Presentation on the topic of the golden ratio in architecture. Presentation "golden ratio". with in-depth study of economics and law

Municipal educational institution "Ilovai-Dmitrievskaya secondary school".

Pervomaisky district, Tambov region

Historical and mathematical conference.

"Golden Section" in the architecture of Russian churches.

Full name of teacher: Ryzhkova Vera Ivanovna

Year of study: 2009-2010

Children's age: 14-15 years.

Target: consideration of the “golden section” from a theoretical point of view (the proportions of the “golden section” and their relationships) and in objects of the surrounding world (the architecture of Russian churches).

Tasks:

Expand students’ understanding of the “golden” proportion as the basis for the proportional structure of architectural masterpieces;

Show children the scope of application of mathematics not only in natural sciences, but also in such an area real life, like architecture;

Expand the general cultural horizons of students through acquaintance with temples Ancient Rus' and pearl architecture - the Church of the Intercession on the Nerl.

Diverse development of children; aesthetic perception of temples;

Development of cognitive motivation and cognitive interest in the subject from the point of view of the future (the possibility of applying the acquired knowledge in the professions of an architect, civil engineer);

Transfer of historical experience of generations.

Event participants: members of the circle “Ilovai-Dmitrievskaya Secondary School”.

Design and equipment:

Statements (posted on the board):

“The spirit of geometric, mathematical order will be the master of the destinies of architecture.” Le Corbusier (famous architect).

“There is no ideal beauty without some strangeness in passers-by.” F Bacon.

Illustrations of temples of Ancient Rus':

St. Sophia Cathedrals in Kyiv and Novgorod, the Church of the Ascension in Kolomenskoye, St. Basil's Cathedral in Moscow;

Reproductions:

Portrait of Andrei Bogolyubsky, icon of “Our Lady of Vladimir”;

Historical map: Vladimir-Suzdal Principality.

Application: Presentation “Golden section in the architecture of Russian churches” (slides 1-27).

Introduction

"Golden ratio" in mathematics and architecture:

a) the concept " golden ratio»;

b) algebraic determination of the “golden ratio”;

V) geometric construction"golden ratio";

d) “golden section” in the proportions of the Parthenon, “golden section” and ancient Russian fathoms.

3. Architecture of Ancient Rus':

a) the “golden ratio” in the construction of cross-domed churches of Orthodox Rus';

b) white stone architecture in the construction of Russian churches in Vladimir-Suzdal Rus' (the reign of Andrei Bogolyubsky);

c) Church of the Intercession on the Nerl - the pearl of architecture of Vladimir-Suzdal Rus'.

Reference material:"Proportion" (from Latin word Proportio) means “proportionality”, a certain relationship between parts.

Progress of the event.

Introduction

Student reads: Oh, bright and beautifully decorated, Russian land!

You are famous for many beauties...

You are filled with everything, Russian land...

You are strong with your shrines, ancient Russian culture.

Illustrations of Russian churches are hung on the boardX- XIIV. V.:

St. Sophia Cathedral in Kyiv, St. Sophia Cathedral in Novgorod, Church of the Ascension in Kolomenskoye, St. Basil's Cathedral in Moscow.

Teacher. Guys, look carefully at the illustrations... Before us are Russian churches, masterpieces of world architecture, built in the 10th-12th centuries. Take a closer look at them... They amaze us with their beauty and perfection... The longer you look at them, the deeper you become imbued with a sense of pride in our Motherland - Russia - Rus', its history.

Today we learn that the beauty of these masterpieces, their greatness lies at the basis of the use of mathematical calculations in the construction - proportional relationships.

A long time ago, before the beginning of our era, people built beautiful buildings with very appropriate proportions. By relentlessly following the eternal laws of geometry, the architects of antiquity achieved harmony and perfection in the temples they built, which can only be called pearls of architectural art.

For a long time it was believed that ancient architects built everything by eye, without special calculations. But research by scientists showed that they knew the proportions and built using certain calculations containing complex system mathematical relations.

