Ideal section. Golden ratio in design

Any person who has at least indirectly encountered the geometry of spatial objects in interior design and architecture is probably well aware of the principle of the golden ratio. Until recently, several decades ago, the popularity of the golden ratio 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 viruses to humans, have basic body or limb proportions very close to the value of the golden ratio;
The dependence of 0.63 or 1.62 is typical only for biological creatures and some types of crystals; inanimate objects, from minerals to landscape elements, have the geometry of the golden ratio extremely rarely;
Golden proportions in body structure 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 of a fairly large number of 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 conditions of their existence.

The structure of the shell of Astreae Heliotropium, one of the marine mollusks, is usually given as an example. The shell is a coiled calcite shell with a geometry that practically coincides with the proportions of the golden ratio.

A more understandable and obvious example is an ordinary chicken egg.

The ratio of the main parameters, namely, the large and small focus, or the distances from equidistant points of the surface to the center of gravity, will also correspond to the golden ratio. At the same time, the shape of a bird's egg shell is the most optimal for the survival of the bird as a biological species. In this case, the strength of the shell does not play a major role.

For your information! The golden ratio, also called the universal proportion of geometry, was obtained as a result of a huge number of practical measurements and comparisons of the sizes of real plants, birds, and animals.

Origin of universal proportion

The ancient Greek mathematicians Euclid and Pythagoras knew about the golden ratio of the section. In one of the monuments of ancient architecture - the Cheops pyramid, the ratio of sides and base, individual elements and wall bas-reliefs are made in accordance with universal proportion.

The golden section technique was widely used in the Middle Ages by artists and architects, while the essence of universal proportion was considered one of the secrets of the universe and was carefully hidden from the common man. The composition of many paintings, sculptures and buildings was built strictly in accordance with the proportions of the golden ratio.

The essence of universal proportion was first documented in 1509 by the Franciscan monk Luca Pacioli, who had brilliant mathematical abilities. But 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, certain body dimensions 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 body and form. We are talking, first of all, about living beings; for dead nature, the golden ratio is much less common.

How to get the golden ratio

The golden ratio is most easily thought of as the ratio of two parts of the same object of different lengths separated by a point.

Simply put, how many lengths of a small segment will fit inside a large one, or the ratio of the largest part to the entire length of a linear object. In the first case, 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, 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 harmonic proportion:

For your information! Unlike the classic golden ratio, the architectural version implies an aspect ratio of 44:56.

If the standard version of the golden ratio for living beings, paintings, graphics, sculptures and ancient buildings was calculated as 37:63, then the golden ratio in architecture from the end of the 17th century began to be increasingly used as 44:56. Most experts consider the change in favor of more “square” proportions to be 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 12th-19th centuries came as a certain surprise. Moreover, the famous ancient Greek Parthenon was built in compliance with universal proportions; many houses and castles of wealthy nobles and wealthy people in the Middle Ages were deliberately built with parameters very close to the golden ratio.

Golden ratio in architecture

Many of the buildings that have survived to this day indicate that the architects of the Middle Ages knew about the existence of the golden ratio, and, of course, when building a house, they were guided by their primitive calculations and dependencies, with the help of which they tried to achieve maximum strength. The desire to build the most beautiful and harmonious houses was especially evident in the buildings of residences of reigning persons, churches, town halls and buildings of special social significance in society.

For example, the famous Notre Dame Cathedral in Paris has many sections and dimensional chains in its proportions that correspond to the golden ratio.

Even before the publication of his research in 1855 by Professor Zeising, at the end of the 18th 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 have been built in strict compliance with the golden ratio rule before. It is worth mentioning the ancient architectural monument 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, by the presence of the golden ratio in the proportions of the building. The amazing beauty of the building becomes even more mysterious if we take into account its age. The building of the Church of the Intercession 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 reconstruction.

Features of the golden ratio for humans

The ancient architecture of buildings and houses of the Middle Ages remains attractive and interesting for modern people for many reasons:

An individual artistic style in the design of facades allows us to avoid modern cliches and dullness; each building is a work of art;
Massive use for decorating and decorating statues, sculptures, stucco moldings, unusual combinations of building solutions from different eras;
The proportions and composition of the building draw the eye to the most important elements of the building.

