I like to walk around the center of Moscow, where there are many old buildings decorated with geometric shapes containing the golden ratio. They catch the eye of a person and make them admire their beauty. It became interesting for me to look beyond the textbook on geometry, and look at the role of the golden section in the cultural sphere of life.
golden ratio(or the proportion of Phidias), according to many researchers, is the most pleasing to the human eye. This can explain its many-sided application by man, for example, such areas as architecture, painting, photography and landscape design widely use this proportion and related properties. This proportion was held in high esteem by the smartest people, such as Leonardo da Vinci and Le Corbusier. The artist and architect Leonardo Da Vinci believed that the ideal proportions of the human body should be associated with the golden ratio. The architect Le Corbusier was guided by him in many of his works. I wanted to get some basic knowledge on this topic.
During the Renaissance, the golden ratio was very popular, for example, it was customary to take the dimensions of the picture so that the ratio of width to height was equal to the number of Phidias. The form of the golden section was given not only to paintings, but also to books, tables, postcards. Therefore, I would like to take a closer look at the use of the golden section in various eras from antiquity, the Renaissance to the XlX century. To do this, you need to read and study the literature related to this topic, find the most Interesting Facts and include them in your abstract.
The purpose of this abstract is to present the information in a clear and interesting way. To achieve the goal, the following tasks were set
1. define the concepts of symmetry and asymmetry, the golden section.
2. describe the golden figures and build some of them
3. talk about the application and use of the divine proportion by man
To write my work, I use the following literature: Azevich A.I. "Twenty Lessons of Harmony", Vedov V. "Health Pyramids", Sagatelova S.S., Studenetskaya V.N. Geometry: beauty and harmony. The simplest tasks analytical geometry on surface. Golden symmetry, Proportion around us. 8-9 grades: elective courses”, N.Ya. Vilenkin “Behind the pages of a mathematics textbook”, articles from the electronic version of the Science and Technology library, an electronic version of the encyclopedia for children in mathematics. Book Azevich A.I. “Twenty Lessons of Harmony”, in my opinion, reveals the topic of symmetry and asymmetry well, and gives clear and detailed initial information about the golden ratio. Sagatelova S. S., Studenetskaya V. N. Geometry: beauty and harmony. The simplest problems of analytic geometry on the plane. Golden symmetry, Proportion around us. Grades 8-9: Elective Courses” describes the golden figures and how to build them well. N.Ya. Vilenkin “Behind the Pages of a Mathematics Textbook” explains in detail the derivation of the golden section formulas and their properties, and also describes the construction of the golden section and the pentagram well. Vedov V. "Pyramids of Health" explains the Fibonacci series and obtaining the number of Phidias in an accessible and understandable way. Articles from the electronic version of the Science and Technology library, the electronic version of the encyclopedia for children in mathematics give detailed description the use of the golden ratio in antiquity, the Renaissance and the 19th century.
Chapter 1 Golden Ratio - Symmetry or Asymmetry?
The most important goal this essay is to show beauty as the main category of aesthetics and mathematics.
Have you ever thought about the meaning of the word "harmony"?
Harmony is a Greek word meaning "consistency, proportion, unity of parts and whole." Outwardly, harmony can manifest itself in melody, rhythm, symmetry and proportion. The last two are related to mathematics. Mathematics is a unique means of knowing beauty. Since beauty is multifaceted and many-sided, it confirms the universality of mathematical laws.
Harmony reigns in everything,
And everything in the world is rhythm, chord and tone.
Let's continue the story in order from largest to smallest.
Symmetry is the fundamental principle of the structure of the world.
Symmetry - in a broad or narrow sense, depending on how you define the meaning of this concept - is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.
G. Weil
Symmetry is a common phenomenon, its universality serves effective method knowledge of nature. Symmetry in nature is needed to maintain stability. Inside the external symmetry lies the internal symmetry of the construction, which guarantees balance. Symmetry is a manifestation of the desire of matter for reliability and strength.
