Research work in mathematics “The Mysterious World of Numbers” (with presentations). Divine proportion - abstract The mysterious world of proportions presentation

Today we will get acquainted with an unusual proportion called the golden ratio and even the divine proportion. You will learn what role this proportion plays in the world around you, how it is related to the concept of harmony and how and why it is used in art (painting, architecture, photography...), design...

In painting, photography, and design, the golden ratio is very often used in the form of classical composition techniques, which you can read about by looking at any website dedicated to these types of art.] The main recommendation is as follows. The object, which is the central figure in the composition, does not always have to be located in the center. Certain points in the composition automatically attract attention. There are 4 such points, and they are located at a distance of 3/8 and 5/8 from the edges of the picture. Having drawn a grid, we get these points at the intersections of the lines (see photo).

The golden ratio refers to such a proportional division of a segment into unequal parts. In which the length of the entire segment is related to its larger part, as the length of the larger part is related to the length of the smaller one. This ratio is equal to the irrational number Ф= The golden ratio is first found in Euclid’s Elements (300 BC). Luca Pacioli, a contemporary of Leonard da Vinci, called it “divine proportion.” The golden ratio is denoted by the symbols PHI or Ф (in honor of the ancient Greek sculptor Phidias, who always used the golden ratio in his works). The mathematician Fibonacci first obtained a sequence of numbers, named after him Fibonacci numbers 1,1,2,3,5,8,13,21,34,55 ... The peculiarity of this number series is that each of its terms, starting from the third, is equal the sum of the previous two: 1+1=2; 1+2=3; 2+3=5; 3+5=8 ...In this case, the ratio of two neighboring terms is equal to the golden ratio, i.e. number F. When considering patterns associated with the manifestation of the golden ratio, they usually use the reciprocal of the number F: 1/1.618 = 0.618 a+ba+b a bb: a = (a+b) : b

Question: What is common in the arrangement of polypeptide chains of nucleic acids, rose petals, mollusk shells, mammal horns, sunflowers, and distant cosmic galaxies? Answer: their structure is based on a golden (logarithmic) spiral. This spiral fits into a golden rectangle (the ratio of its length and width is equal to the number Ф). By successively cutting off squares from it and inscribing a quarter of a circle into each of them, we get a golden spiral (see photo). The role of the spiral in the structure of animal and plant objects was discovered by T. Cook, who proved that the phenomenon of growth is associated with the golden spiral. The carrier of the genetic code - the DNA molecule - consists of two intertwined helices. Not long ago, spiral structures were discovered in inanimate nature.

Phyllotaxis is a peculiar lattice arrangement of leaves, seeds, and scales of many plant species. The rows of nearest neighbors in such lattices unfold in spirals or twist in helical lines around a cylinder. Sunflower seeds are arranged in logarithmic spirals. In this case, the ratio of the number of left and right spirals is equal to the ratio of neighboring Fibonacci numbers. You can find sunflowers with a ratio of the number of spirals of 34/55 and 55/89.

The golden ratio permeates the entire history of art: the pyramids of Cheops, the famous Greek temple of the Parthenon, most Greek sculpture monuments, the unsurpassed Mona Lisa by Leonard da Vinci, paintings by Raphael, Shishkin, etudes by Chopin, music by Beethoven, Tchaikovsky, poems by Pushkin... this is not a complete list of outstanding works of art , filled with wonderful harmony based on the golden ratio. The photograph shows buildings in which the golden ratio was used to divide the main masses of their structures. It is usually believed that such division is used in buildings built in the classical style. However, look at the Smolny Cathedral, built in the Baroque style, and you will easily discover the golden ratio.

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. The ideal female figure is considered to be that of Aphrodite de Milo (see picture). Interestingly, the statistically average body sizes of various people are also subject to the rule of the golden section (this is evidenced by the anthropological studies of Zeising (1855), who measured almost 2000 people. Out of curiosity, you can check for yourself how close your body is to the ideal. Go to the Internet, type “ideal proportions of the human body”, take measurements and draw conclusions.There are certain rules by which the human figure is depicted, based on the concept of proportionality of the sizes of various parts of the body.

The shape of bird eggs is described by the golden ratio. Today it has already been established that with this configuration the strength characteristics of the shell are the highest. The perfect shape of a dragonfly's body is created according to the laws of the golden ratio: the ratio of the length of the tail and the body is equal to the ratio of the total length to the length of the tail. Summary For centuries, scientists have been using the unique mathematical properties of the golden ratio. This relationship is found in all living organisms, plants at all levels of their development. The universality of its manifestation in the structure of organs, systems, and their functional parameters suggests that it plays the role of a brick in the foundation of all life on Earth. Recent research in the field of astronomy and physics shows that this section is related to the entire Universe.

1. Divide a segment 16 cm long in relation to the golden ratio. Use Fibonacci numbers Option 1 – 3 and 5 Option 2 – 2 and 3 2. The length of the rectangle is 20 cm (Option 1), 15 cm (Option 2). Find the width of the rectangle such that the ratio of length to width is the golden ratio Ф = 1.6 Solve the problem by composing equation 3. Check how ideal one of the ratios of your palm is: the ratio of the length of the index finger to the length of its two phalanges from the end of the finger. Measure the indicated lengths using a ruler and find their ratio. Round the resulting number to tenths and compare with Ф=1.6 (determine how much more or less it is than the number Ф)

Divine proportion

1. Introduction

2. “Golden ratio” in mathematics

2.1. "Golden ratio" - harmonic proportion

2.2. Fibonacci series

3. Self-organization of inanimate nature

3.1.Optimal physical parameters of the external environment

3.2. Symphony of the Earth

4. Principles of shape formation in nature

5.Proportions of the human body

Conclusion

Annex 1

Appendix 2

Appendix 3

Appendix 4

List of keywords.