Important! When designing a house and developing its appearance, medieval architects applied the rule of the golden ratio, unconsciously using the peculiarities of perception of the human subconscious.

Modern psychologists have experimentally proven that the golden ratio is a manifestation of a person’s unconscious desire or reaction to a harmonious combination or proportion in sizes, shapes and even colors. An experiment was conducted in which a group of people who did not know each other, did not have common interests, 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 proportion of sides. Based on the testing 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.

Therefore, 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 using the golden proportion 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 proper distribution of energy inside the house.

The modern interpretation of the rule of universal harmony has long spread beyond the usual geometry and shape of an object. Today, the rule is subject to 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.

The easiest way to build a harmonious house is on a modular basis. In this case, most departments and rooms are made in the form of independent blocks or modules, designed in compliance with the rule of the golden ratio. Constructing a building in the form of a set of harmonious modules is much easier than building one box, in which most of the facade and interior must be within the strict framework of the golden ratio proportions.

Many construction companies designing private households use the principles and concepts of the golden ratio to increase the cost estimate and give clients the impression that the house’s design has been thoroughly worked out. As a rule, such a house is declared to be very comfortable and harmonious to use. A correctly selected ratio of room areas 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 redesign the rooms so that the proportions of the room correspond to the ratio of the walls in the proportion 1:1.61. To do this, furniture can be moved or additional partitions installed inside rooms. In the same way, 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, 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% should be given to the shading color, and the rest is allocated to various related tones, designed to enhance the perception of the color scheme.

The interior walls of the room are divided by a horizontal belt or border at a height of 70 cm; 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 partition, and the total area occupied by the furniture relates to the area of the room as 1:1.61.

The golden proportion is difficult to apply in practice on a large scale due to just one cross-sectional 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 ratio, 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 that is not only harmonious, but also truly individual.

Conclusion

In practice, according to reviews from those who decided to build a house according to the golden ratio rule, a well-built building actually turns out to be quite comfortable for living. But the cost of the building due to individual design and the use of building materials of non-standard sizes increases by 60-70%. And there is nothing new in this approach, since most buildings of the last century were built specifically for the individual characteristics of their future owners.

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 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.

Golden ratio - harmonic proportion

In mathematics proportion(lat. proportio) call the equality of two relations: a : b = c : d.

Straight 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 respect (such parts do not form proportions);

thus, when AB : AC = AC : Sun.

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 With : 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.

Rice. 2. Dividing a straight line segment using the golden ratio. B.C. = 1/2 AB; CD = B.C.

From point IN a perpendicular equal to half is restored AB. Received point WITH connected by a line to a point A. A segment is plotted on the resulting line Sun ending with a dot D. Line segment AD transferred to direct AB. The resulting point E divides a segment AB in the golden ratio ratio.

Segments of the golden ratio are expressed as an infinite irrational fraction A.E.= 0.618..., if AB take as one BE= 0.382... For practical purposes, approximate values of 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then the larger part of the segment is equal to 62, and the smaller part is 38 parts.

The properties of the golden ratio are described by the equation:

x 2 - x - 1 = 0.

Solution to this equation:

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

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 another ratio of 44: 56.

This proportion is found in architecture, and also occurs when constructing compositions of images of an elongated horizontal format.

Rice. 3. Construction of the second golden ratio

The division is carried out as follows (see Fig. 3). Line segment AB divided according to the golden ratio. From point WITH the perpendicular is restored CD. Radius AB there is a point D, which is connected by a line to a point A. Right angle ACD is divided in half. From point WITH a line is drawn until it intersects with the line AD. Dot E divides a segment AD in relation to 56:44.

Rice. 4. Dividing a rectangle with the line of the second golden ratio

In Fig. Figure 4 shows the position of the line of the second golden ratio. It is located midway between the golden ratio line and the middle line of the rectangle.

Golden Triangle

To find segments of the golden proportion of the ascending and descending series, you can use pentagram.