Each building was imbued with a mathematical system that determined the shape of the bricks, the thickness of the walls, the radii of the arches and the overall dimensions of the building.

Let's get acquainted with one of the most important proportions, which is often found in works of art - architecture.

A student appears in the clothes of the Queen of Mathematics, with the emblem of proportion.

Proportion. I am not just a proportion, I am the “golden proportion” or “golden ratio”, that’s what they called me famous artist Leonardo da Vinci. And his friend, the mathematician monk Luca Pacioli, called me “divine proportion.” For the Greeks, I replaced the theory of real numbers and thus helped them create their scientific masterpiece - geometry.

I bring harmony to architecture. More precisely, I am the soul of harmony. It is impossible to extol my importance enough: I have the glory of an architect, the strength of a structure and the wonders of art. In general, I hear a lot of compliments addressed to me. Thus, when I enter the image of the “golden ratio,” one of my most ardent admirers, the German poet and philosopher Adolf Zeising, assures me that I simply dominate nature. And the famous Johannes Kepler said: “Geometry has two treasures: one of them is the Pythagorean theorem, and the other is the division of a segment in the average and extreme ratio... The first can be compared with the measure of gold; the second is more like a precious stone.”

2. “Golden ratio” in mathematics and architecture.

Teacher. (Slide show 1,2)

a) consider the basic information regarding the famous proportion. “Golden proportion” or “golden section” is the division of a segment in average and extreme ratio, i.e. division of a segment into two unequal parts, in which most relates to the whole as the smaller part relates to the larger. How does it work?

Explanation on the board.

Teacher.

b) take an arbitrary segment AB. Let's find point C on it, which divides the segment in the following ratio: AC:AB=CB:AC

If the length of the segment AB is denoted by a, and the length of the segment AC by x, then the length of the segment CB is equal to a-x. The proportion will take the form

x\a=(a-x)\x

In a proportion, as is known, the product of the extreme terms is equal to the product of the middle terms and we rewrite the proportion in the form x 2 = a(a-x). We get a quadratic equation:

X 2 + Oh- A 2 = Oh.

The length of a segment is expressed as a positive number, so from two roots

X 1.2 =(-а±√а 2 +4 а 2)/2

you should choose positive x=(-a+√5a 2)/2 or x=(√5-1)a/2

This is the golden ratio.

It is denoted by the Greek letter φ in honor of ancient Greek sculptor Phidias (born at the beginning of the 5th century BC), in whose works the golden ratio appears many times.

The number is irrational, but in practice they use a rounded value equal to 0.62. If AB = a, then AC = 0.62a, CB = 0.38a.

Thus, the parts of the golden ratio make up approximately 62% and 38% of the entire segment.

c) how to geometrically, using a compass and a ruler, divide the segment AB in relation to the “golden ratio”. After all, the ancient architects did not know algebra? (Show slide 3).

On the segment AB from point B we restore a perpendicular to AB, the length of which is half the length of AB, i.e. BD=1/2AB. Next, connect points A and D. From point D as the center, draw a circle of radius BD. It will intersect the hypotenuse at point E. The length of the hypotenuse is 5 (according to Pythagoras). The length of the segment AE is √ 5-1. From point A we draw a circle of radius AE. It will intersect the circle at point C. If we now find the ratio AC:AB, then it will be equal to (√5-1)/2.

Student message

Student. It is generally accepted that the concept of the “golden ratio” was introduced into scientific use by Pythagoras, who borrowed knowledge about it from the Egyptians and Babylonians during his travels. Plato devoted his dialogue “Timaeus” to the mathematical and aesthetic views of the Pythagorean school, in particular to the issues of the golden ratio. (Show slide 4).

One of the most beautiful works Ancient Greek architecture is the Parthenon (5th century BC) – a temple in Athens.

This ancient building with its harmonious proportions gives us pleasure. The secret of Parferon's harmony lies in the relationships of its parts. “Golden proportions” are present in the dimensions of the facade of the ancient Greek temple of Partheron. During its excavations, compasses were discovered that were used by architects and sculptors ancient world. (Slide show 5, 6).