Bibliography

Introduction

The history of the “Golden Section” is the history of human knowledge of the world. The concept of the “Golden Section” has passed through all stages of cognition in its development. The first stage of knowledge is the discovery of the “golden ratio” by the ancient Pythagoreans. They moved from simple contemplation of reality to expressing it in the world of numbers, but they confused the cause-and-effect concepts of the world and the guess about the global significance of the “Golden Section” remained only a guess for centuries. And yet, in his life, a person begins to use the “Golden Ratio” in his works of art.

The entire ancient Greek culture developed under the sign of the golden proportion. The Greeks were the first to establish: the proportions of a well-built human body obey its laws, which is especially clearly seen in the example of ancient statues (Apollo Belvedere, Venus de Milo). The Phrygian tombs and the ancient Parthenon, the Theater of Dionysus in Athens - they are all filled with the harmony of the golden proportion. Nowadays, interest in the golden ratio has increased with renewed vigor. A number of musicological works emphasize the presence of the golden ratio in the composition of works by Bach, Chopin, and Beethoven.

During the Renaissance, the golden proportion was elevated to the rank of the main aesthetic principle. Leonardo da Vinci, Raphael, Michelangelo, Titian and other great Renaissance artists composed their paintings consciously using the golden ratio. The 15th-century Dutch composer Jacob Obrecht makes extensive use of the “Golden Ratio” in his musical compositions, which are still likened to a “cathedral” created by a brilliant architect.

The practical needs of trading lead Fibonacci to open its series, which no one has yet associated with the “Golden Ratio”. In the 19th century, it was no longer artists, but experimental scientists who studied the patterns of phyllataxis (the arrangement of flowers), who again turned to the golden proportion. It turned out that the flowers and seeds of sunflowers, chamomiles, scales in pineapple fruits, conifer cones, etc. are “packed” in logarithmic spirals, curling towards each other. In this case, the numbers of “right” and “left” spirals always relate to each other, like neighboring Fibonacci numbers (13:8, 21:13, 34:21, 55:34), the limit of the sequence of which is the golden ratio.

Scientists discover “Golden proportions” in living and non-living matter, and already on the basis of this experience, amazing discoveries by our contemporaries A. P. Stakhov and I. V. Vitenko occur. Generalized golden proportions and generalized Fibonacci series. Their analysis leads researchers to results that are stunning in their simplicity and therefore even more significant: the “Golden Ratio” has redundancy and stability, which allow self-organizing systems to organize.

Topic of the work: the golden ratio is the basis of structural harmony of natural and artificial systems. A person distinguishes objects around him by their shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the shape. The form, the construction of which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a feeling of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole.

The purpose of the work is to prove that 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. To fully cover the topic, the author should consider the following topics:

1) What is the golden ratio? What is its connection with the Fibonacci series?

2) Clarification of the general patterns of development of living and inanimate nature.

3) Find mathematical patterns in the proportions of the human body.

4) Consider the effect of the law of the golden proportion in the physical and biological world.

5) Consider the historical process in accordance with the laws of growth “according to Fibonacci”

7) The golden ratio as a criterion of harmony and beauty in nature, art, architecture, etc.

The author found answers to the questions in the weekly educational and methodological supplement to the newspaper “First of September” Mathematics, in the books of Voloshinov V.A. , Vorobyova N.N., Stakhova A.P., Kovaleva F.V. To study this topic in more depth, the author of the work was forced to resort to Internet technologies.

1.1. "Golden ratio" - harmonic proportion

“Geometry has two treasures: one of them is the Pythagorean theorem, the other is the division of a segment in extreme and mean ratio. The first can be called a measure of gold, but the third is more like a precious stone.”

In mathematics, proportion (lat. proportio) is the equality of two ratios:

A straight line segment AB can be divided into two parts in the following ways:

into two equal parts – AB: AC = AB: BC;

into two unequal parts in any respect (such parts do not form proportions);

thus, when AB: AC = AC: BC.

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

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

Rice. 1. Geometric image of the golden ratio

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

Constructing division of a segment in the golden ratio.

Rice. 2. Division of a straight line segment according to the golden ratio. BC = 1/2 AB; CD = BC

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

Proof:

From DABC by the Pythagorean theorem we have: AC2 = AB2 + CB2, since AC = AD + DC then

(AD + DC)2 = AB2 + CB2,

by construction AD = AE, DC = CB= ½ AB.

From these equalities it follows (AE + ½ AB)2 = AB2 + AB2/4

AB – AE = EB => it follows that point E is the golden ratio of the segment AB.

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:

Pairs of rabbits

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division. So, 21: 34 = 0.617, and 34: 55 = 0.618. This ratio is denoted by the symbol F. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden proportion, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to everything.

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

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

Self-organization of inanimate nature

3.1.Optimal physical parameters of the external environment

A person’s senses give him the opportunity to perceive all the diversity of the external world, react sensitively even to minor changes in the external environment, and choose a way of behavior that provides him with a safe existence. However, the senses cannot perceive the entire range of relevant environmental parameters that can arise in nature. There are certain boundaries of sensation, characterized by the minimum and maximum parameters of the external environment that a person is able to perceive. These boundaries are called the absolutely lower and absolutely upper thresholds of sensations.

In the book of the Russian scientist V.I. Korobko “The Golden Proportion and Problems of Harmony of Systems” (1998) made an interesting attempt to show that the lower and upper thresholds are connected through the golden proportion.