Symmetrical forms provide repeatability of successful forms, therefore they are more resistant to various influences. Symmetry is multifaceted.
The immutability of certain objects can be observed in relation to various operations - rotations, reflections, transfers.
There are three main types of symmetry studied at school: symmetry about a point (central symmetry), symmetry about a line (axial symmetry) and symmetry about a plane.
Central symmetry of the flower
Central symmetry in a man-made ornament.
Symmetry relative to a straight line on the example of the building of Moscow State University
Symmetry with respect to a plane in a ball.
These are not the only types of symmetry, there is also screw symmetry. If we consider the arrangement of leaves on a tree branch, we will notice that the leaf is separated from the other, but also rotated around the axis of the trunk. The leaves are arranged on the trunk along a helical line, so as not to obscure sunlight from each other.
Screw symmetry in nature on the example of a shell .
The use of helical symmetry by a person on the example of a ladder .
Symmetry is multifaceted. It has properties that are both simple and complex at the same time, capable of manifesting once and infinitely many times.
If a person who is not very familiar is offered several figures, he will intuitively choose the most symmetrical ones. Most likely, being in such a situation, we will choose equilateral triangle or square.
Man instinctively strives for stability, convenience and beauty. The world is so chaotic and unpredictable that figures and things that contain order, harmony, symmetry are the most pleasant for a person to perceive. Working with shapes that have more symmetries is easier.
According to how many symmetries the figures have, one can classify them. The most perfect figure is a ball that has all kinds of symmetry.
Symmetry is hardworking. It gives each of its species the power to generate more and more new figures.
Symmetry can be observed in all areas of our life: the symmetry of the construction of buildings, music and the symmetry of images in literature, the symmetry of dance.
Symmetry is one of the principles of building the world.
Symmetry is the guardian of peace,
Asymmetry is the engine of life.
Asymmetric can also be harmonious. Symmetry evokes a feeling of peace, stillness, while asymmetry evokes a feeling of movement and freedom.
The researchers who received the Nobel Prize showed that our world is not symmetrical, the laws of symmetry in the Universe are not observed. The world is asymmetric at all levels: from elementary particles to biological species.
The most famous example of asymmetric harmony is the golden ratio. There are words belonging to Johannes Kepler: “Geometry has two treasures: one of them is the Pythagorean theorem, the other is the division of a segment in the middle and extreme ratio.” . It is this proportion that is the subject of my essay. In the following chapters, I will talk about the application of the golden ratio, and below I will give a definition of this concept and how to obtain it.
Any person who at least indirectly had to deal with the geometry of spatial objects in interior design and architecture is probably well aware of the principle of the golden section. Until recently, several decades ago, the popularity of the golden section was so high that numerous supporters of mystical theories and the structure of the world call it the universal harmonic rule.
The essence of universal proportion
Surprisingly different. The reason for the biased, almost mystical attitude towards such a simple numerical dependence was several unusual properties:
- A large number of objects in the living world, from a virus to a person, have basic proportions of the body or limbs that are very close to the value of the golden ratio;
- The dependence of 0.63 or 1.62 is typical only for biological beings and some varieties of crystals, inanimate objects, from minerals to landscape elements, have the geometry of the golden section extremely rarely;
- The golden proportions in the structure of the body turned out to be the most optimal for the survival of real biological objects.
Today, the golden ratio is found in the structure of the body of animals, the shells and shells of mollusks, the proportions of leaves, branches, trunks and root systems in enough a large number shrubs and herbs.
Many followers of the theory of the universality of the golden section have repeatedly made attempts to prove the fact that its proportions are the most optimal for biological organisms in the context of their existence.
Usually, the structure of the shell of Astreae Heliotropium, one of the marine mollusks, is given as an example. The shell is a calcite shell rolled up in a spiral with a geometry that almost coincides with the proportions of the golden section.
A more understandable and obvious example is an ordinary chicken egg.
