurate, like all Euclidean constructions. Euclid does not attempt to divide a given arc of a circle into three equal parts, and Dürer knew, although the proof was not found until the 19th century, that this problem was insoluble.

The construction of a regular pentagon proposed by Euclid includes the division of a straight line segment in the mean and extreme ratio, which was later called the golden section and attracted the attention of artists and architects for several centuries.

Point B divides the segment ABE in the average and extreme ratio or forms the golden ratio if the ratio of the larger part of the segment to the smaller is equal to the ratio of the entire segment to the larger part.

The golden ratio written in the form of equality of ratios has the form

AB/BE= AB/AE

If we put AB=a, and BE=a/F so that the golden ratio is equal to AB/BE=F, then we get the ratio

That is, Ф satisfies the equation

This equation has one positive root

Ф=(√5+1)/2=1.618034….

Note that 1/Ф = (√5 -1)/2, since (√5-1)(√5+1) =5-1=4. 1/F is considered to be φ=0.618034….

Ф and φ are the uppercase and lowercase forms of the Greek letter "phi".

This designation was adopted in honor of the ancient Greek sculptor Phidias (5th century BC). Phidias supervised the construction of the Parthenon Temple in Athens. The number φ is repeatedly present in the proportions of this temple.

2.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 Ramesses, 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.

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 Parthenon has 8 columns on the short sides and 17 on the long sides. The ratio of the building's height to its length is 0.618. If we divide the Parthenon according to the “golden section”, we will get certain protrusions of the facade. 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 the Elements, a geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD) and others studied the golden division. In medieval Europe, they became familiar with the golden division from 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 a lack of 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 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 whole 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.

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.

Introduction……………………………………………………….………3

1. Dynamic symmetry in nature and architecture………………3

2. Golden ratio – harmonic proportion…………………..6

3. Second golden ratio…………………………………………..7

4. History of the golden ratio………………………………………..7

5. Fibonacci series……………………………………………………11

6. Nature………………………………………………………12

Conclusion……………………………………………………………13

References……………………………………………………...15

Introduction.

The idea that the physical world is dominated by harmony and order, which can be expressed mathematically, goes back to ancient Greece. In Europe during the Renaissance, Galileo said that the book of the universe was written in the language of mathematics. Scientists who lived after him also expressed amazement at the fact that all the laws of the universe could be translated into mathematical language.

Realizing this “universal applicability” of mathematics, unknown to chemical and biological sciences, the great physicist James Jones said: “The architect of the universe must have been a mathematician.” It is known that Einstein's theory of relativity is not just the result of reflection; it was put forward after certain mathematical developments.

Bearing in mind the intelligibility that physical laws acquire when translated into the language of mathematics, Einstein said: “The only incomprehensible quality of the universe is its comprehensibility.”

And how not to be amazed even at the simplest example - the expression of the force of mutual attraction of bodies in the form of a mathematical formula:

F = Y-mi-1712/r

In this formula, the constant value of the constant “Y” in all cases - from the force of attraction between electrons and protons in an atom to the mutual attraction of stars, from our planet to worlds billions of light years distant from us, demonstrates amazing simplicity, that is, the phenomenality of the formula and its enduring value, as a kind of universal currency.

The extremely effective and unexpected results of the application of mathematics to other branches of science still seem to us a mystery. Some scientists associate this with the orientation of other sciences towards the development of mathematical knowledge.

1. Dynamic symmetry in nature and architecture

Term "dynamic symmetry" was first used by the American architectural researcher D. Hambidge, denoting a certain principle of proportionation in architecture. Later, this term independently appeared in physics, where it was introduced to describe physical processes characterized by invariants. Finally, the term dynamic symmetry a pattern of natural morphogenesis is named, which in terms of origin also turns out to be unrelated to Hambidge’s idea and, even more so, to the appearance of this term in physics. However, all three options are deeply interconnected in content.

First, let us note the strategic similarity of our research direction with Hambidge. This is a well-known historically established direction, which in the field of architecture and art is motivated by the search for patterns of harmony, and therefore focused on the study of natural objects. Typically, architects are interested in the structural patterns of natural shape formation and especially in the golden ratio and Fibonacci numbers

Patterns that are notable for their intriguing role in architectural formation. It is no coincidence that architect-researchers so often pay attention to the botanical phenomenon phyllotaxis, which is characterized by these patterns.