Many art historians, who sought to uncover the secret of the powerful emotional impact that the temple has on the viewer, sought and found the “golden proportion” in the relationships of its parts. The figure shows a number of patterns associated with the “golden ratio” coefficient. If the width of the end façade of the Parferon is taken as 1, we can obtain a geometric progression consisting of eight members: the distance between the second and seventh columns is equal, between the third and sixth, between the fourth and fifth. Similar patterns can be traced in the construction of the building in height. The ratio of the building's height to its length is 0.618. Combining these patterns, we get progression 1.

Architecture of Ancient Rus'.

a) the “golden ratio” in the construction of cross-domed churches

Student. Russian art The era of the Middle Ages, starting from the 10th century and up to the 12th century, is inextricably linked with the Church and the faith of Christ, which our people called Orthodox.

How many magnificent churches, decorated with mosaics, paintings (frescoes), and icons, were erected in Rus'. IN In the countries of Orthodox Christianity in the 10th-12th centuries, cross-domed churches with four or six pillars were built inside. What is the peculiarity of the architecture of such temples? (Slide show 7,8).

The pillars, dividing the internal space, seem to inscribe a cross into the rectangle of the temple, they divide the internal space, as if inscribing a cross into the rectangle of the temple, they divide interior space into three longitudinal and three transverse corridors (galleries) called naves. The central naves are wider than the side naves. A drum with a dome is supported on the pillars, and semi-cylindrical vaults are supported on them, facing the facades in the form of arches that complete them, the so-called zakomar.

Adjacent to the eastern side of the building are three altar semicircles, called apses. These are semi-cylinders protruding strongly on the plane of the walls. The structure is crowned with a cross.

If we design the drum and dome on the base of the temple, they will be depicted as a circle placed in the central part of the symbolic square. It feels the presence of a cross that intersects the circle - a reflection of the dome.

The architecture of the temples is deeply symbolic: the cube embodies the earth, and the dome the sky. In the temple itself, earth and sky are united both in the architectural structure and in the minds of people. But they are not easily united, they create a single space in which believers find peace and hope, compassion, consolation, love and faith.

When analyzing the proportions of the temple, the “golden ratio” can be found in the structure of the temple more than once. The main verticals of the temple are subject to the law of the “golden ratio”, determining its silhouette, the height of the base and the height of the drum, the ratio of the drum to its height, the shoulders to the diameter of the drum, etc.

As a result of such mathematical analysis, how perfect the creations of ancient architects seem, how much subtle harmonious elegance they contain. How firmly architecture and mathematics merge here.

b) White stone architecture of Vladimir-Suzdal Rus'

Teacher. But the most significant in the construction of temples is the white stone architecture of Vladimir-Suzdal Rus', which has survived to this day. The temples of Vladimir-Suzdal Rus' amaze with their nobility of shapes and proportions, and unique stone carvings.

A historical map of the Vladimir-Suzdal Principality is posted

(slide 9).

Student3. The city of Vladimir, the capital of the Vladimir-Suzdal principality, became the largest center of Russian culture during the reign of Prince Andrei Bogolyubsky, the son of Yuri Dolgorukov. The big and obese Prince Yuri Dolgoruky least of all liked to study state affairs. He preferred noisy feasts and riotous fun. He planted his sons in the cities to guard the borders. And to the bravest and most fearless Andrei Yuryevich he gave the important fortress of Vyshgorod.

Prince Andrei was 44 years old at that time, having lived his entire life in Suzdal, he felt uncomfortable and unusual in the fortress.

In the end, one night, without notifying his father, Andrei Yuryevich secretly rode north, taking with him the stolen icon of the Mother of God, well known in the area. Andrei was on his way to the Vladimir fortress on the Klyazma.

It is not known how the story would have ended, but Yuri Dolgoruky was poisoned at a feast and died.

So Andrei Yuryevich became an independent prince and left Vladimir as the capital of the principality.

Reproductions of the portrait of Andrei Bogolyubsky, the icon of the Mother of God (slides 10-13).

