Design and research work in mathematics on the topic "golden ratio". Divine proportion - abstract The world of proportions in human life

Class: 6

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Lesson type: generalization lesson

Equipment: computer, interactive whiteboard.

Lesson objectives:

Educational:

• generalization and systematization of students’ knowledge on this topic;
• strengthening the applied and practical orientation of the studied topic;
• establishing intra-subject and inter-subject connections with other topics in the course of mathematics, geography, physics, astronomy, biology, chemistry.

Educational:

• broadening the horizons of students,
• vocabulary replenishment;

Educational:

• nurturing interest in the subject and related disciplines,
• to cultivate a sense of beauty, a sense of patriotism.

I. Organizational moment:

1) message about the topic of the lesson (slide 1);

2) communicating the goals and objectives of the lesson.

II. Updating knowledge on the topic “Proportions”:

1. What is the ratio of two numbers called?
2. What does the ratio of two numbers show?
3. What is proportion?
4. What are the terms of this proportion called?
5. What basic property do the members of a proportion have?
6. Which two quantities are called directly proportional? (give examples of directly proportional quantities).
7. Which two quantities are called inversely proportional? (examples).

III. From the history of proportion. (slides 2-5)

Word "proportion" comes from the Latin word proportionio, meaning proportionality, a certain relationship between parts. Proportions have been used since ancient times to solve various problems in mathematics.

Even in ancient Greece, mathematicians used such a device as PROPORTION.

Proportion is the equality of the ratios of two or more pairs of numbers or quantities.

In Babylon, plans of ancient cities were drawn using proportions. The drawing shows a plan of the ancient Babylonian city of Nippur found during excavations. When scientists compared the results of excavations of the city with this plan, it turned out that it was made with great accuracy.

IV. Practical application of proportions. (slide 6-7)

Mathematics is used in almost all areas of human life. And in everyday life we ​​use mathematical skills, including proportion.

1. Architecture (slides 8-11)

When building a temple in honor of the goddess Diana, the Romans took the proportion that distinguishes slender women: the thickness of the column was only 1/8 of its height. Thanks to this, the columns seemed taller than they actually were, precisely due to the reduction in thickness. The architecture included both types of columns, one maintaining male and the other female proportions in the relationship between base and height.

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.

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

Solve problems.

1. 4 thousand bricks are used to build a house. How many thousand bricks are needed to build 15 similar houses?

2. To transport sand during construction, 14 vehicles with a carrying capacity of 4.5 tons were required. How many vehicles with a carrying capacity of 7 tons would be required to transport the same sand?

2. Cooking (slides 12-13)

The concept of proportion is used in cooking. When we prepare a dish, we try to use the amount of ingredients indicated in the cookbook. This is done so as not to spoil the dish. If we take more salt, we will oversalt it, and if we take less, it will not be tasty. Another proportion allows you to calculate the amount of ingredients for preparing the same dish for a different number of guests.

Solve problems

3. To make jam from 2 kg of gooseberries, you need 3 kg of sugar. How many kg of sugar are needed to make jam from 4.4 kg of gooseberries.

4. During drying, the mass of apples changed from 20 kg to 18.2 kg. By what % did the mass of apples decrease during drying?

3. Medicine (slides 14-16)

In medical practice, doctors monitor how much and when to give medicine to the patient. In the right doses, the medicine gives a therapeutic effect, in smaller doses it is useless, and in large doses it is harmful. When making medicines, proportions are also observed. Accuracy is necessary here, since if the proportions of the ingredients that make up the medicine are violated, the result may be not medicine, but poison. Ratios and proportions are also used in pharmacies in the manufacture of medicines and medicinal drinks. To make a drug, you need to know exactly how many parts there are for any part.

Solve problems

5. For a medicinal chamomile decoction, 20 g of dry chamomile are needed per 100 g of boiling water. How many g of chamomile are needed for 500 g of decoction?

6. The patient is prescribed a course of medication, which must be taken 250 mg twice a day for 7 days. One package of the medicine contains 10 tablets of 125 mg. What is the smallest number of packages needed for the entire course of treatment?

4. Chemistry (slides 17-19)

The theory of proportions has occupied a well-deserved place in solving problems in chemistry.

For example. What is the percentage concentration of the solution obtained by dissolving 5 g of table salt in 45 g of water?

Solve problems

7. 100 g of salt was dissolved in 2.4 liters of water. What is the concentration of the resulting solution?