Sound volume. It is known that the maximum sound volume that causes pain is 130 decibels. If we divide this interval by the golden ratio of 1.618, we get 80 decibels, which are typical for the volume of a human scream. If we now divide 80 decibels by the golden ratio, we get 50 decibels, which corresponds to the volume of human speech. Finally, if we divide 50 decibels by the square of the golden ratio 2.618, we get 20 decibels, which corresponds to a human whisper. Thus, all characteristic parameters of sound volume are interconnected through the golden proportion.

Air humidity. At a temperature of 18-20°, a humidity range of 40-60% is considered optimal. The boundaries of the optimal humidity range can be obtained if the absolute humidity of 100% is divided twice by the golden ratio:

100/2.618 = 38.2% (lower limit); 100/1.618 = 61.8% (upper limit).

Air pressure. When the air pressure is 0.5 MPa, a person experiences unpleasant sensations and his physical and psychological activity worsens. At a pressure of 0.3 - 0.35 MPa, only short-term work is allowed, and at a pressure of 0.2 MPa, work is allowed for no more than 8 minutes. All these characteristic parameters are interconnected by the golden ratio:

0.5/1.618 = 0.31 MPa; 0.5/2.618 = 0.19 MPa.

Outside air temperature. The boundary parameters of the outside air temperature, within which the normal existence (and, most importantly, the origin) of a person is possible, is the temperature range from 0 to + (57-58) ° C. Let us divide the indicated range of positive temperatures by the golden section. In this case we get two boundaries:

Both boundaries are temperatures characteristic of the human body: the first corresponds to a human body temperature of 36.6 ° C, the second is the most favorable temperature for the human body. The last limit can be obtained from human body temperature using the golden ratio: 36.6/1.618 = 22.62°C.

Although all these calculations, at first glance, seem artificial, they nevertheless force us to think about them, and sometimes even use them practically.

3.2. Symphony of the Earth

The cosmic body called Earth, in the process of global self-organization, turned into a “Beautiful Symphony” based on the “golden ratio”.

Let's start with the relationship between land and water on the Earth's surface. It turns out that the area of ​​the oceans is close to 62%, the rest of the planet's surface is occupied by continents and seas. Is it a coincidence that the relationship between these two main formations, which determine the appearance of the planet, the topography of the earth’s crust and its complex geomorphological life, corresponds to the golden proportion? Obviously not. Over a long period of evolution, lasting about 4.5 billion years, the structure of the planet should have reached some optimal state. And this harmony was expressed in the fact that, on the one hand, the Earth turned into a geododecahedron, and on the other hand, the ratio of land and water on its surface became equal to the ratio of the golden ratio.

Just as in the human body, blood carries substances throughout the body, ensuring metabolism, the creation of new structures, and the removal of toxins, so the branched system of water arteries transports substances on the planet, feeds plants, cleanses soils, and transports substances. The solar pump, like a heart, moves water, lifting it from the surface of the ocean into the atmosphere, and then irrigates the surface of the biosphere with rain. The entire water supply system - from the seepage of water through the capillaries of the soil and the saturation of rocks, to the formation of small streams, rivulets and huge water arteries - doesn’t all this resemble the circulatory system of humans and other higher organisms of the Earth

Let's start with the composition of the air. If there were 25% oxygen in the Earth's air, and not 21, as it is now, then the forest could burn in the rain, scientists believe. And if there were only 10% oxygen, then even dry wood would not burn. It seems that 21% of oxygen in the current atmosphere is not a random value, but the result of the life activity of the biosphere, the result of the self-organization of the planet.

The earth's crust is composed of rocks of sedimentary and igneous origin. Over the long history of the Earth, a variety of igneous rocks have been formed. Among the various varieties of rocks, two groups predominate - acidic (granites, granodiorites) and basic (gabbaro, balsates), the rest are found in tenths of a percent. 61% of Post-Cambrian rocks are felsic and 38.5% are mafic. For igneous rocks of all ages, felsic rocks account for 62.2% and mafic rocks account for 34.7%. The ratio of felsic to mafic rocks is 1.6 for Precambrian rocks and 1.66 for Post-Cambrian rocks. Within the limits of accuracy, all these ratios correspond to the golden proportion! Isn’t this where the basic principle of the earth’s crust is revealed, based on the harmonic relationship between acidic and basic igneous rocks? The question arises: is the formation of igneous rocks a “game of chance” or is it subject to some fundamental pattern, a “striving” for a harmonious, most expedient organization?

The elucidation of the general fundamental laws of the development of the Earth as a cosmic body is just beginning. Now the urgent task is to combine knowledge, create a general science of the Earth as an organically integral system, and the laws of the golden section can play an important role in the creation of science of the Earth.

4. "Golden" spirals are widespread in the biological world. This growth occurs in a logarithmic spiral. In the book "Curved Lines in Life" T. Cook explores the different types of spirals that appear in the horns of rams, goats, antelopes and other horned animals. Among many spirals, he chooses the “golden” spiral (“harmonic increase curve”) and considers it as a symbol of evolution and growth.

Spirals widely manifest themselves in living nature. The tendrils of plants twist in a spiral, tissue growth in tree trunks occurs in a spiral, seeds in a sunflower are located in a spiral, spiral movements (nutations) are observed during the growth of roots and shoots. Obviously, this shows the heredity of plant organization, and its roots should be sought at the cellular and molecular level.

The shape of the shells is striking in its perfection and the cost-effectiveness of the funds spent on its creation. The idea of ​​a spiral in shells is not expressed approximately, but in a perfect geometric form, in an amazingly beautiful, “honed” design

In some mollusks, the number of parts that form conical shells corresponds to Fibonacci numbers. Thus, the shells of foraminifera have 13 parts, the shells of the clawed snail - 8, the number of chambers of the nautilus shell - 34, the body of the nautiloids is divided into 13 parts, the shell of the giant tridacna is assembled into 5 folds. The number of ribs in a fossil brachiopod shell is 34. Tiny tectaculite shells have the same number of ribs. Along the edges of the spotted Cyprian shell from the Indian Ocean there are small teeth, the number of which is 21. From the above examples it is clear that the shell designs of many fossil and modern mollusks prefer the numbers 5, 8, 13, 21, 34.