The ratio of the main parameters, namely, large and small focus, or distances from equidistant points of the surface to the center of gravity, will also correspond to the golden section. At the same time, the shape of the shell of a bird's egg is the most optimal for the survival of a bird as a biological species. In this case, the strength of the shell plays a far from the main role.
For your information! The golden ratio, also called the universal proportion of geometry, was obtained as a result of huge amount practical measurements and comparisons of the sizes of real plants, birds, animals.
Origin of the universal proportion
The ancient Greek mathematicians Euclid and Pythagoras knew about the golden section ratio. In one of the monuments ancient architecture- The Cheops pyramid has a ratio of sides and base, individual elements and wall bas-reliefs are made in accordance with the universal proportion.
The golden section technique was widely used in the Middle Ages by artists and architects, while the essence of the universal proportion was considered one of the secrets of the universe and was carefully hidden from the average layman. The composition of many paintings, sculptures and buildings was built strictly in accordance with the proportions of the golden section.
For the first time, the essence of the universal proportion was documented in 1509 by the Franciscan monk Luca Pacioli, who had brilliant mathematical ability. But the real recognition took place after the German scientist Zeising conducted a comprehensive study of the proportions and geometry of the human body, ancient sculptures, works of art, animals and plants.
In most living objects, some body sizes are subject to the same proportions. In 1855, scientists concluded that the proportions of the golden section are a kind of standard for the harmony of the body and form. We are talking, first of all, about living beings; for dead nature, the golden ratio is much less common.
How did you get the golden ratio
The golden ratio is easiest to imagine as the ratio of two parts of the same object of different lengths, separated by a dot.
Simply put, how many lengths of a small segment will fit inside a large one, or the ratio of the largest of the parts to the entire length of a linear object. In the first case, the ratio of the golden ratio is 0.63, in the second case, the aspect ratio is 1.618034.
In practice, the golden ratio is just a proportion, the ratio of segments of a certain length, the sides of a rectangle or other geometric shapes, related or conjugate dimensional characteristics of real objects.
Initially, the golden proportions were derived empirically using geometric constructions. There are several ways to construct or derive a harmonic proportion:
For your information! Unlike the classic golden ratio, the architectural version implies the aspect ratio of the segment in the proportion of 44:56.
If the standard version of the golden section for living beings, painting, graphics, sculptures and ancient buildings was calculated as 37:63, then the golden section in architecture from the end of the 17th century began to be used more and more often 44:56. Most experts consider the change in favor of more "square" proportions as the spread of high-rise construction.
The main secret of the golden ratio
If the natural manifestations of the universal section in the proportions of the bodies of animals and humans, the stem base of plants can still be explained by evolution and adaptability to the influence of the external environment, then the discovery of the golden section in the construction of houses of the XII-XIX centuries was a certain surprise. Moreover, the famous ancient Greek Parthenon was built in compliance with the universal proportion, many houses and castles of wealthy nobles and wealthy people in the Middle Ages were built deliberately with parameters very close to the golden ratio.
The golden ratio in architecture
Many of the buildings that have survived to this day testify that the architects of the Middle Ages knew about the existence of the golden section, and, of course, when building a house, they were guided by their primitive calculations and dependencies, with which they tried to achieve maximum strength. The desire to build the most beautiful and harmonious houses in the buildings of the residences of the reigning persons, churches, town halls and buildings of particular social importance in society was especially manifested.
For example, the famous Notre Dame Cathedral in its proportions has many sections and dimensional chains corresponding to the golden section.
Even before the publication of his research in 1855 by Professor Zeising, in late XVIII century, the famous architectural complexes of the Golitsyn hospital and the Senate building in St. Petersburg, the Pashkov house and the Petrovsky Palace in Moscow were built using the proportions of the golden section.
Of course, houses with strict observance of the rule of the golden section were built earlier. It is worth mentioning the monument of ancient architecture of the Church of the Intercession on the Nerl, shown in the diagram.
