Goal: Find the patterns of the “golden ratio” in literary works, analyze world-famous examples of the use of the golden ratio in painting, music, etc. Work of students: Efimova Ekaterina, 7th grade, Teplova Anna, 8th grade, Yushkevich Maxim, 10th grade “Where there is beauty, the laws of mathematics apply” (G.G. Hardy).

Golden proportions in literature. Poetry and the golden ratio. Much in the structure of poetic works makes this art form similar to music. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry the sister of musical works. Each verse has its own musical form - its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden proportion will appear. Let's start with the size of a poetic work, that is, the number of lines in it. It would seem that this parameter of poetic works can be changed arbitrarily. However, it turned out that this was not the case. For example, N. Vasyutinsky’s analysis of the poems of A.S. Pushkin from this point of view showed that the sizes of poems are distributed very unevenly; it turned out that Pushkin clearly prefers the sizes of 5, 8, 13, 21 and 34 lines (Fibonacci numbers).

Many researchers have noticed that poetic works are similar to musical works; they also have culminating points that divide the poem in proportion to the golden ratio. Consider, for example, the poem by A.S. Pushkin's "Shoemaker": Let's analyze this parable. The poem consists of 13 lines. It has two semantic parts: the first in 8 lines and the second (the moral of the parable) in 5 lines (13, 8, 5 are Fibonacci numbers).

One of Pushkin’s last poems, “I value loud rights not dearly...” consists of 21 lines and has two semantic parts: 13 and 8 lines. It is characteristic that the first part of this verse (13 lines), according to its semantic content, is divided into 8 and 5 lines, that is, the entire poem is structured according to the laws of the golden proportion.

The analysis of the novel "Eugene Onegin" made by N. Vasyutinsky is of undoubted interest. This novel consists of 8 chapters, each with an average of about 50 verses. The eighth chapter is the most perfect, most polished and emotionally rich. It has 51 verses. Together with Eugene’s letter to Tatiana (60 lines), this exactly corresponds to the Fibonacci number 55! N. Vasyutinsky states: “The ending of the chapter is Eugene’s explanation of his deep feelings for Tatyana - the line “To turn pale and fade away... this is bliss!” This line divides the entire eighth chapter into two parts - in the first there are 477 lines, and in the second there are lines. Their ratio is 1.617! The finest correspondence to the value of the golden proportion! This is a great miracle of harmony, perfected by the genius of Pushkin!" Lermontov's famous poem "Borodino" is divided into two parts: an introduction addressed to the narrator and occupying only one stanza ("Tell me, uncle, it's not without reason..."), and the main part, which represents an independent whole, which falls into two equal parts. The first of them describes the anticipation of the battle with increasing tension, the second describes the poetic work itself with a gradual decrease in tension towards the end. The boundary between these parts is the culmination point of the work and falls exactly at the point of division by the golden section. The main part of the poetic work consists of 13 seven-line lines, that is, 91 lines. Having divided it by the golden ratio (91:1.618 = 56.238), we are convinced that the division point is located at the beginning of the 57th verse, where there is a short phrase: “Well, it was a day!” It is this phrase that represents the “culmination point of excited anticipation”, completing the first part of the poetic work (anticipation of the battle) and opening its second part (description of the battle). Thus, the golden ratio plays a very meaningful role in poetry, highlighting the climax of poetic works

Is it possible to talk about the golden ratio in music? It is possible if you measure a piece of music by the time it was performed. In music, the golden ratio reflects the peculiarities of human perception of temporal proportions. The golden ratio point serves as a guideline for shaping. Often it is the climax. It can also be the brightest moment, or the quietest, or the highest pitch of the place. Back in 1925, art critic L.L. Sabaneev, having analyzed 1,770 musical works by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or by intonation system, or by modal system, which are in relation to each other. relation to the "golden ratio". Moreover, the more talented the composer, the more “golden ratios” are found in his works.