But an even more convincing demonstration of the manifestation of the golden ratio in the plant world is the phenomenon of “phyllotaxis”

5.Proportions of the human body

For thousands of years, people have been trying to find mathematical patterns in the proportions of the human body, especially a well-built, harmonious person. For many centuries, individual parts of the human body served as units of length. So, the ancient Egyptians had three units of length: a cubit (466 mm), which was equal to seven palms (66.5 mm), a palm, in turn, equal to four fingers. The main measures of length in Russia were the fathom and the cubit, associated with human height; in addition, an inch was used - the length of the joint of the thumb, a span - the distance between the spread thumb and index fingers, a palm - the width of the hand.

Even in Ancient Egypt, the length of the foot was taken as a unit of body measurement. At the same time, the height of a person was on average 7 the length of his foot. In accordance with the aesthetic canon of the Greek sculptor Polycletus, the head was the unit of measurement for the body; The length of the body should be equal to eight times the size of the head.

The golden proportion occupies a leading place in the artistic canons of Leonardo da Vinci and Durer. In accordance with these canons, the golden proportion corresponds not only to the division of the body into two unequal parts by the waist line. The height of the face (to the roots of the hair) refers to the vertical distance between the arches of the eyebrows and the bottom of the chin, just as the distance between the bottom of the nose and the bottom of the chin refers to the distance between the corners of the lips and the bottom of the chin, this ratio is equal to the golden ratio.

Let us now take an “inventory” of the human body. He has one body, one head, one heart, etc.; many parts of the body are paired, for example, arms, legs, eyes, kidneys. The legs, arms, and fingers consist of three parts. The hands and feet have five fingers, and the hand and fingers consist of eight parts. A person has 12 pairs of ribs (one pair is atrophied and is present in the form of a rudiment). Obviously, in the past, humans had 13 ribs, but during the process of evolution, during the transition to an upright position, the number of ribs decreased.

As can be seen from the above list of parts of the human body, in its division into parts there are Fibonacci numbers from 1 to 34. Note that the total number of bones of the human skeleton is close to 233, that is, it corresponds to another Fibonacci number.

But the Fibonacci pattern is not limited to bones. For example, the structure of the brain is divided into seven parts: cortex, corpus callosum, cerebellum, cerebral ventricle, mosi, medulla oblongata, pituitary gland. At the base of the brain there are 8 parts that perform different functions. There are 8 different endocrine glands in the human body. The intestines and adjacent organs (stomach, liver, gall bladder, etc.) make up a total of 13 organs. The human respiratory organs consist of 8 parts. The liver also consists of 8 parts; The kidneys consist of 5 parts, and the heart of 13.

This list of human parts in which Fibonacci numbers are found could be continued. Is this a coincidence? Most likely no. Man, like other creations of nature, is subject to the universal laws of development. The roots of these laws must be sought deeper - in the structure of cells, chromosomes and genes, and further - in the emergence of life itself on Earth.

Conclusion

Nature, understood as the whole world in the diversity of its forms, consists of two parts: living and inanimate nature. Creations of inanimate nature are characterized by high stability and low variability, judging on the scale of human life. A person is born, lives, ages, dies, but the granite mountains remain the same and the planets revolve around the Sun in the same way as in the time of Pythagoras.

The world of living nature appears to us completely different - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and uniqueness of creative combinations! The world of inanimate nature is, first of all, a world of symmetry, which gives his creations stability and beauty. The natural world is, first of all, a world of harmony, in which the “law of the golden ratio” operates.

In the modern world, science is of particular importance due to the increasing impact of humans on nature. Important tasks at the present stage are the search for new ways of coexistence of man and nature, the study of philosophical, social, economic, educational and other problems facing society.

Analyzing all of the above, you can once again marvel at the enormity of the process of understanding the world, the discovery of more and more of its laws and conclude: 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. It can be expected that the laws of development of various natural systems, the laws of growth, are not very diverse and can be traced in a wide variety of formations. This is where the unity of nature is manifested. The idea of ​​such unity, based on the manifestation of the same patterns in heterogeneous natural phenomena, has retained its relevance from Pythagoras to the present day. When teaching school subjects, it is possible to demonstrate the relationships between concepts accepted in various fields of knowledge and processes occurring in the natural environment and in human society using the example of the properties of the “golden ratio”. When studying proportions, right triangles, the Pythagorean theorem, rectangles and regular pentagons, there is an opportunity to become familiar with the concept of the golden ratio. At the same time, an approach can be found to creating a holistic picture of the world in the minds of schoolchildren.

Human body proportions - was developed... this principle is golden proportions. Golden center proportions the structure of the human body... the ancient ideal of beauty - images divine twins Apollo and Artemis. Their...

Research project

"Golden ratio

around us"

1. Introduction…………………………………………………….3

2. Chapter 1 . History of the golden ratio

Golden ratio in mathematics. …………...……………….5

1. Chapter 2. Golden ratio in art…………………….….7

2. Chapter 3. Golden ratio in nature……………………..…..10

3. Chapter 4. The golden ratio around us………………….……..11

4. Chapter 4. Experiment……………………………………………………14

5. Conclusion…………………………………………………………….....15

6. Literature……………………………………………………..15

7. Appendix………………………………………………………..……16

Introduction.