Each nation has its own shrine, the possession of which promises security and prosperity. The icon of the Mother of God, brought from Vyshgorod, became such a shrine. The clergy close to the prince begin to willingly and a lot talk about the miracles allegedly performed by her. One of them, as the legend says, happened not far from Vladimir. 10 km from the city, the horses that were carrying the icon stopped and could not budge. And then the prince decided to build a temple on this site and build his palace nearby. And name the place "Bogolyubovo"- "God's Favorite". A temple (Assumption Cathedral) and a castle were built, and the prince was nicknamed Andrei Bogolyubsky.

Prince Andrey begins large construction in the city of Vladimir. He erects fortress walls around it, and in the center of Vladimir he builds a new temple and the main entrance gate to the city, which is called “Golden”.

Scientists studying the reign of Andrei Bogolyubsky are amazed by his feverish activity to expand, strengthen and equip his capital.

The architects invited by Andrei Bogolyubsky understood perfectly well that they were participating in a great political matter - establishing the strength and power of the new center of the Russian land. It was a stronghold that was respected by other European sovereigns. And this stronghold was so wonderfully decorated that even now we see in its monuments one of high achievements artistic genius of our people. More than eight centuries have passed, but the memory of Andrei Bogolyubsky has not faded. They continue their lives and famous monuments his era. During the reign of Andrei Bogolyubsky, masterpieces of world art were erected - the palace complex in Bogolyubovo, the Assumption Cathedral, Dmitrievsky Cathedral, the Golden Gate in Vladimir and a unique church on the Nerl River near the city of Vladimir (Slide show 14,15,16).

The Church of the Intercession on the Nerl is a pearl of architecture of Vladimir-Suzdal Rus'.

Teacher. The Church of the Intercession on the Nerl is the most perfect temple created in Rus'. And now we will take a short trip to the Church of the Intercession on the Nerl (Slide show 17,18).

Two students take turns commenting on the slide show.

Student 1. An unfading, white stone temple, like a swan song.

Student 2. Graceful, slender, perfect, indescribable, obligatory, weightless - these and other enthusiastic epithets accompany the description of the famous Church of the Intercession on the Nerl.

Student 1. He stands among the flooded meadows above a quiet lake in which his overturned reflection lives.

Student 2. The Church of the Intercession on the Nerl is a masterpiece of world architecture, the pinnacle of creativity of Vladimir meters from the heyday of the Vladimir-Suzdal principality (Show slide 19).

Student 1. Tradition says that Prince Andrei Bogolyubsky built the Church of the Intercession on the Nerl in honor of the victorious campaign of the Vladimir regiments against the Bulgarians and in memory of the death of his son Izyaslav in this campaign. This is probably why this church, standing alone on the banks of the Nerl, emanates a light sadness. (Show slide 20).

Student 2. At the same time, the temple was dedicated to the new feast of the Intercession of the Virgin Mary in Rus'. This holiday was intended to testify to the special patronage of the Mother of God for the Vladimir land.

Thus, the temple, dedicated simultaneously to various events, became a monument of royal beauty (Show slide 21).

Student 1. The location for the church, a floodplain meadow at the confluence of the Nerl and the Klyazma, was indicated by Prince Andrei Bogolyubsky himself. Since there was widespread flooding here, a high foundation was built especially for the temple - an artificial hill made of clay and cobblestones, in which the foundation of the future building was laid (Slide show22).

Student 2. Structurally, the Church of the Intercession on the Nerl is very simple - it is a single-domed cross-domed four-pillar temple, typical for ancient Russian architecture. But the builders of the church managed to implement in it a completely new artistic image. Unnoticed by the eye, the walls of the church are tilted inward and thus visually increase the height (Slide show23).

Student 1. The church is large and surprisingly harmonious. The semi-cylinders (apses) are recessed into the body of the temple, and the eastern (altar) part does not outweigh the western (Slide show24).