According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition. Sabaneev checked this result on all 27 Chopin etudes. He discovered 178 “golden ratios” in them. It turned out that not only large parts of the studies are divided by duration in relation to the “golden ratio”, but also parts of the studies inside are often divided in the same ratio. Composer and scientist M. A. Marutaev counted the number of bars in the famous sonata "Appassionata" and found a number of interesting numerical relationships. In particular, in the development - the main structural unit of the sonata, where themes intensively develop and tones replace each other - there are two main sections. The first has 43.25 measures, the second – 26.75. The ratio 43.25:26.75=0.618:0.382=1.618 gives the “golden ratio”. The largest number of works in which the Golden Ratio is present are by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Chopin (92%), Schubert (91%)

As an example of constructing a violin based on the law of the golden ratio, let's take a violin made by Antonio Stradivari, created by him in 1700. Stradivari wrote that using the golden ratio he determined the locations for f-shaped cutouts on the bodies of his famous violins. Case length 355 mm Upper oval width 167.5 mm Lower oval width 207 mm Middle part width 109 mm

Having analyzed some works, we saw that the melody develops in accordance with the law of the golden ratio. Classic works are created according to strict rules and canons. Great composers, creating their immortal works, were guided only by their feelings and knowledge of musical notation, knowledge of the laws of musical notation. Upon closer examination of these works, it became clear that the laws of musical notation echo the laws of the golden ratio.

IN PAINTING Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it is completely unimportant what format the picture has - horizontal or vertical.

“The Appearance of Christ to the People” by Alexander Ivanov. The clear effect of the Messiah approaching people arises due to the fact that he has already passed the point of the golden section (the cross of orange lines) and is now entering the point that we will call the point of the silver section (this is a segment divided by the number π, or a segment minus segment divided by the number π)..

I.I. Shishkin. Ship Grove The proportion of the golden ratio is evident in Shishkin’s painting. A brightly sunlit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio.

Accent points fall not only on two of the four golden intersections (the butts of the two central birches), but also on 2 (yellow grid - along the lower horizontal, the border of the shadow and butt of four more trees, and vertically, the trunk of one of the birches) and two horizontals 5 ( highlighted in red - horizontally the far edge of the clearing and the height of distant trees, vertically the border of the crowns of the left group of trees). A. Kuindzhi Birch Grove

“Pictures background” - Prepared by: Vsevolod Tsurikov, www.2bzy.net For the TangoCamp team, Kyiv. Select the “Format Background” option from the context menu (1). Changing background images in PowerPoint presentations and the Word text editor (MS Office 2007). In the window that appears, select a standard background template (2) or a prepared image (3).

“Hyperlink” - A graphic package for preparing presentations and slide films is called ... Hyperlinks that allow you to make transitions within a given document. If necessary, configure the link to be followed using the Action Settings... External command. MS Word. In the window that opens, select the object to which the transition will be made.

“Presentation text” - Tool capabilities of the “Drawing” panel. Finish the slide - include visual examples. That is, the text should clearly stand out against the background of the slide. PowerPoint presentation software is very easy to use. DIAGRAM, GRAPH for slide illustration. This will create a new presentation.

"Powerpoint Animation" - View the result. Make changes to the animation settings. Multimedia – devices that allow you to present information in audio and video form. Let's look at the animation. What we learned: Save your work in your folder under the name Animation. Multimedia programs are software tools that allow you to process audio and video information.

“Creating presentations in Power Point” - Subject. Depends on the goal. To view the resulting presentation, click: Slide Show Start Show. Animation effects. Use of presentation: will attract listeners' attention to the topic. You will learn about what a presentation is. The presentation can be used: for students from grades 1 to 11 (during class hours).

“Creating PowerPoint” - 33. Place the mouse pointer in the Outline Area and enter text. Operating modes in PowerPoint. 11. Structure area. Use the resulting final slide to create a Contents slide. 19. 8. 1. Menu Insert - Caption.

There are 7 presentations in total

Completed by a 6th grade student: Anton Stafeev. Golden ratio.

What is the Golden Ratio? The “golden ratio” is the division of a segment AC into two parts in such a way that its larger part AB relates to the smaller BC in the same way as the entire segment AC relates to AB (i.e. AB: BC = AC: AB). This ratio is approximately 8:5.

History of the Golden Ratio. In the ancient literature that has come down to us, the golden ratio is first encountered in Book II of Euclid’s Elements, where the geometric construction of the golden ratio X (A+ X) = A 2 is given. Euclid uses the golden ratio when constructing regular 5 and 10 squares. There is no doubt that the golden ratio was known before Euclid. It is very likely that the problem of the golden section was solved by the Pythagoreans, who are credited with the construction of a regular 5-gon and geometric constructions equivalent to solving quadratic equations. After Euclid, Hypsicles studied the golden ratio.