8. There are 90 g of 80% vinegar essence. What is the largest amount of 9% table vinegar that can be obtained from it?

5. Technology (slides 20-23)

In technology lessons we also use proportion. When we want to sew something smaller or larger, we reduce or increase the pattern to the size we need. For example, a pattern for an apron for yourself and a doll. The sizes of the elements of the doll's apron differ from the corresponding sizes of my apron by the same number of times.

Solve problems

9. The overcasting machine processes 0.6 m of fabric in 2.16 minutes. How many meters can you sweep in 1.44 minutes?

10. It takes 1.2 m to make a children’s dress. How much fabric is needed for a dress for adults if the cost is 40% more?

6. Physics (slides 24-25)

Since ancient times, people have used various levers. Paddle, crowbar, scales, scissors, swing, wheelbarrow, etc. - examples of levers. The gain that the lever gives in the applied effort is determined by the proportion, where M and m are the masses of the loads, and L and l are the “arms” of the lever.

Solve problems

11. Using the lever rule, find M if l=2 m, L=8 m, m=4 kg.

12. In the city of Zhukovsky, demonstration flights of aircraft take place at the MAKS air show. A fighter aircraft such as the MIG-29 requires about 7.5 tons of kerosene for 3 hours of flight. How many tons of kerosene will the MIG-29 need for 7 hours of flight?

7. Modeling (slides 26-27)

Solve problems

13. The length of the car model is 42 cm. What is the length of the car if its dimensions are reduced by 10,000 times.

14. The sailboat model uses 60 cm of fabric. How many meters of fabric are needed to make three similar sailboats?

8. Geography. (slides 28-30)

In geography, proportion is also used - scale . Scale is the ratio of the length of a segment on a map or plan to the length of the corresponding segment on the ground. The scale shows how many times the distance on the plan is less than the indicated distance in reality.

Solve problems

15. Find the distance from Moscow to the North Pole, if on the map this distance is 3.5 cm, and M is 1:100000000.

16. Find the distance on the map between the cities of Rostov-on-Don and Moscow, if the distance between them is 1200 km, and M is 1:50000000.

V. Student reports on the application of proportion.

9. Fine arts. (slides 30-37)

10. Biology.(slides 38-39)

11. Music (slides 40-41)

12. Literature (slides 42-44)

VI. Conclusion (slide 45)

Since ancient times, people have used mathematics in everyday life. One of them is proportion. It is used from cooking to works of art such as sculpture, painting, architecture, and also in wildlife.

VII. Homework.

Literature

1. From the experience of conducting extracurricular work in mathematics in high school. Sat. articles edited by P. Stratilatova. – M.: Uchpedgiz, 1955.
2. D.Pidow. Geometry and art. – M.: Mir, 1989.
3. Magazine “Quantum”, 1973, No. 8.
4. Magazine “Mathematics at School”, 1994, No. 2, No. 3.
5. G. Mishkevich “Doctor of Entertaining Sciences” - M.: Knowledge, 1986
6. I. Ageeva “Entertaining materials on computer science and mathematics” – M.: Creative Center, 2005.
7. CD-ROM “From Plow to Laser 2.0”, New disc, 1998
8. Standard basic software package for educational institutions First Aid 1.0 Disc No. 56 New generation electronic educational resources Disc 1/1 DVD
9. http://www.sak.ru/reference/famous-buildings/famous-building5-1f.html Parthenon
10. http://www.foxdesign.ru/legend/apollo1.html Apollo Belvedere
11. http://www.sunhome.ru/journal/184 Mona Lisa
12. http://www.beseder.co.il/image-gallery/11897/1/1/ Leonardo da Vinci

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 the 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)

At school, in the lessons of natural sciences: physics, chemistry, biology, astronomy, geography and in the lessons of the humanities: history, literature, native and foreign languages, we study nature and society. In music, drawing, drawing, and gymnastics lessons, we are introduced to the world of art. In addition to these disciplines, these subjects, throughout all school years we study mathematics: arithmetic, algebra, geometry, trigonometry. What sciences should these disciplines be classified as? What is the subject of their study? Many scientists classify mathematics as a natural science, since mathematics studies the world around us: objects and phenomena of nature, society and human thinking. Physics, chemistry, biology study objects and phenomena of the world around us from the aspect of their quality. Mathematics studies the same objects, phenomena from the side of their quantity, space and time, they say - from the side of their form.