Phyllotaxis turned out to be the object of attention of the author of the first version of the concept of dynamic symmetry, D. Hambidge. As a result of studying this phenomenon, D. Hambidge concludes law so-called monotonous growth, and offers its geometric interpretation - spiral of monotonous growth, or else

- golden spiral (Fig. 1).

Fig 1. Construction of the golden spiral according to Hambidge.

However, the main generalization made by D. Hambidge as a result of studying the laws of natural morphogenesis (phyllotaxis), as well as the proportions of classical architecture, comes down to the idea of ​​architectural proportioning, called dynamic symmetry. Hambidge illustrates it using a simple geometric diagram (Fig. 2).

Fig 2. Proportional system “Dynamic symmetry” by D. Hambidge.

This is a sequential system of rectangles, the first of which is a square, and each subsequent one is built on the side of the original square, equal to 7, and on the diagonal of the previous rectangle. The result is a series of rectangles, the ratio of the sides of which is expressed by the series. In this series, Hambidge distinguishes between two types of rectangles - static and dynamic. For static rectangles, the aspect ratios are expressed as integers, while for dynamic rectangles they are expressed as irrational ones. Dynamic rectangles, according to D. Hambidge, express the idea of ​​growth, movement and development. Of these, he first of all singles out three whose long sides are equal But it attaches special importance to the rectangle that is directly related to "golden rectangle" Hambidge conducts a thorough geometric study, discovering various manifestations of the golden section in the rectangle system. Studying the geometric properties of this rectangle, he shows the possibility of its use for analyzing the proportions of objects of classical architecture and art (Fig. 3, 4).

This, in a nutshell, is the essence of D. Hambidge's idea of ​​dynamic symmetry. As we see, it does not directly follow from the properties of phyllotaxis. Hambidge, generally speaking, does not delve into the mathematics of phyllotaxis. In his various diagrams illustrating the patterns of uniform growth, or some ideas of proportioning, he uses well-known numerical relationships characteristic of phyllotaxis, incl. golden ratio.

2. GOLDEN RATIO - harmonic proportion.

In mathematics, a proportion is the equality of two ratios: a: b = c: d.
A straight line segment AB can be divided by point C 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.

Golden ratio- this is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part as the larger part itself relates to the smaller; or in other words, the smaller segment is to the larger as the larger is to the whole

Segments of the golden proportion are expressed as an infinite irrational fraction 0.618..., if c is taken as one, a = 0.382. The numbers 0.618 and 0.382 are the Fibonacci sequence ratios. The basic geometric figures are based on this proportion.
A rectangle with this aspect ratio became known as the golden rectangle. It also has interesting properties. If you cut a square from it, you will again be left with a golden rectangle. This process can be continued indefinitely. And if you draw a diagonal of the first and second rectangles, then the point of their intersection will belong to all the resulting golden rectangles.
Of course there is also a golden triangle. This is an isosceles triangle whose side length to base length ratio is 1.618.
There is also a golden cuboid - this is a rectangular parallelepiped with edges having lengths of 1.618, 1 and 0.618.

In a star pentagon, each of the five lines that make up the figure divides another in relation to the golden ratio, and the ends of the star are golden triangles.

3. 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.

Dividing a rectangle with the line of the second golden ratio

4. 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 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 devoted 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.

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 the Elements, a 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 (3rd century AD), and others. In medieval Europe, they became acquainted with the golden division 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.
IN Renaissance Interest in the golden division is increasing among scientists and artists due to its application both in geometry and in 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 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 He 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. 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). Kepler called the golden proportion self-continuing. “It is structured in such a way,” he wrote, “that the two lowest terms of this endless 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."
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”. 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 did 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, 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. 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

Golden proportions in the human figure

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.

5. 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:

Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...

6. Nature.

Now let's move on to Nature, which gives a huge number of manifestations of the Golden Section and Fibonacci numbers. Let us give several illustrative examples of the manifestation of the Golden Section in Nature.

"Golden" spirals in sea shells

These illustrative examples could be continued ad infinitum. One thing is clear: The Golden Ratio and Fibonacci numbers reflect some fundamental patterns of living nature.