Student 2. The vertical aspiration gradually and imperceptibly turns into the semicircular outlines of mosquitoes. The semicircles of the zakomar are echoed by the completion of gracefully elongated windows, the elongated drum of the dome, and the arcature belt of elongated stripes enhance the impression of elongation and elongation of the temple. (Slide show26).

Student 1. Res. The chicks who decorated the Church of the Intercession took the first, but brilliant steps on the path of Vladimir-Suzdal plastic arts from single relief images to grandiose sculptural and decorative ensembles on the walls of the Dmitrievsky Cathedral in Vladimir. The walls of the temple are decorated with white stone carvings, characteristic of Vladimir-Suzdal architecture (Show slide 26).

Student 2. The Church of the Intercession on the Nerl is compared to ancient Greek temples in its brevity and perfection of forms.

Student 1. In all of Russian poetry, which has given the world so many unsurpassed masterpieces, there is no more lyrical monument than the Church of the Intercession on the Nerl.

Student 2. How accurately and naturally the building fits into the surrounding landscape - the Central Russian meadow expanse, where fragrant herbs and azure flowers grow and the endless songs of larks sound...

Student 1.“Music frozen in stone” is the name given to the Church of the Intercession of the Virgin Mary, standing on the picturesque bank of the Nerl River. The pearl of ancient Russian architecture amazes with its perfection... How firmly architecture and mathematics merged in it.

Student 2. Exact proportions and ancient measures form a kind of “mathematical frame” of the church. And a detailed analysis of the building using geometric tools and calculations confirms the inextricable unity of mathematics and art.

Student 1. Let's take a break from mathematics and look at the church as wonderful work art that harmoniously fits into the natural landscape. The church stands on an island that was formed as a result of melting snow. There is water all around, the trees stand frozen, and only the church, like a fragile white boat, floats on the wide surface of the formed sea.

Student 2. The air smells like spring. There is amazing silence, peace and tranquility all around. They seem to protect people from the dark evil forces. And the standing water does not dare to flood and destroy its architectural splendor. Math melody architectural forms frozen in static chastity (Show slide 27, pause).

The student is reading. We came with you and froze

And they forgot all the words

Before the white miracle on the Nerl,

In front of the Church of the Intercession,

That is not stone, but all made of light,

From love, from prayers...

Teacher. Such masterpieces could only have arisen on Russian soil, personifying the ideal of beauty that had developed and reached such a remarkable flowering in the then main center of this land. After all, it is these monuments that reveal the soul of our people, the love for native land, the beauty of which they were called upon to crown not only for their time, but also for all subsequent generations of Russian people, glorifying in it the beauty of the Universe.

The student is reading. Russia, Rus-

Everywhere I look!

For all your suffering and battles

I love your old Russia,

Your forests, graveyards and prayers,

I love your huts and flowers,

And the skies are burning with heat

And the whisper of willows by the muddy waters,

I love you forever, until eternal peace.

Russia, Rus-

Protect yourself, protect yourself!

During this aesthetic-mathematical conference, members of the circle get acquainted with the relationship between mathematics and architecture. In preparation for the event, the children did a little independent research on the conference issues, where they had to conduct an independent search for information. Children worked with reference books, popular science literature, and Internet information.

The role of the manager consists of consulting work and joint processing of theoretical materials.

When familiarizing yourself with theoretical material concerning the concept of the “golden ratio,” the teacher’s message, accompanied by a demonstration of the necessary reproductions and information from the Internet, is most effective.

When familiarizing yourself with the architecture of the churches of Vladimir-Suzdal Rus' and, in particular, with the Church of the Intercession on the Nerl, children's performances are most effective. Independent coverage of these issues will expand ideas about the areas of application of mathematics and increase general cultural horizons. It is important that this event becomes a kind of impetus for the development of interest in the subject, arouses a desire to know more and arouses children’s interest in future professional activities.

Literature.

1. Teacher's newspaper. No. 13, 2006. A. Azevich. Music frozen in stone.

2. “Mathematics at school”. Magazine No. 8, 2007 O.B. Vergazova. Golden proportion: from ancient Russian fathoms to modern design.