All of them are united not only by a harmonious combination of forms and high quality construction, but also, first of all, the presence of the golden section in the proportions of the building. The amazing beauty of the building becomes even more mysterious if you take into account the age, the building of the Intercession Church dates back to the 13th century, but the building received its modern architectural appearance at the turn of the 17th century as a result of restoration and restructuring.
Feature of the golden section for a person
The ancient architecture of buildings and houses of the Middle Ages remains attractive and interesting for modern man for many reasons:
- Individual art style in the design of the facades, it avoids the modern stamp and dullness, each building is a work of art;
- Mass use for decorating and decorating statues, sculptures, stucco, unusual combinations of building solutions from different eras;
- The proportions and compositions of the building draw the eye to the most important elements of the building.
Important! When designing a house and developing appearance Medieval architects used the rule of the golden section, unconsciously using the peculiarities of the perception of the human subconscious.
Modern psychologists have experimentally proven that the golden ratio is a manifestation of an unconscious desire or human reaction to a harmonious combination or proportion in size, shape, and even color. An experiment was conducted during which a group of people who were unfamiliar with each other, who did not have common interests, of different professions and age categories, were offered a series of tests, among which was the task of bending a sheet of paper in the most optimal aspect ratio. According to the test results, it was found that in 85 cases out of 100 the sheet was bent by the subjects almost exactly according to the golden ratio.
That's why modern science believes that the phenomenon of universal proportion is a psychological phenomenon, and not the action of any metaphysical forces.
Using the Universal Section Factor in Modern Design and Architecture
The principles of applying the golden ratio have become extremely popular in the construction of private houses in the last few years. The ecology and safety of building materials have been replaced by harmonious design and correct distribution energy inside the house.
The modern interpretation of the rule of universal harmony has long spread beyond the boundaries of the usual geometry and shape of an object. Today, not only the dimensional chains of the length of the portico and pediment, individual elements of the facade and the height of the building, but also the area of rooms, window and door openings, and even the color scheme of the interior of the room are subject to the rule.
The easiest way is to build a harmonious house on a modular basis. In this case, most departments and rooms are made in the form of independent blocks or modules, designed in accordance with the rule of the golden section. Building a building as a set of harmonious modules is much easier than building a single box, in which most of the facade and interior must be within the strict limits of the golden ratio.
Many private home construction firms use the principles and concepts of the golden ratio to increase the estimate and give clients the impression of a deep study of the design of the house. As a rule, such a house is declared as very comfortable and harmonious in use. The right ratio of the areas of the rooms guarantees spiritual comfort and excellent health of the owners.
If the house was built without taking into account the optimal ratios of the golden section, you can redevelop the rooms so that the proportions of the room correspond to the ratio of the walls in a ratio of 1: 1.61. To do this, furniture can be moved or additional partitions inside the rooms can be installed. Similarly, the dimensions of window and door openings are changed so that the width of the opening is 1.61 times less than the height of the door leaf. In the same way, planning of furniture, household appliances, wall and floor decoration is carried out.
It is more difficult to choose a color scheme. In this case, instead of the usual ratio of 63:37, the followers of the golden rule adopted a simplified interpretation - 2/3. That is, the main color background should occupy 60% of the space of the room, no more than 30% is given to the shading color, and the rest is reserved for various related tones, designed to enhance the perception of the color solution.
The internal walls of the room are divided by a horizontal belt or border at a height of 70 cm, the installed furniture should be commensurate with the height of the ceilings according to the golden ratio. The same rule applies to the distribution of lengths, for example, the size of the sofa should not exceed 2/3 of the length of the wall, and the total area occupied by the furniture is related to the area of the room as 1: 1.61.
The golden ratio is difficult to apply en masse in practice due to only one section value, therefore, when designing harmonious buildings, they often resort to a series of Fibonacci numbers. This allows you to expand the number of possible options for proportions and geometric shapes of the main elements of the house. In this case, a series of Fibonacci numbers, interconnected by a clear mathematical relationship, is called harmonic or golden.