Now let’s talk about another modern scientific discovery that establishes connection of the genetic code with Fibonacci numbers and the Golden Ratio. In 1990, French researcher Jean-Claude Perez, who was working at that time as a researcher at IBM, made a very unexpected discovery in the field of genetic coding. He discovered a mathematical law governing the self-organization of bases T, C, A, G inside DNA. He discovered that successive sets of DNA nucleotides are organized into long-range order structures called RESONANCES . Resonance represents a special proportion that ensures the division of DNA in accordance with the Fibonacci numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...).

The key idea behind Jean-Claude Perez's discovery, called DNA SUPRA code , is as follows. Let us consider any segment of the genetic code consisting of bases of the type T, C, A, G, and let the length of this segment be equal to the Fibonacci number, for example, 144. If the number of bases is like T in the DNA segment under consideration is 55 (Fibonacci number) and the total number of bases of the type A, C And G is equal to 89 (Fibonacci number), then the segment of the genetic code in question forms resonance, that is, resonance there is a proportion between three adjacent Fibonacci numbers (55-89-144). The discovery is that each DNA forms many resonances of the considered type, that is, as a rule, segments of the genetic code with a length equal to the Fibonacci number Fn, are divided by the golden ratio into many bases like T(the number of which in the considered segment of the genetic code is equal to Fn- 2) and the total set of remaining bases (the number of which is equal to Fn- 1). If we carry out a systematic study of all possible “Fibonacci” segments of the genetic code, then we will obtain a certain set resonances, called SUPRA DNA code .

Since 1990, this pattern has been repeatedly tested and confirmed by many outstanding biologists, in particular Professors Montagniere and Sherman, who studied the DNA of the AIDS virus.

There is no doubt that the discovery in question belongs to the category of outstanding discoveries in the field of DNA that determine the development of genetic engineering. According to the author of the discovery, Jean-Claude Perez, the DNA SUPRA code is a universal bio-mathematical law, which indicates the highest level of self-organization of nucleotides in DNA according to the principle of the “Golden Section”.

Conclusion.

So, when the Lord created the universe, he was not content with merely caring for the perfection of his laws, which were to be established, but also gave them a beauty that elevates the human spirit. He wove a beautiful and graceful pattern into this grandiose lace, woven by the power of science. And as the son of the human race revealed the secrets of the pattern on this lace, mathematical science was born. Each was initiated into the secret of one thread, different from the others, and a grandiose picture appeared to us in its present form. Having gained this knowledge, we will either concentrate it at a single point and lock it in the human brain, or scatter it across the tablets of the books of the universe. The fact that we become familiar with existing truths only at a certain level of development indicates that mathematics belongs to the primordial.

Bibliography.

1. D. Pidou. Geometry and art. – M.: Mir, 1999

2. Stakhov A. Codes of the golden proportion.

3. Kepler I. About hexagonal snowflakes. – M., 1982.

4. Magazine “Mathematics at School”, 1994, No. 2; No. 3.

5. Tsekov-Pencil Ts. About the second golden ratio. – Sofia, 1983.

6. www.trinitas.ru/rus/doc/0232/004a/02321053.htm

7. http://www.noviyegrani.com/archives_show.php?ID=13&ISSUE=3

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

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

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

So, the numbers forming the sequence:

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

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

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

Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series... It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden ratio, the golden mean, or the golden proportion. . In algebra, this number is denoted by the Greek letter phi (Ф)

So, Golden ratio = 1:1.618

233 / 144 = 1,618

377 / 233 = 1,618

610 / 377 = 1,618

987 / 610 = 1,618

1597 / 987 = 1,618

2584 / 1597 = 1,618

The human body and the golden ratio

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

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

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

M/m=1.618

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

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

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

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

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

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

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

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

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

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

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

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

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

*Face height/face width;

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

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

*Mouth width/nose width;

* Nose width / distance between nostrils;

* Distance between pupils / distance between eyebrows.

Human hand

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

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

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

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

The golden ratio in the structure of the human lungs

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

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

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

Structure of the golden orthogonal quadrilateral and spiral

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

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

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

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

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

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

In nature

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

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

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

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

The structure of sea shells

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

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

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

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

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

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

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

Sir Thompson makes this comment:

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

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

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

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

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

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

Golden ratio in the human ear

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

Animal horns and tusks developing in a spiral shape

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

Golden ratio in the structure of microcosms

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

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous . For example, many viruses have the three-dimensional geometric shape of an icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a certain sequence. At each corner of the icosahedron there are 12 units of protein cells in the shape of a pentagonal prism and spike-like structures extend from these corners.

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

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

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