3. Bendukidze A.D. Magazine “Quantum”, No. 8, 1973.

4. L.S. Sagatelova, V.N. Studenetskaya. Geometry: beauty and harmony. Publishing house "Teacher", 2006.

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temples

Performed by ancient Russians artists. “I look at the majestic paintings of ancient Russian temples, and me... in the pre-war years, books were published about goldsection V architecture: N. Vrunov. Proportions of ancient and medieval...

School-gymnasium No. 33

with in-depth study of economics and law

Golden ratio

Project manager: O. V. Bukaneva

Completed by: Bayizkan uulu Ali

Project goal:

• Knowledge of mathematical patterns in the surrounding world;
• Determining the meaning of mathematical patterns in nature and in world culture;
• Supplementing the knowledge system with ideas about the “Golden Section” as the harmony of the surrounding world.

Relevance:

The relevance of the study is dictated by the ubiquitous application of the principle of the golden ratio, which is found almost everywhere: in science, nature, humans, music, art, photography and much more, uniting the whole world into a single harmonious whole. There is an opinion that the events that happen to us also happen according to the golden ratio, the golden section.

Project objectives:

• Give a formulation to the concept of the golden ratio, its geometric application;
• Get acquainted with the history of the golden ratio;
• Find evidence of the presence of the golden ratio in nature;
• Explore the proportions of the human body;
• Consider the use of the golden ratio in art (sculpture, painting);
• Familiarize yourself with the use of the golden ratio in architecture;
• Conduct an analysis of architectural objects in Kyrgyzstan;

• Draw conclusions on the topic under study.

Introduction.

« 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."

Johannes Kepler

The concept of the Golden Ratio

The golden ratio is 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:

a: b = b: c

The parts of the golden ratio are approximately 62% And 38%

Golden ratio number - 0,618 And 1,6

Golden geometric shapes

IN

Golden Triangle

The golden triangle is an isosceles triangle whose base and side are in the golden ratio. AC/AB=0.62. One of its remarkable properties is that the length of the bisectors of the angles at its base is equal to the length of the base itself.

A

WITH

golden rectangle

M

L

A rectangle whose sides are in the golden ratio i.e. the ratio of length to width gives the number 1: 1.618 = 0.62; called the golden rectangle. KL/KN=0.62.

N

TO

Golden pentagon

The pentagram represents the container of golden proportions!

From the similarity of triangles ACD and ABE we can derive the known proportion AB/AC=AC/BC .

It is interesting that all the diagonals of the pentagon divide each other into segments connected by the golden ratio.

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.

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. 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, 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,

History of the golden ratio

Fibonacci series

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci, is indirectly connected with the history of the golden ratio. He traveled extensively in the East and introduced Arabic numerals to Europe. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time.

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 terms, 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 attitude - 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.

History of the golden ratio

Archimedes spiral

Archimedes' Spiral - a spiral constructed using a series of Fibonacci numbers

According to Archimedes himself: “A spiral is a trajectory of uniform motion of a point along a ray uniformly rotating around its origin.”

History of the golden section It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek and mathematician (VI century BC). There is a pre-philosophy that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians.

However, without the concept of the “golden ratio” we will not be able to trace the connection between the Fibonacci number series and the Archimedes spiral.

Let's imagine a watch dial with a long hand. The arrow moves around the circumference of the dial. And at this time a small bug moves along the arrow at a constant speed. The trajectory of the bug's movement is an Archimedes spiral. Goethe called the spiral “the curve of life.”

In nature, most shells have the shape of an Archimedes spiral. Sunflower seeds are arranged in a spiral. The spiral can be seen in cacti and pineapples. The hurricane is spiraling. A herd of deer scatters in a spiral. The DNA molecule is twisted in a double helix. Even galaxies are formed according to the principle of a spiral.

Let's imagine a watch dial with a long hand. The arrow moves around the circumference of the dial. And at this time a small bug moves along the arrow at a constant speed. The trajectory of the bug's movement is an Archimedes spiral.