In the modern method of designing housing based on the principle of the golden section, in addition to the Fibonacci series, the principle proposed by the famous French architect Le Corbusier is widely used. In this case, the height of the future owner or the average height of a person is chosen as the starting unit of measurement, by which all parameters of the building and interior are calculated. This approach allows you to design a house not only harmonious, but also truly individual.
Conclusion
In practice, according to the reviews of those who decided to build a house according to the rule of the golden section, a well-built building really turns out to be quite comfortable for living. But the cost of the building due to individual design and the use of building materials custom sizes increases by 60-70%. And there is nothing new in this approach, since most of the buildings of the last century were built specifically for individual characteristics future owners.
A person distinguishes objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a sense 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 principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.
Golden Ratio - Harmonic Proportion
In mathematics proportion(lat. proportio) call the equality of two relations: a : b = c : d.Line segment AB can be divided into two parts in the following ways:
into two equal parts AB : AC = AB : Sun;
into two unequal parts in any ratio (such parts do not form proportions);
so when AB : AC = AC : Sun.
The golden section is such a proportional division of a segment into unequal parts, in which the entire segment relates to the larger part in the same way as the larger part itself relates to the smaller one; or in other words, the smaller segment is related to the larger one as the larger one is to everything
a : b = b : c or from : b = b : but.
Rice. one. Geometric representation of the golden ratio
Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden ratio using a compass and ruler.
Rice. 2. Division of a line segment according to the golden ratio. BC = 1/2 AB; CD = BC
From a point IN a perpendicular is restored equal to half AB. Received point FROM connected by a line to a dot BUT. A segment is drawn on the resulting line Sun, ending with a dot D. Section AD transferred to a straight line AB. The resulting point E divides the segment AB in the golden ratio.
Segments of the golden ratio are expressed by an infinite irrational fraction AE= 0.618... if AB take as a unit BE\u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then the largest part of the segment is 62, and the smaller is 38 parts.
The properties of the golden section are described by the equation:
x 2 - x - 1 = 0.
Solution to this equation:
The properties of the golden section created a romantic aura of mystery and almost mystical worship around this number.
The second golden ratio
The Bulgarian magazine "Fatherland" (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives a different ratio of 44: 56.Such a proportion is found in architecture, and also takes place in the construction of compositions of images of an elongated horizontal format.
Rice. 3. Construction of the second golden section
The division is carried out as follows (see Fig. 3). Section AB is divided according to the golden ratio. From a point FROM the perpendicular is restored CD. Radius AB there is a point D, which is connected by a line to a point BUT. Right angle ACD is divided in half. From a point FROM a line is drawn until it intersects with a line AD. Dot E divides the segment AD in relation to 56:44.
Rice. 4. Division of a rectangle by a line of the second golden ratio
On fig. 4 shows the position of the line of the second golden section. It is located in the middle between the golden section line and the middle line of the rectangle.
Golden Triangle
To find segments of the golden ratio of the ascending and descending series, you can use pentagram.Rice. five. Building regular pentagon and pentagrams
To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471...1528). Let be O- the center of the circle A- a point on the circle and E- middle of the segment OA. Perpendicular to Radius OA, restored at the point ABOUT, intersects the circle at a point D. Using a compass, set aside a segment on the diameter CE = ED. The length of a side of a regular pentagon inscribed in a circle is DC. Putting segments on the circle DC and get five points to draw a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36° at the top, and the base laid on the side divides it in proportion to the golden section.
Rice. 6. Construction of the golden triangle
We draw a straight line AB. from point BUT lay a segment on it three times ABOUT arbitrary value, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside segments ABOUT. Received points d And d 1 connect with straight lines to a point BUT. Section dd 1 set aside on the line Ad 1 , getting a point FROM. She split the line Ad 1 in proportion to the golden ratio. lines Ad 1 and dd 1 is used to build a "golden" rectangle.