Golden ratio in nature. Biological studies have shown that starting from viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems. One can note two types of manifestations of the golden section in living nature: irrational relations according to Pythagoras and integer, discrete relations according to Fibonacci.

Golden ratio in a spiral. Goethe emphasized nature's tendency toward spirality. 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, 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 frightened herd of reindeer scatters in a spiral. Goethe called the spiral the “curve of life.”

Golden ratio in technology. The shell is twisted in a spiral. Spirals are very common in nature. 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. Currently, the Archimedes spiral is widely used in technology.

Golden ratio in architecture. One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC). The figure shows a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

Golden ratio in music. At the beginning of the 20th century, at one of the meetings of the Moscow Scientific and Musical Circle, the Russian Soviet musicologist E.K. Rosenov made a presentation on “The Law of the Golden Section in Poetry and Music.” This work can be considered one of the first mathematical studies of musical works. Thus, comparing the manifestation of the law of the golden section in Bach and Beethoven, Rosenov writes: “We find in Bach a comparatively more detailed and organic cohesion. In Beethoven, the manifestation of the law of the golden section is deeply logical in relation to the sizes of the parts of the form, but mainly indicates the strength of the temperament of this the author in terms of the accuracy of the coincidence of all moments of the highest tension of feelings and the resolution of prepared expectations with the moments of the golden sections...”

Golden ratio in literature. Much in the structure of poetic works makes this art form similar to music. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry the sister of musical works. Each verse has its own musical form - its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden proportion will appear. The analysis of poems by A.S. carried out by N. Vasyutinsky. Pushkin from this point of view showed that the sizes of poems are distributed very unevenly; it turned out that Pushkin clearly prefers Fibonacci numbers.

The analysis of the novel "Eugene Onegin" made by N. Vasyutinsky is of undoubted interest. This novel consists of 8 chapters, each with an average of about 50 verses. The eighth chapter is the most perfect, most polished and emotionally rich. It has 51 verses. Together with Eugene’s letter to Tatiana (60 lines), this exactly corresponds to the Fibonacci number 55! N. Vasyutinsky states: “The culmination of the chapter is Eugene’s declaration of love for Tatyana - the line “To turn pale and fade away... this is bliss!” This line divides the entire eighth chapter into two parts - in the first there are 477 lines, and in the second - 295 lines ". Their ratio is 1.617! The finest correspondence to the value of the golden proportion! This is a great miracle of harmony, perfected by the genius of Pushkin!" Golden ratio in literature.

Golden ratio in painting. Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. It doesn’t matter what format it is. There are only four such points. The portrait of Monna Lisa has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles.

Examples. In a regular five-pointed star, each segment is divided by the segment intersecting it in the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is 1.618).

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 formula of the golden ratio is visible when looking at the index finger. Each finger of the hand consists of three phalanges. The sum of the first two phalanges of the finger in relation to the entire length of the finger = golden ratio (excluding the thumb)

Thank you for your attention.

Completed the presentation

The presentation was made by a student of class 6 “A” of Municipal Educational Institution Secondary School No. 5 in Kstovo Krasilnikov Vladimir Teacher Gushchina T.L. 2011

Golden ratio (golden ratio)

Dividing a continuous quantity into two parts

in such a way that

the greater part is to the lesser as the whole quantity is to the greater.

The term "golden ratio"

(goldener Schnitt)

was put into use

Martin Ohm in 1835.

The golden section of the segment AB can be constructed as follows: at point B, a perpendicular to AB is restored, a segment BC equal to half AB is laid on it, a segment AD equal to AC - CB is laid on the segment AC, and finally a segment AE equal to A.D.

Cutting a square from a rectangle,

built on the principle of the golden ratio,

we get a new, smaller rectangle

with the same aspect ratio

Each end of a pentagonal star

represents a golden triangle.

Its sides form an angle of 36° at the apex,

and the base, laid aside on the side,

divides it in proportion to the golden ratio.

Pythagoras - ancient Greek philosopher and mathematician

Vl in. BC e.