Goethe called the spiral “the curve of life.” In nature, most shells have the shape of an Archimedes spiral. Sunflower seeds are arranged in a spiral. The spiral can be seen in cacti and pineapples. The hurricane is spiraling. A herd of deer scatters in a spiral. The DNA molecule is twisted in a double helix. Even galaxies are formed according to the principle of a spiral.

Human body proportions and the golden ratio

There are certain rules by which the human figure is depicted, based on the concept of proportionality of size various parts bodies.

An ideal, perfect body is considered to have proportions equal to the golden ratio. The basic proportions were determined by Leonardo da Vinci, and artists began to consciously use them. The main division of the human body is the navel point. The ratio of the distance from the navel to the foot to the distance from the navel to the crown is the golden ratio.

Golden ratio in the human body

Human bones are kept in proportion close to the golden ratio. And the closer the proportions are to the golden ratio formula, the more ideal a person’s appearance looks.

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 - φ

The distance from fingertips to wrist and from wrist to 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 shoulder level 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 exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze.

the top line of the eyebrows and from the top line

eyebrows to crown equals 1:1.618

Distance from tip of chin to

the top line of the eyebrows and from the top

eyebrow line to crown equals 1:1.618

Face height/face width

The central point where the lips connect to the base of the nose/length of the nose.

Face height / distance from the tip of the chin to the center point of the lips

Mouth width/nose width

Nose width / distance between nostrils

Interpupillary distance/eyebrow distance

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 = the golden ratio (excluding the thumb).

Middle finger/little finger ratio = 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 biphalangeal thumbs, only 8 fingers are created according to the principle of the golden ratio (the numbers 2, 3, 5 and 8 are the numbers of the Fibonacci sequence).

Also worth noting is the fact that for most people, the distance between the ends of their outstretched arms is equal to their height.

“The human body is the best beauty on earth” N. Chernyshevsky

Golden ratio in art

Golden ratio in painting

"Let no one

being a mathematician,

works."

Leonardo da Vinci.

Golden ratio in the picture

Leonardo da Vinci "La Gioconda"

The portrait of Mona Lisa is attractive because the composition of the drawing is built on “golden triangles” (more precisely, on triangles that are pieces of a regular star-shaped pentagon).

Painting "The Holy Family" by Michelangelo

Recognized as one of the masterpieces of Western European art of the Renaissance. Harmonic analysis showed that the composition of the painting is based on a pentacle.

.

Golden spiral in Raphael's painting "Massacre of the Innocents"

The “rule of the golden ratio” in architecture and art usually refers to compositions containing proportions close to the golden ratio of 3/8 and 5/8.

Golden ratio and visual centers

Painting “12 Apostles of Jesus Christ”

“Everything in the world is afraid of time, and time is afraid of the pyramids.” Arabic proverb.

Golden proportions of the Parthenon

The creation of the Parthenon follows the golden ratio, and therefore we are pleased to look at it

Golden proportions

cathedral Notre Dame of Paris

Intercession Cathedral

The proportions of the Intercession Cathedral on Red Square in Moscow are determined by eight members of the golden ratio series; many members of the golden ratio series are repeated many times in the intricate elements of the temple.

“..., but perhaps it would be even better to call such a cathedral “fossilized mathematics”

Jung D.

Government House (" White House»)

Golden ratio in the architecture of Kyrgyzstan

Burana Tower

Golden ratio in the architecture of Kyrgyzstan

Kyrgyz national academic theater opera and ballet named after Abdylas Maldybaev

Golden ratio in the architecture of Kyrgyzstan

Kyrgyz State Circus named after. A. Izibaeva

Golden ratio in the architecture of Kyrgyzstan

Gumbez Manas

"Golden ratio" and happiness

Research by sociologists confirm that the number of people satisfied and dissatisfied with their circumstances is subject to the proportions of the famous “golden ratio”.

According to the results of a survey of domestic and foreign psychologists, it turned out that they consider themselves happy 63% respondents. An amazing figure, since the golden ratio falls on 62% .