History of the golden section
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek 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 decorations from the tomb of Tutankhamun indicate that the Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from the tomb of his name, holds measuring instruments in his hands, in which the proportions of the golden division are fixed.The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.
Rice. 7. Dynamic Rectangles
Plato (427...347 BC) also knew about the golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the questions of the golden division.
In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.
Rice. 8. Antique golden ratio compasses
In what has come down to us ancient literature the golden division was first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (II century BC), Pappus (III century AD) and others were engaged in the study of the golden division. medieval Europe they got acquainted with the golden division from the Arabic translations of the "Beginnings" of Euclid. The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.
During the Renaissance, interest in the golden division among scientists and artists increased in connection with its use both in geometry and in art, especially in architecture Leonardo da Vinci, an artist and scientist, saw that Italian artists had great empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is considered the creator of descriptive geometry.
Luca Pacioli was well aware of the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. In 1509, Luca Pacioli's Divine Proportion was published in Venice, with brilliantly executed illustrations, which is why they are believed to have been made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden ratio, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity of God the Son, God the Father and God the Holy Spirit (it was understood that the small segment is the personification of God the Son, the larger segment is the personification of God the Father, and the entire segment - the god of the holy spirit).
Leonardo da Vinci also paid much attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in golden division. So he gave this division the name golden ratio. So it is still the most popular.
At the same time, in northern Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches an introduction to the first draft of a treatise on proportions. Durer writes. “It is necessary that the one who knows something should teach it to others who need it. This is what I set out to do."
Judging by one of Dürer's letters, he met with Luca Pacioli during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. Dürer assigned an important place in his system of ratios to the golden section. The height of a person is divided in golden proportions by the belt line, as well as by the line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face - by the mouth, etc. Known proportional compass Dürer.
Great astronomer of the 16th century Johannes Kepler called the golden ratio one of the treasures of geometry. He is the first to draw attention to the significance of the golden ratio for botany (plant growth and structure).
Kepler called the golden ratio continuing itself. “It is arranged in such a way,” he wrote, “that the two junior terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains until infinity."
The construction of a series of segments of the golden ratio can be done both in the direction of increase (increasing series) and in the direction of decrease (descending series).
If on a straight line of arbitrary length, postpone the segment m, put aside a segment M. Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending series
Rice. nine. Building a scale of segments of the golden ratio
In subsequent centuries, the rule of the golden ratio turned into an academic canon, and when, over time, a struggle began in art with the academic routine, in the heat of the struggle, “they threw the child out with the water.” The golden section was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden section, Professor Zeising, published his work Aesthetic Research. With Zeising, exactly what happened was bound to happen to the researcher who considers the phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be "mathematical aesthetics".
Rice. 10. Golden proportions in parts of the human body
Zeising did a great job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden section. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and approach the golden ratio somewhat closer 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 the male. The proportions of the golden section are also manifested in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Rice. eleven. Golden proportions in the human figure
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, poetic sizes. Zeising defined the golden ratio, showed how it is expressed in line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction and the other. His next book was entitled "Golden division as the basic morphological law in nature and art." In 1876, a small book, almost a pamphlet, was published in Russia, outlining Zeising's work. The author took refuge under the initials Yu.F.V. Not a single painting is mentioned in this edition.
At the end of XIX - beginning of XX centuries. a lot of purely formalistic theories appeared about the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.
Fibonacci series
The name of the Italian mathematician monk Leonardo from 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, in which all the problems known at that time were collected. One of the tasks read "How many pairs of rabbits in one year from one pair will be born." Reflecting on this topic, Fibonacci built the following series of numbers: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 the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers of the series approaches the ratio of the golden division. So, 21:34 = 0.617, and 34:55 = 0.618. This relationship is symbolized F. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden ratio, its increase or decrease to infinity, when the smaller segment is related to the larger one, as the larger one is to everything.
Fibonacci also dealt with the practical needs of trade: what is the smallest number of weights that can be used to weigh a commodity? Fibonacci proves that the following system of weights is optimal: 1, 2, 4, 8, 16...