Therefore, scientists consider mathematics to be a natural science that studies our material world. Mathematics permeates all branches of knowledge, including the humanities. Nowadays economics, philology and other sciences cannot do without mathematics. Therefore, some scientists consider mathematics to be a layer between the natural sciences and the humanities.

The great German mathematician Carl Friedrich Gauss once called mathematics “the queen of all sciences” and “the queen and servant of all sciences.” This is what she is called for her noble service to almost all sciences.

There are many methods in mathematics that allow you to solve certain problems. Even in ancient Greece, mathematicians used such a device as PROPORTION.

Proportion is the equality of the ratios of two or more pairs of numbers or quantities. For example, the dimensions of a model of a machine or structure differ from the dimensions of the original by the same factor, which specifies the scale of the model. Therefore, if you select 4 points A, B, C and D on the original and designate the corresponding points on the model as A1, B1, C1 and D1, then the equality == will hold. This equality of relations is called proportion. It shows that the ratio of distances between points on the original is the same as the ratio of distances between corresponding points on the model.

In ancient times, the idea of ​​proportionality was used in an implicit form when solving problems using the complex position method: they gave the desired quantity a value, calculated what value one of these quantities should have, and compared it with the condition of the problem. The ratio of the values ​​gave the coefficient by which the selected value must be multiplied to obtain the correct answer.

Proportions began to be studied systematically in Ancient Greece. At first, only proportions made up of natural numbers were considered, and therefore it was believed that the numbers a, b, c, d form a proportion if a is the same multiple, the same fraction or the same fraction of b as c of d. In the 4th century. BC e. The ancient Greek mathematician Eudoxus gave a definition of proportion, composed of quantities of any nature. Ancient Greek mathematicians solved problems that are solved today using equations, and algebraic transformations were replaced by the transition from one proportion to another.

In modern mathematics, various PROPERTIES OF PROPORTIONS are used.

The main property of proportion. If a: b = c: d, then a∙d = b∙c

Reversal of proportion. If a: b = c: d, then b: a = d: c

Rearrangement of middle and extreme terms. If a: b = c: d, then a: c = b: d (rearrangement of the middle terms of the proportion), d: b = c: a (rearrangement of the extreme terms of the proportion).

Increasing and decreasing proportions. If a: b = c: d, then

(a + b) : b = (c + d) : d (increase in proportion),

(a – b) : b = (c – d) : d (decreasing proportion).

Making proportions by adding and subtracting. If a: b = c: d, then

(a + c) : (b + d) = a: b = c: d (composing proportions by addition),

(a – c) : (b – d) = a: b = c: d (composing proportions by subtraction)

Mathematics is used in almost all areas of human life. And in everyday life we ​​use mathematical skills, including proportion.

COOKING

The concept of proportion is used in cooking. When we prepare a dish, we try to use the amount of ingredients indicated in the cookbook. This is done so as not to spoil the dish. If we take more salt, we will oversalt it, and if we take less, it will not be tasty. Another proportion allows you to calculate the amount of ingredients for preparing the same dish for a different number of guests.

MEDICINE

In medical practice, doctors monitor how much and when to give medicine to the patient. In the right doses, the medicine gives a therapeutic effect, in smaller doses it is useless, and in large doses it is harmful. When making medicines, proportions are also observed. Accuracy is necessary here, since if the proportions of the ingredients that make up the medicine are violated, the result may not be medicine, but poison.

TECHNOLOGY

In technology lessons we also use proportion. When we want to sew something smaller or larger, we reduce or increase the pattern to the size we need. For example, a pattern for an apron for yourself and a doll. The sizes of the elements of the doll's apron differ from the corresponding sizes of my apron by the same number of times.

GEOGRAPHY

In geography, proportion is also used - scale. Scale is the ratio of the length of a segment on a map or plan to the length of the corresponding segment on the ground. The scale shows how many times the distance on the plan is less than the indicated distance in reality.

There are different types of scale: numerical, linear and named. The numerical scale is written as a fraction, the numerator of which is one, and the denominator is the degree of reduction in the projection. For example, a scale of 1:5,000 shows that 1 cm on the plan corresponds to 5,000 cm (50 m) on the ground. The larger scale is the one whose denominator is smaller. For example, a scale of 1:1,000 is larger than a scale of 1:25,000. The numerical scale is used to determine how many times all distances on the plan are reduced. The larger the number in the denominator of the fraction, the greater the number of times the actual distance is reduced, the smaller the map.