First introduced the concept of the golden ratio

The Pyramid of Cheops

The area of ​​the lateral surface of the Pyramid is related to the area of ​​the base, just as the area of ​​the total surface of the Pyramid is to the area of ​​the lateral surface.

Tutankhamun's tomb

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: 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.

Applied the golden ratio

creating geometry

He said that the Universe is arranged according to the golden ratio

Aristotle

Found the correspondence of the golden ratio to the ethical law

Luca Pacioli

1509 published a book

"Divine Proportion"

1 escape - 100 units.

The size of the chest and abdominal parts of the body corresponds to

golden ratio

The bird's egg has

golden proportions

The length of the lizard's tail is compared to the length of the rest of the body as 62 to 38

Emphasized nature's tendency toward spirality

Spirals in

Wildlife

Human body proportion

has a golden ratio

Golden ratio

in sculpture

Famous statue

Apollo Belvedere

Sculptor Phidias

Used the golden ratio in statues

Athens Parthenos and Olympian Zeus

Golden ratio

in architecture

Parthenon 5th century BC e.

Senate building in the Kremlin

Architect M. Kazakov

First Clinical Hospital

Pirogov

Architect M. Kazakov

Pashkov House

Architect Bazhov

Golden ratio

in painting

Leonardo da Vinci

Portrait of Monna Lisa

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 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.

Golden ratio

L.L. Sabaneev

Arensky Beethoven Borodin Haydn

Mozart Scriabin Chopin Schubert

90% of all their works are the Golden Ratio

"There are two treasures in geometry - the Pythagorean theorem and the division of a segment in extreme and mean ratio. The first can be compared to the value of gold, the second can be called a precious stone." "There are two treasures in geometry - the Pythagorean theorem and the division of a segment in extreme and mean ratio. The first can be compared to the value of gold, the second can be called a precious stone."