Generalized golden ratio
The Fibonacci series could have remained only a mathematical incident if it were 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 golden division law.Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. There are elegant methods for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden section. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.
One of the achievements in this area is the discovery of generalized Fibonacci numbers and generalized golden ratios.
The Fibonacci series (1, 1, 2, 3, 5, 8) and the "binary" series of weights 1, 2, 4, 8, 16 discovered by him... are completely different at first glance. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... Is it possible to find a common mathematical formula, from which both the "binary" series and the Fibonacci series are obtained? Or maybe this formula will give us new numerical sets with some new unique properties?
Indeed, let us set the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider a number series, S+ 1 whose first terms are units, and each of the subsequent ones is equal to the sum of the two terms of the previous one and the one that is separated from the previous one by S steps. If n we denote the th term of this series by φ S ( n), then we get general formulaφ S ( n) = φ S ( n- 1) + φ S ( n - S - 1).
It is obvious that at S= 0 from this formula we get a "binary" series, with S= 1 - Fibonacci series, with S\u003d 2, 3, 4. new series of numbers that are called S-Fibonacci numbers.
IN general view golden S-proportion is the positive root of the golden equation S-sections x S+1 - x S - 1 = 0.
It is easy to show that when S= 0, we get a division of the segment in half, and when S= 1 - the familiar classical golden ratio.
Relationships of neighbors S-Fibonacci numbers with absolute mathematical accuracy coincide in the limit with golden S-proportions! Mathematicians in such cases say that gold S-sections are numerical invariants S-Fibonacci numbers.
Facts confirming the existence of gold S-sections in nature, the Belarusian scientist E.M. Soroko in the book "Structural Harmony of Systems" (Minsk, "Science and Technology", 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermally stable, hard, wear-resistant, oxidation-resistant, etc.) only if the specific weights of the initial components are related to each other by one of gold S-proportions. This allowed the author to put forward a hypothesis that gold S-sections are numerical invariants of self-organizing systems. Being confirmed experimentally, this hypothesis can be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.
With golden codes S-proportions can express any real number as a sum of degrees of gold S-proportions with integer coefficients.
The fundamental difference between this method of encoding numbers is that the bases of the new codes, which are golden S-proportions, S> 0 turn out to be irrational numbers. Thus, the new number systems with irrational bases, as it were, put the historically established hierarchy of relations between rational and irrational numbers “upside down”. The fact is that at first the natural numbers were "discovered"; then their ratios are rational numbers. And only later - after the Pythagoreans discovered incommensurable segments - irrational numbers appeared. For example, in decimal, quinary, binary and other classical positional number systems, natural numbers - 10, 5, 2 - were chosen as a kind of fundamental principle, from which all other natural numbers, as well as rational and irrational numbers, were constructed according to certain rules.
A kind of alternative to the existing methods of numbering is a new, irrational system, as the fundamental principle, the beginning of which is chosen as an irrational number (which, we recall, is the root of the golden section equation); other real numbers are already expressed through it.
In such a number system, any natural number always representable in the form of a finite - and not infinite, as previously thought! - sums of degrees of any of the golden S-proportions. This is one of the reasons why "irrational" arithmetic, with its amazing mathematical simplicity and elegance, seems to have absorbed best qualities classical binary and "Fibonacci" arithmetic.
Principles of shaping in nature
Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds realization mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral.The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral.
Rice. 12. Spiral of Archimedes
The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.
Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."
Among the roadside herbs, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf.
Rice. 13. Chicory
The process makes a strong ejection into space, stops, releases a leaf, but is shorter than the first one, again makes an ejection into space, but of less force, releases an even smaller leaf and ejection again. If the first outlier is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section.
Rice. fourteen. viviparous lizard
In a lizard, at first glance, proportions that are pleasant to our eyes are captured - the length of its tail relates to the length of the rest of the body as 62 to 38.