The entry “1 cm - 10 m” is called a named scale, and the distance on the ground corresponding to 1 cm on the plan is called the scale value. Using the scale value is very convenient for determining distances.

A linear scale is also placed on the plans. Linear scale is a graphic scale in the form of a scale bar divided into equal parts. This is a straight line divided into equal parts (usually centimeters). At each division of the line, the distance on the ground corresponding to it is indicated. The first division to the left of 0 is divided into smaller parts. Using a linear scale, you can find out the exact sizes of objects depicted on the terrain plan and the distances between them.

Task. Find the distance from Moscow to the North Pole, if on the map this distance is 3.5 cm, and M is 1:100000000.

Let's make a proportion: x=, i.e. x= 350000000cm=3500km.

Answer. The distance on the ground from Moscow to the North Pole is 3500 km.

ART

Alexey Petrovich Stakhov, Doctor of Technical Sciences (1972), Professor (1974), Academician of the Academy of Engineering Sciences of Ukraine writes about harmony:

“For a long time, people have been striving to surround themselves with beautiful things. Already the household items of the inhabitants of antiquity, which, it would seem, pursued a purely utilitarian purpose - to serve as a storage place for water, a weapon for hunting, etc., demonstrate man’s desire for beauty. At a certain stage of his life development, man began to wonder: why is this or that object beautiful and what is the basis of beauty? Already in Ancient Greece, the study of the essence of beauty, beauty, formed into an independent branch of science - aesthetics, which among ancient philosophers was inseparable from cosmology. At the same time, the idea was born that the basis of beauty is harmony.

Beauty and harmony have become the most important categories of knowledge, to a certain extent even its goal, because ultimately the artist seeks truth in beauty, and the scientist seeks beauty in truth. The beauty of a sculpture, the beauty of a temple, the beauty of a painting, a symphony, a poem. What do they have in common? Is it possible to compare the beauty of the temple with the beauty of the nocturne? It turns out that it is possible if common criteria for beauty are found, if general formulas of beauty are discovered that unite the concept of beauty of a wide variety of objects - from a daisy flower to the beauty of a naked human body? ".

The famous Italian architectural theorist Leon Battista Alberti, who wrote many books on architecture, said the following about harmony:

“There is something more, made up of the combination and connection of three things (number, limitation and placement), something with which the whole face of beauty is miraculously illuminated. We call this harmony, which, without a doubt, is the source of all charm and beauty. After all, the purpose and goal of harmony - to arrange parts, generally speaking, different in nature, by some perfect relationship so that they correspond to one another, creating beauty. It embraces the whole of human life, permeates the entire nature of things. For everything that nature produces is all measured by the law of harmony. And "Nature has no greater concern than that what she produces is perfect. This cannot be achieved without harmony, for without it the highest harmony of the parts disintegrates."

The Great Soviet Encyclopedia gives the following definition of the concept 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.”

The “golden proportion” is a mathematical concept and its study is primarily a task for science. But it is also a criterion of harmony and beauty, and this is already a category of art and aesthetics, which studies harmony and beauty from a mathematical point of view.

In the classics of fine art, for many centuries, a technique for constructing proportions has been traced, called the golden ratio, or the golden number. (this term was introduced by Leonardo da Vinci). 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.

In art, the number 1:1.62 or

That is, an approximate expression of the ratio of a smaller value in proportion to its larger value.

The golden number is observed in the proportions of a harmoniously developed person: the length of the head divides the distance from the waist to the top of the head in the golden ratio.

In addition, there are several more basic golden proportions of our body: the distance from the fingertips to the wrist and from the wrist to the elbow is 1:1. 618 the distance from shoulder level to the top of the head and the size of the head is 1:1. 618 the distance from the navel point to the top of the head and from shoulder level to the top of the head is 1:1. 618 the distance of the navel point to the knees and from the knees to the feet is 1:1. 618 the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1. 618 the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1. 618 the distance from the tip of the chin to the top line of the eyebrows and from the top line of the eyebrows to the crown is 1:1. 618

In works of fine art, artists and sculptors, consciously or subconsciously, trusting their trained eye, often use the ratio of sizes in the golden ratio.

The same phenomenon is observed in other structures of nature: in the spirals of mollusks, in the corollas of flowers and in many other things familiar to us, for example, the arrangement of leaves on a shoot also obeys the golden number!

Since ancient times, people have used mathematics in everyday life. One of them is proportion. It is used from cooking to works of art such as sculpture, painting, architecture, and also in wildlife.

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.

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