"Geometry has two great

treasures. The first is the Pythagorean theorem,

second - divisions of the segment at the extreme and middle

relationships. The first is comparable to a measure of gold,

the second one looks more like a precious stone"

Johannes Kepler

While discussing the topic “Proportion” in a mathematics lesson, the teacher gave examples of the golden ratio, calling it “divine proportion.” Having become fascinated by this topic, I learned that the golden ratio was called the “Divine Proportion” by the medieval Italian mathematician Luca Pacioli, who wrote a book about the golden ratio, which he called “The Divine Proportion.” In his opinion, even God used the principle of the golden ratio to create the Universe.

The golden ratio is found everywhere: in art, nature, and the world around us. The topic is interesting and modern, it has not been lost in time. And therefore is the topic of my research.

My parents and math teacher supported me in my choice. In our research work, we tried to study this topic in more detail, to prove the presence of the golden ratio in the world around us.

What is the “golden ratio”?

Hypothesis:"Golden ratio" - harmonic proportion.

Object of research: reproductions of paintings, photographs and drawings of famous architectural structures, sculptures, modern buildings and the world around us.

Subject of study: shape and structure of the objects under study.

Target: Show that the great discovery - the GOLDEN RATIO, having passed through many centuries, is alive, relevant and in demand to this day.

Job Objectives:

We will try to analyze the history of the “golden ratio”.

We examine reproductions of paintings by famous artists, architectural structures and sculptures for the “golden ratio”.

We will try to find the “golden ratio” in nature and the world around us.

Let's conduct an experiment to identify preferences for the proportions of the “golden section”.

The novelty of the research: the disclosure to students of our school of the concept of the “golden ratio” in the world around us.

Progress of the study:

Select materials on the history of the “golden ratio” in the library and on the Internet.

Study the selected material.

Select photographs and drawings.

Find the “golden ratio” in the world around us.

Conduct an experiment and analyze the collected material.

Draw conclusions.

Practical significance:

Using acquired knowledge and research skills in the study of geometry, biology, fine arts, history, and astronomy.

Using the work to create a stand: “The Golden Ratio Around Us” in the mathematics classroom.

Research methods: observation, measurement, analysis, experiment.

Skills and abilities: select the necessary literature and draw conclusions based on the information collected, work on the Internet, conduct an experiment, and complete the work.

Chapter 1 . History of the golden ratio. Golden ratio in mathematics.

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

Denoted by the Greek letter phi (φ), the golden ratio is expressed as a number

͌ 0.618 (its inverse is 1.618) and has a number of interesting properties. φ is the first letter in the name of the great ancient Greek sculptor Phidias, who often used the golden ratio in his works, and the term was introduced by the great artist, scientist and inventor Leonardo da Vinci (1452-1519)

The history of the “Golden Section” is the history of human development of the world.

It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras and his students borrowed their knowledge of the golden ratio from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, and household items indicate that Egyptian craftsmen used the golden section ratios when creating them.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The entire ancient Greek culture developed under the sign of the golden proportion. The Greeks were the first to establish: the proportions of a well-built human body obey its laws, which is especially clearly seen in the example of ancient statues (Apollo Belvedere, Venus de Milo). The ancient Parthenon is filled with the harmony of the golden proportion. Nowadays, interest in the golden ratio has increased with renewed vigor. Let's consider the basic geometric shapes in which the “golden ratio” is present.

Fibonacci numbers and the golden ratio.

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, Leonardo Fibonacci built the following series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.
It is 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 two previous ones 2 + 3 = 5; 3 + 5= 8; 5 + 8= 13, 8 + 13= 21; 13 + 21= 34, etc., and if you divide each of them by the previous one, you get: 1:1=1; 2:1=2; 3:2=1.5; 5:3=1.666 666; 8:5=1.6; 13:8=1.625; 21:13= 1,615384 ... If you divide larger and larger Fibonacci numbers, you can get closer to the golden ratio ratio. Despite the fact that the book was published in 1202, Fibonacci numbers still attract mathematicians.

Golden ratio in mathematics.

Chapter 2. Golden ratio in art.

The golden ratio in I. I. Shishkin’s painting “Pine Grove”

Statue of Apollo Belvedere Statue of Venus de Milo

It is known that the famous sculpture was created according to the principle of the golden ratio. Point C divides segment AD, point B divides segment AC in a ratio approximately equal to 1.618.

Parthenon (ancient Greek architecture)

The ancient Greeks Iktinas, Collicrates, and Phidias, jointly created the Parthenon, in Athens around 440 BC. If the facade of the Parthenon is inscribed in a rectangle, then the sides of the rectangle form the golden ratio. The length of a rectangle is approximately 1.6 times greater than its width.

Great Pyramid of Pharaoh Cheops

Among the grandiose pyramids of Egypt, it occupies a special place Great Pyramid of Pharaoh Cheops (Khufu). The ingenious creators of the Egyptian pyramids sought to amaze distant descendants with the depth of their knowledge, and they achieved this by choosing the “golden” right triangle as the “main geometric idea” for the Cheops pyramid.

Frozen music of Russian churches

It is difficult to find a person who does not know and have not seen St. Basil's Cathedral on Red Square in Moscow. The composition of the cathedral buildings is characterized by a harmonious combination of symmetrical and asymmetrical proportions. The temple, symmetrical in its core, contains many geometric "irregularities". Thus, the central volume of the tent is shifted 3 m to the west from the geometric center of the entire composition. However, inaccuracy makes the composition more picturesque, “live” and it wins overall. The architectural decoration of the cathedral is characterized by an upward growth of decorative forms; the forms grow from one another, stretch upward, sometimes rising in large elements, sometimes forming groups consisting of smaller decorative parts.

Researchers discovered a proportion in it based on the golden ratio series:

1: 0,618:0,618 2:0,618 3:0,618 4:0,618 5:0,618 6:0,618 7

Pentagram and Pentagon

The pentagram was especially admired by the Pythagoreans and was considered their main identification sign.