Conclusions:

The laws of the golden ratio have been known since ancient times and were used in science and art.

A beautiful (harmonious) combination of sounds contains the “golden” proportion (Pythagorean scale). Built according to the law of the golden ratio solar system. The planet Earth has five-pointed symmetry, the crust of which is made of pentagonal plates. There is reason to think that the whole world is built according to the principle of the golden proportion. In this sense, the Universe as a whole is a grandiose living organism, the similarity with which gives us the right to be called living organisms ourselves.

“ The “golden ratio” seems to be that moment of truth, without which, in general, nothing existing is possible. 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.

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.

Thank you

for your attention!

The presentation reveals the topic of the Golden Ratio in architecture Ancient world, architecture different countries world, architecture of Russia and the city of Bataysk, Rostov region. The work can be used in mathematics lessons in grades 5-9.

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Golden ratio Mathematics teacher of Municipal Educational Institution Secondary School No. 4 with in-depth study of individual subjects Priyma T.B. in architecture

Project goals: Understanding mathematical patterns in the world, determining the meaning of mathematics in world culture and supplementing the knowledge system with ideas about the “Golden Section” as the harmony of the surrounding world. Formation of independent skills research activities. Formation of skills to solve a key problem in the process of cooperation and creation of a product useful to society. Training in working with information and media to expand horizons and develop creative abilities.

Problem: The existence of harmony in the world around us. Application of knowledge about the golden ratio in the study of objects in the city of Bataysk.

Project objectives: Select literature on the topic. Conduct research on the following directions: Formulate the concept of harmony and mathematical harmony Familiarize yourself with the application of the Golden Ratio in architecture Study of the school yard Analysis of architectural objects and sculpture in Bataysk Conclusions on the topic under study

Mathematical understanding of harmony “Harmony is the proportionality of parts and the whole, the merging of various components of an object into a single organic whole. In harmony, internal orderliness and measure of being are externally revealed" - Big Soviet Encyclopedia Mathematical harmony is the equality or proportionality of parts with each other and parts with the whole. The concept of mathematical harmony is closely related to the concepts of proportion and symmetry.

Golden ratio in architecture 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. Pyramid of Cheops

Golden proportions of the Parthenon

We can also see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris)

Golden ratio in Russian architecture

The golden ratio in the architecture of the city of Bataysk The symbol of the city of Bataysk fits into the “golden triangle”

Height to width ratio is 1.67

Golden proportions of the Holy Trinity Church in Bataysk

Eternal Flame Monument to Soldiers Liberators Golden proportion of the Monument to Soldiers Liberators. Ratio 1.68

The golden ratio of the sculpture passes in front of the girl, focusing attention on her and reinforcing the impression that she is waiting for someone...

The Romeo and Juliet sculpture also fits into the golden rectangle

IN modern design cars: ratio of length to length of the car up to the second door is 1.61; side doors fit into a golden rectangle 1.62 Proportion of the height of the building in the center of Bataysk 1.62

Railway station The golden ratio of the central part of the railway station building in Bataysk is 1.66

Municipal educational institution secondary school No. 4. The ratio of the height of the building to the height of the porch is 1.61. The porch cut is a rectangle (aspect ratio 1.55)

The school fence section is close to the golden rectangle (1.58)

Well Ratio is 1.7, close to the golden ratio

Harmonious design of a school flower bed. Plants are planted near points of increased attention (3/8 from the edges of the flowerbed).

The design of this flowerbed does not correspond to the proportions of the golden ratio

In the process of harmonic analysis of architectural objects in the city of Bataysk, it was established that not all buildings under consideration obey the principle of the golden section. Many buildings built in Soviet era And modern buildings, forming the face of our city, gravitate towards the laws of beauty. Our city has its own harmonious face, thanks to its architecture, monuments, sculpture... We hope that the appearance hometown will bring aesthetic pleasure to more than one generation of Batayans.

Conclusion Having conducted research on this topic, we were able to provide answers to all the questions that were posed at the beginning of the project

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