Both in the plant and animal world, the form-building tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.
Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.
Rice. 15. bird egg
The great Goethe, a poet, naturalist and artist (he drew and painted in watercolor), dreamed of creating a unified doctrine of the form, formation and transformation of organic bodies. It was he who introduced the term morphology into scientific use.
Pierre Curie at the beginning of our century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.
The laws of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human organs and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.
Golden ratio and symmetry
The golden ratio cannot be considered in itself, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulff (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the golden division is an asymmetric symmetry. The science of symmetry includes such concepts as static And dynamic symmetry. Static symmetry characterizes rest, balance, and dynamic symmetry characterizes movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments, equal magnitudes. Dynamic symmetry is characterized by an increase in segments or their decrease, and it is expressed in the values of the golden section of an increasing or decreasing series.
Mystery golden section tried to comprehend Plato, Euclid, Pythagoras, Leonardo da Vinci, Kepler. Created long ago, the Golden Section still excites the minds of many scientists.
Since ancient times, people have sought to understand how our world is organized and arranged by nature.
Pythagoras He believed that the world is organized according to strict geometric laws and that the universe is based on a number. There are suggestions that he borrowed his knowledge of the golden division from the Egyptians and Babylonians. This is evidenced by the proportions of the pyramid of Cheops, temples, household items and decorations from the tomb of Tutankhamen.
One of the tasks of the ancients was to divide a segment into 2 equal parts so that the length of the larger segment was related to the length of the smaller one in the same way as the length of the entire segment was to the length of the larger one.
Or this proportion can be turned over and the ratio of the smaller to the larger can be found. As a result, it was calculated that the ratio of the larger to the smaller = 1.61803 ..., and the smaller to the larger = 0.61803 ...
IN Ancient Greece such a division was called a harmonic ratio. In 1509, an Italian mathematician, monk Luca Pacioli wrote a whole book On Divine Proportion».
2. Golden triangle and pentagram
« Gold" triangle is an isosceles triangle, the ratio of the side to the base is 1.618 ( Attachment 1).
golden ratio can also be seen in the pentagram - the Greeks called the star polygon.
The pentagon with drawn diagonals forming a five-pointed star was called a pentagram, which was considered a revered figure from ancient times.
It was an ancient magical sign of goodness, and the brotherhood of the five principles underlying the world-fire, earth, water, wood and metal. A pentagram is a regular pentagon, on each side of which are built isosceles triangles, equal in height.
The five-pointed star is very beautiful; it is not for nothing that many countries place it on their flags and coats of arms. The perfect shape of this figure pleases the eye.
The pentagon is literally woven from proportions, and above all the golden ratio ( application 2).
The Bulgarian magazine "Fatherland" (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives another ratio of 44: 56.
Such a proportion is found in architecture, and also takes place in the construction of compositions of images of an elongated horizontal format.
The figure shows the position of the line of the second golden section. It is located in the middle between the golden section line and the middle line of the rectangle.
Golden Triangle
To find segments of the golden ratio of the ascending and descending series, you can use pentagram.
To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Dürer (1471...1528). Let be O- the center of the circle A- a point on the circle and E- middle of the segment OA. Perpendicular to Radius OA, restored at the point ABOUT, intersects the circle at a point D. Using a compass, set aside a segment on the diameter CE = ED. The length of a side of a regular pentagon inscribed in a circle is DC. Putting segments on the circle DC and get five points to draw a regular pentagon. We connect the corners of the pentagon through one diagonal and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.
Each end of the pentagonal star is a golden triangle. Its sides form an angle of 36° at the top, and the base laid on the side divides it in proportion to the golden section.
We draw a straight line AB. from point BUT set aside on it three times a segment O of arbitrary size, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside segments ABOUT. Received points d And d1 connect with a straight line BUT. Section dd1 put on the line Ad1, getting a point FROM. She split the line Ad1 in proportion to the golden ratio. lines Ad1 And dd1 used to build a "golden" rectangle.