The building of the US military department is shaped like a pentagram and is called the “Pentagon”, which means a regular pentagon.

Having studied the literature on this topic, we came to the conclusion that since ancient times people have been interested in the “golden ratio”. We also learned that people have long used the “golden ratio” in practice in the construction of various home buildings and temples, they were used in the manufacture of household utensils, and they were used to create mechanisms that made human work easier. In this chapter we have given the most interesting, in our opinion, examples related to the golden ratio.

Chapter 3. Golden ratio in nature.

Having studied the literature, we learned that in many butterflies the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding its wings, the moth forms a regular equilateral triangle. But if you spread your wings, you will see the same principle of dividing the body into 2,3,5,8. The dragonfly is also created according to the laws of the golden proportion: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

At first glance, the lizard has 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. You can notice the golden proportions if you look closely at the bird’s egg. “Pentagonal” symmetry occurs in nature:

Chinese rose cross-section of apple starfish cactus

We learned that the human body is created according to the laws of the golden ratio. It turns out that the waist divides the perfect human body in relation to the golden ratio. Scientists have found that for adult men this ratio

equals on average approximately 13/8 = 1.625, and for adult women it is 8/5 = 1.6. So men's proportions are closer to the "golden ratio" than women's proportions (however, a woman wearing high heels may be closer to the "golden" proportions). In a newborn, the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 in men it is 1.625. 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..

Thus, nature obeys the principle of the “golden proportion” (the structure of the human body, lizards, butterflies, leaves and flowers of many plants, etc.).

Chapter 4. The Golden Ratio Around Us

We examined the world around us for the presence of the golden ratio. And we found many interesting facts confirming that the golden ratio lives near us.

We study a 6th grade student:

The table shows that the body proportions are close to the golden ratio, but it is still necessary to grow to the ideal proportion.

Exploring the Solar System

Leafing through reference materials and encyclopedias, we found tables with characteristics of the planets of the solar system (Appendix 1). And they decided to check the solar system for the content of the golden ratio. To our surprise, we found it. The data is given in table No. 2

Table No. 2.

Note that the relationship between the planets Mars and Jupiter is noticeably different from others. Almost twice. The literature indicates that there is an asteroid belt between these planets. I was interested in this question. Having considered various sources of information, it turned out that the German physicist and mathematician I. Titius in 1766 found a numerical pattern in the distances of planets from the Sun. According to this

As a rule, some planet must have existed between the orbits of Mars and Jupiter. It is believed that the ancient Greeks called it Phaethon and its orbit was between the orbits of Mars and Jupiter. There is still debate about its existence.

We, relying on this table (the arithmetic mean ratio of the distances of the planets from the Sun, including the planet Phaethon, is 1.6) - We believe that there was a planet!!!

Chapter 4. Experiment

In this chapter, we conduct experiments to identify the preference for the proportions of the “golden section” by students of our school.

Experiment No. 1. We asked subjects to choose the most attractive among 11 rectangles, and only two of them were gold. The data is given below in table No. 3. Table No. 3

The table shows that the choice of the golden rectangle increased among high school students.

Experiment No. 2. We invited students in grades 5-6 (17 people in total) to be a little artists and depict the horizon of their future painting. When calculating the results, we noticed that the horizon line divides all the figures on average ͌ 1.65.

Experiments have confirmed that greater preference is given to golden proportions.

Conclusion.

The theme “The Golden Ratio Around Us” is interesting and modern, it has not been lost in time.The Golden Ratio can indeed be called the “Divine Proportion”. It not only surrounds us around us and is widespread in the solar system, but the events that happen to us also occur according to the golden proportion. For example, age-related crises of people. In social science there is a law of compaction of history - with each new stage the speed of development of society increases. This is a topic for a separate research paper.

An important result of studying this topic is that the principle of the golden ratio is used everywhere: in art, science, nature, man, harmoniously uniting everything in the world into a single whole. The accumulated material will be useful in further research work. You can study in more detail the buildings located on the territory of the Rebrikha district for the golden ratio. A journey into the world of the golden ratio in mathematics will be no less exciting for high school students.

Literature.

Weekly educational and methodological newspaper “Mathematics” No. 13, 2008 - Publishing house “First of September” ch. ed. A. Soloveichik.

I explore the world: Mathematics: Det. encycl./automatic comp. A.P. Savin, 2004.

What do we know about the planets? Mn., “People's Asveta”, 1977

Earth / Per. with it. I. Goreloy; - M.: JSC “Planet of Childhood”, 2001

Langdon N, Snape Ch. Let's go with mathematics: Transl. from English-M.: Pedagogy, 1987

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Municipal educational institution "Parfenyevskaya secondary school"

Supervisor Smirnova L.A., mathematics teacher

2010-2011 academic year

Introduction

There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you produced the “golden ratio”. The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. An endless series after the decimal point - 1.6180339887... A strange, mysterious, inexplicable thing: this divine proportion mystically accompanies all living things. Inanimate nature does not know what the “golden ratio” is. But you will certainly see this proportion in the curves of sea shells, and in the shape of flowers, and in the appearance of beetles, and in the beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”. So what is the “golden ratio”?.. What is this ideal, divine combination? Maybe this is the law of beauty? Or is he still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The “golden ratio” is both, and the third. Only not separately, but simultaneously... And this is his true mystery, his great secret.

The concept of the "golden ratio".

Golden ratio - this is such a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger one, as the larger one is to the whole.

a: b = b: c orc: b = b: a.

This proportion is:

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

For example, in a regular five-pointed star, each segment is divided by a 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 equal1.618 It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge 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.