astronomer Johannes Kepler

1. 1. Completed by: student of class 11A of MBOU Secondary School No. 23 in Dimitrovgrad Arthur Harutyunyan Scientific supervisor: higher category mathematics teacher Lena Rubenovna Avakyan
2. 2. Goals and objectives of the project: Deepening students’ knowledge on the topic “Ratios and Proportions”. Expanding the concept of mathematical patterns in the world. Increasing students’ interest in mathematics, determining the meaning of mathematics in world culture. Supplementing students’ knowledge system with ideas about the “Golden Section" as harmony of the surrounding world. Identification of the connection between mathematics and other subjects: literature, computer science, natural science, art.
3. 3. ABSTRACT: The project material can be used in mathematics, geometry, history and fine arts lessons; in extracurricular activities, the information will be interesting and useful when conducting subject evenings and intellectual competitions. This work discusses the theoretical foundations of the concepts: proportion, golden ratio, golden triangle, golden rectangle .Historical information about the development of the golden section is of interest. Material about the golden section in painting is presented in detail: sections dedicated to Leonardo da Vinci, I.I. Shishkin and description of their paintings; the presence of the golden section in the paintings of Leonardo da Vinci “La Gioconda”, “The Last Supper” and I.I. is convincingly proven. Shishkin “Ship Grove”. The presentation presents succinctly presented, illustrated material that is interesting for reading and studying.
4. 4. INTRODUCTION For a long time, people have strived to surround themselves with beautiful things. Already the household items of the ancient inhabitants, which, it would seem, pursued a purely utilitarian goal - to serve as a storage facility for water, a weapon for hunting, etc., demonstrate man’s desire for beauty. At a certain stage of his development, a person began to wonder: why is this or that object beautiful and what is the basis of beauty? Already in Ancient Greece, the study of the essence of beauty, beauty, formed into an independent branch of science - aesthetics, which among ancient philosophers was inseparable from cosmology. At the same time, the idea was born that the basis of beauty is harmony. Beauty and harmony have become the most important categories of knowledge, to a certain extent even its goal, because ultimately the artist seeks truth in beauty, and the scientist seeks beauty in truth.
5. 5. GOLDEN RATIO The whole part is to the larger as the larger is to the smaller. 1-XIf the height of a person is taken as 1, then we get the proportion 1:X=X:(1-X). Having solved this equation, X we obtain the irrational number 0.618... (1, 618) This number Ф (phi) is named after the ancient Greek sculptor Phidias, who calculated the proportions of the Parthenon temple.
6. 6. GOLDEN SECTION Dividing a segment according to the golden ratio using a compass and ruler. From point B, a perpendicular equal to half AB is drawn. The resulting point C is connected by a line to point A. On the resulting line, a segment BC is laid out, ending in point D. The segment AD is transferred to the straight line AB. The resulting point E divides the segment AB in the golden proportion ratio. The golden proportion segments are expressed by an 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 the segment AB is taken to be 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 section are described by the equation: x2 – x – 1 = 0. The solution to this equation: The properties of the golden section have created a romantic aura of mystery around this number and almost not mystical worship.
7. 7. GOLDEN RECTangle The sides of the Golden Rectangle are in the proportion of 1.618 to 1. To construct the Golden Rectangle, start with a square with sides of 2 units and draw a line from the middle of one of its sides to one of the corners of the opposite side.
8. 8. Triangle EDB is right. Pythagoras, around 550 BC, proved that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. In this case:
9. 9. CONNECTION OF THE GOLDEN RATIO WITH THE FIBONACCI SERIES The history of the golden ratio is indirectly connected with the name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci). He traveled extensively throughout the East and introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, in which all the problems known at that time were collected. The Fibonacci sequence (nearby) is a sequence in which the first two terms are equal to 1, and each subsequent one is the sum of the previous two ( 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13.8 + 13 = 21; 13 + 21 = 34). Thus, this sequence (we denote it by (u), n) is defined as follows: u =1, u =1, u =u +u, n. Here are the first numbers of this sequence: 1, 1, 2, 3, 5 , 8, 13, 21, 34, 55, 89,144, ...The connection with the golden ratio here is that the ratio of adjacent numbers in a series approaches the ratio of the golden division (21: 34 = 0.617, and 34: 55 = 0.618). 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... 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 in art, they invariably came to this series as an arithmetic expression of the law of gold division.
10. 10. GOLDEN RATIO IN ARCHITECTURE The proportions of the Intercession Cathedral on Red Square in Moscow are determined by eight members of the golden section series: Many members of the golden section series are repeated many times in the intricate elements of the temple d d 2 1; d 2 d 3 d ; d 3 d 4 2 d ; etc.
11. 11. PARTHENON – THE MAIN TEMPLE OF THE ACROPOLIS OF ATHENS. The facade of the ancient Greek temple of the Parthenon contains golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world.
12. 12. The figures show a number of patterns related to the golden ratio. The proportions of the building can be expressed through various degrees of the number Ф 0.618... =
13. 13. GOLDEN RATIO IN THE HUMAN BODY To identify the golden proportions in the human body, Professor Zeising did a tremendous amount of work. 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 section. 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 value of the proportion is expressed in the ratio 8: 5 = 1.6.
14. 14. GOLDEN RATIO IN PAINTING AND PHOTOGRAPHY Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane. Visual centers are also used in photography and web design.
15. 15. The portrait of Monna Lisa (La Gioconda) has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon.
16. 16. GOLDEN RATIO IN NATURE 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. 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.
17. 17. In a lizard, at first glance, we can see proportions that are pleasing 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 makes its way - symmetry relative to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.
18. 18. Nature has carried out division into symmetrical parts and golden proportions. In the parts, the repetition of the structure of the whole is manifested.
19. 19. Conclusion The “Golden Ratio” seems to be that moment of truth, without which, in general, anything existing is impossible. Whatever we take as an element of research, the “golden ratio” will be everywhere; even if there is no visible observance of it, then it certainly takes place at the energetic, molecular or cellular levels.
20. CONCLUSION: The golden ratio is a very interesting and deep concept, which contains the basics of symmetry and asymmetry. Using the “golden ratio” you can perform interesting experiments in any conditions (find the F ratio in people’s faces, in the facades of buildings). And in my opinion, the concept of the “golden ratio” should be known to any person interested in mathematics, architecture, and painting.
21. 21. Literature Kovalev F.V. Golden ratio in painting. K.: Vyshcha Shkola, 1989.  Kepler I. About hexagonal snowflakes. - M., 1982. Durer A. Diaries, letters, treatises - L., M., 1957. Tsekov-Karandash Ts. About the second golden ratio - Sofia, 1983. Stakhov A. Codes of the golden proportion. A. D. Berdukidze. Golden ratio-