Examples of using the golden ratio

Golden ratio in mathematics

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler. From the pointIN a perpendicular equal to half is restoredAB . Received pointWITH connected by a line to a pointA . A segment is plotted on the resulting lineSun ending with a dotD . Line segmentAD transferred to directAB . The resulting pointE divides a segmentAB in the golden ratio ratio. Segments of the golden ratio are expressed as an infinite irrational fractionA.E. = 0.618..., ifAB take as oneBE = 0.382... For practical purposes, approximate values ​​of 0.62 and 0.38 are often used. If the segmentAB taken as 100 parts, then the larger part of the segment is equal to 62, and the smaller part is 38 parts.

The properties of the golden ratio are described by the equation:x 2 – x – 1 = 0. Solution of this equation:

Golden ratio in art

in music

The most extensive study of the manifestations of the golden section in music was undertaken in 1925 by art critic L. Sabaneev. He studied two thousand works by various composers. In his opinion, the temporal extent of a musical work is divided by “certain milestones” that stand out during the perception of music and facilitate the contemplation of the form of the whole. All these musical milestones divide the whole into parts, usually according to the law of the golden ratio.

According to the observations of L. Sabaneev, in the musical works of various composers, not one golden ratio is usually stated, but a whole series of similar sections. Each such section reflects its own musical event, a qualitative leap in the development of the musical theme. In the 1770 works of 42 composers he studied, 3275 golden sections were observed. The number of works in which at least one golden ratio was observed was 1338. The largest number of musical works in which there is a golden ratio are by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%) , Scriabin (90%), Chopin (92%), Schubert (91%).
All 27 etudes by Chopin have been studied in most detail. 154 golden ratios were discovered in them; in only three studies the golden ratio was absent. In some cases, the structure of a musical work combined symmetry and the golden ratio at the same time; in these cases it was divided into several symmetrical parts, in each of which the golden ratio manifests itself. Beethoven's works are also divided into two symmetrical parts, and within each of them manifestations of the golden proportion are observed.
Moreover, the more talented the composer, the more golden sections are found in his works. In Arensky, Beethoven, Borodin, Haydn, Mozart, Scriabin, Chopin and Schubert, golden sections were found in 90% of all works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition. It can be recognized that the golden proportion is a criterion for the harmony of the composition of a musical work.

to the cinema

In cinema, S. Eisenstein artificially constructed the film Battleship Potemkin according to the rules of the “golden ratio”. He broke the tape into five parts. In the first three, the action takes place on a ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city occurs exactly at the golden ratio point. And each part has its own fracture, which occurs according to the law of the golden ratio.

Golden ratio in painting

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci.His personality is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.”Let's look closely at the painting "La Gioconda".The portrait of Mona 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.

Also, the proportion of the golden ratio appears in Shishkin’s painting. In this famous painting by I. I. Shishkin, the motifs of the golden ratio are clearly visible. A brightly sunlit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio.

In Raphael's painting "The Massacre of the Innocents" another element of the golden proportion is visible - the golden spiral. In Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman holding him close, the warrior with his sword raised, and then along the figures of the same group on the right side of the sketch . It is unknown whether Raphael built the golden spiral or felt it.

T. Cook used the golden ratio when analyzing Sandro Botticelli’s painting “The Birth of Venus.”

Golden ratio in architecture

One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).

The figures show 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...

On the floor plan of the Parthenon you can also see the "golden rectangles"

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the Pyramid of Cheops:

Not only were the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids.

I decided to look at the plans for Parfenyev's churches and see if there was a golden ratio there. The result is an application (multimedia presentation).

Golden ratio in sculpture

The golden proportion was used by many ancient sculptors. The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

Athena Parthenos Olympian Zeus

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.

Golden ratio in fonts and household items

Golden proportions in parts of the human body

In 1855, the German researcher of the golden ratio, Professor Zeising, published hiswork "Aesthetic Research". Zeising 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.

I conducted a similar study in 11th grade. The measurement results are shown in the table.Application (multimedia presentation).

Golden ratio in biology and wildlife

Biological studies have shown that, starting with 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.

Consider a chicory shoot. 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 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 ejects 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. It was found that the numerical series of Fibonacci numbers characterizes the structural organization of many living systems. For example, the helical leaf arrangement on a branch forms a fraction (number of revolutions on the stem/number of leaves in a cycle, eg 2/5; 3/8; 5/13), corresponding to the Fibonacci series. The “golden” proportion of five-petal flowers of apple, pear and many other plants is well known. The carriers of the genetic code - DNA and RNA molecules - have a double helix structure; its dimensions almost completely correspond to the numbers of the Fibonacci series. Goethe emphasized nature's tendency toward spirality.

The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. Goethe called the spiral the “curve of life.” The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. In many butterflies, the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding its wings, the moth forms a regular equilateral triangle. But if you spread your wings, you will see the same principle of dividing the body into 2,3,5,8. The dragonfly is also created according to the laws of the golden proportion: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail

In a lizard, the length of its tail is related to the length of the rest of the body as 62 to 38. You can notice the golden proportions if you look closely at a bird's egg.

All living things are created in accordance with the proportion of the Golden Section

Some discoveries and theories of modern science,
related to the "golden ratio"

1.Penrose tiles

In ancient science, the “parquet problem” was widely known, which boils down to densely filling a plane with geometric figures of the same type. As is known, such filling can be done usingtriangles , squares Andhexagons . By usingpentagons ( Pentagons ) such filling is impossible.

Parquet problem

Let's take a closer look againregular pentagon also calledPentagon orpentagram , a flat geometric figure based on the "golden ratio".

Regular pentagon or pentagon

As is known, after drawing diagonals in the pentagon, the original pentagon can be represented as a set of three types of geometric figures. In the center there is a new pentagon formed by the intersection points of the diagonals. The rest of the pentagon includes five isosceles triangles, colored yellow, and five isosceles triangles, colored red. Yellow triangles are "golden" because the ratio of the hip to the base is equal to the golden ratio; they have sharp corners of 36 at the apex and sharp corners at 72 at the foundation. Red triangles are also “golden”, since the ratio of the hip to the base is equal to the golden ratio; they have an obtuse angle of 108 at the apex and sharp corners at 36 at the foundation. Now let's connect two yellow triangles and two red triangles with their bases. As a result we get two"golden" rhombus . The first one (yellow) has an acute angle of 36 and an obtuse angle of 144 . We will call the left rhombusthin rhombus, and the right rhombus isthick rhombus.

"Golden" diamonds

The English mathematician and physicist Rogers Penrose used “golden” diamonds to construct “golden” parquet, which was calledPenrose tiles. Penrose tiles are a combination of thick and thin diamonds.

Penrose Tiles

It is important to emphasize thatPenrose tiles have “pentagonal” symmetry or 5th order symmetry, and the ratio of the number of thick rhombuses to thin ones tends to the golden proportion!

2.Quasicrystals

On November 12, 1984, a short paper published in the prestigious journal Physical Review Letters by Israeli physicist Dan Shechtman provided experimental evidence for the existence of a metal alloy with exceptional properties. When studied by electron diffraction methods, this alloy showed all the signs of a crystal. Its diffraction pattern is composed of bright and regularly spaced dots, just like a crystal. However, this picture is characterized by the presence of “icosahedral” or “pentangonal” symmetry, which is strictly prohibited in the crystal for geometric reasons. Such unusual alloys were calledquasicrystals. In less than a year, many other alloys of this type were discovered. There were so many of them that the quasicrystalline state turned out to be much more common than one might imagine.The discovery of quasicrystals is another scientific confirmation that, perhaps, it is the “golden proportion”, which manifests itself both in the world of living nature and in the world of minerals, that is the main proportion of the Universe.

3.Fullerenes

The term "fullerenes"» are called closed molecules of type C 60 , WITH 70 , WITH 76 , WITH 84 , in which all carbon atoms are located on a spherical or spheroidal surface. In these molecules, the carbon atoms are arranged at the vertices of regular hexagons or pentagons that cover the surface of a sphere or spheroid. The central place among fullerenes is occupied by the C molecule 60 , which is characterized by the greatest symmetry and, as a consequence, the greatest stability. This molecule, which resembles the tire of a soccer ball and has the structureregular truncated icosahedron, The carbon atoms are arranged on a spherical surface at the vertices of 20 regular hexagons and 12 regular pentagons, so that each hexagon is bordered by three hexagons and three pentagons, and each pentagon is bordered by hexagons. "Fullerenes" are essentially "man-made" structures arising from fundamental physics research. They were first synthesized in scientists G. Kroto and R. Smalley (who received the Nobel Prize in 1996 for this discovery). But in they were unexpectedly discovered in rocks , that is, fullerenes turned out to be not only “man-made”, but also natural formations. Now fullerenes are being intensively studied in laboratories in different countries, trying to establish the conditions for their formation, structure, properties and possible areas of application.

4. Resonance theory of the solar system

The revolution frequencies of the planets and the differences in revolution frequencies form a spectrum with an interval equal to the golden ratio.

5. Fibonacci resonances of the genetic code

The establishment by science of the now widely known fact of the amazing simplicity of the basic principles of encoding hereditary information in living organisms is one of the most important discoveries of mankind. This simplicity lies in the fact that hereditary information is encoded by texts of three-letter words -triplets orcodons compiled on the basis of an alphabet of four letters - nitrogenous bases A (adenine), C (cytosine), G (guanine), T (thymine). This recording system is essentially the same for the entire vast variety of diverse living organisms and is calledgenetic code.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 basesT, C, A, G inside DNA. He discovered that successive sets of DNA nucleotides are organized into long-range order structures calledRESONANCES . 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, ...). For example, the genetic code -insulin chain has the following form:

AT G-TT G-GT C-AAT -CAG-CAC-CTT - T GT -GGT - T CT -CAC-CT C-GTT - GAA-GCT
- TT G-T AC-CTT -GTT - T GC-GGT -GAA-CGT -GGT - TT C-TT C-T AC-ACT -CCT -AAG-
A.C.T

6. Golden proportion in Cantor’s theory of transfinite sets and quantum physics (E-infinity theory)

In recent years, there has been an increased interest in theoretical physics in the “golden section”. The works of the English physicist of Egyptian origin Mohammed El Nashieh show the connection of the “golden ratio” with quantum physics.

Conclusion

The dramatic history of the Golden Ratio, which lasted several thousand years, at the beginning of the 21st century - the “Age of Harmony” - may end with a great triumph for the Golden Ratio. Penrose tiles, the resonant theory of the Solar system (Molchanov, Butusov), quasicrystals (Shekhtman), fullerenes (Croto and Smalley, Nobel Prize 1996) were only the harbingers of this triumph. “Mathematics of Harmony” (Stakhov), Fibonacci and Lucas hyperbolic functions (Stakhov, Tkachenko, Rozin), “Bodnar geometry”, “Law of structural harmony of systems” (Soroko), “E-infinity theory” (El Nashie), Fibonacci matrices and “golden” square matrices (Stakhov) and, finally, “golden” genematrices (Petukhov) - this is not a complete list of modern scientific discoveries based on the Golden Section. These discoveries give reason to suggest that the Golden Ratio is some kind of “metaphysical” knowledge, a “proto-number”, a “universal code of Nature”, which can become the basis for the further development of science, in particular mathematics, theoretical physics, genetics, and computer science.