Rice. 5. Construction of a regular pentagon and 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 Durer (1471...1528). Let O- center of the circle, A- a point on a circle and E- the middle of the segment OA. Perpendicular to radius OA, restored at the point ABOUT, intersects the circle at the point D. Using a compass, plot a segment on the diameter C.E. = ED. The side length of a regular pentagon inscribed in a circle is DC. Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals 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 represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

Rice. 6. Construction of the golden triangle

We carry out a direct AB. From point A 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 the segments ABOUT. Received points d And d 1 connect with straight lines to a point A. Line segment dd put 1 on the line Ad 1, getting a point WITH. She split the line Ad 1 in proportion to the golden ratio. Lines Ad 1 and dd 1 is used to construct a “golden” rectangle.

History of the golden ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (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 Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I 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 a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for the construction of 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 Pythagorean school and, in particular, to the issues of the golden division.

The façade of the ancient Greek temple of the Parthenon features golden proportions. During its excavations, compasses were discovered that 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 compass

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. In the 2nd book of the “Principles” the geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd century BC), Pappus (III century AD), and others. In medieval Europe, with the golden division We met through Arabic translations of Euclid’s Elements. The translator J. Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates.

During the Renaissance, interest in the golden division increased among scientists and artists due to its use in both geometry and art, especially in architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of 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 of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry.

Luca Pacioli perfectly understood 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 in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “The Divine Proportion” was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity - God the son, God the father and God the holy spirit (it was implied that the small segment is the personification of God the son, the larger segment - God the father, and the entire segment - God of the Holy Spirit).

Leonardo da Vinci also paid a lot of 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 the golden division. That's why he gave this division the name golden ratio. So it still remains as the most popular.

At the same time, in the north of Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches the introduction to the first version of the treatise on proportions. Dürer writes. “It is necessary that someone who knows how to do 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 while in Italy. Albrecht Durer develops in detail the theory of proportions of the human body. Dürer assigned an important place in his system of relationships to the golden section. A person's height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is well known.

Great astronomer of the 16th century. Johannes Kepler called the golden ratio one of the treasures of geometry. He was the first to draw attention to the importance of the golden proportion for botany (plant growth and their structure).

Kepler called the golden proportion self-continuing. “It is structured in such a way,” he wrote, “that the two lowest terms of this never-ending proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion is maintained until infinity."

The construction of a series of segments of the golden proportion 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, set aside the segment m, put the segment next to it M. Based on these two segments, we build a scale of segments of the golden proportion of the ascending and descending series

Rice. 9. Construction of a scale of golden proportion segments

In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Studies”. What happened to Zeising was exactly what should inevitably happen to a researcher who considers a 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 tremendous 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 ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man. The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

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 the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied. Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the numbers expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction or the other. His next book was titled “The Golden Division as the Basic Morphological Law in Nature and Art.” In 1876, a small book, almost a brochure, was published in Russia outlining this work of Zeising. The author took refuge under the initials Yu.F.V. This edition does not mention a single work of painting.

At the end of the 19th - beginning of the 20th centuries. Many purely formalistic theories appeared about the use of the golden ratio 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 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:

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 relationship 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 or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to the whole.

Fibonacci also dealt with the practical needs of trade: what is the smallest number of weights that can be used to weigh a product? Fibonacci proves that the optimal system of weights is: 1, 2, 4, 8, 16...

Generalized golden ratio

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.

Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio. 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 field 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 discovered by him 1, 2, 4, 8, 16... at first glance are completely different. 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 = 2 + 2..., in the second - this is the sum of the two previous numbers 2 = 1 + 1, 3 = 2 + 1, 5 = 3 + 2.... Is it possible to find a general mathematical formula from which we obtain “ binary series and Fibonacci series? Or maybe this formula will give us new numerical sets that have 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 of the first terms of which are units, and each of the subsequent ones is equal to the sum of two terms of the previous one and separated from the previous one by S steps. If n We denote the th term of this series by φ S ( n), then we obtain the general formula φ S ( n) = φ S ( n- 1) + φ S ( n - S - 1).

It is obvious that when S= 0 from this formula we get a “binary” series, with S= 1 - Fibonacci series, with S= 2, 3, 4. new series of numbers, which are called S-Fibonacci numbers.

Overall 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, the segment is divided in half, and when S= 1 - the familiar classical golden ratio.

Relations between neighbors S- Fibonacci numbers coincide with absolute mathematical accuracy in the limit with gold 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, cites 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 (thermal stable, hard, wear-resistant, resistant to oxidation, etc.) only if the specific gravities of the original components are related to each other by one of gold S-proportions. This allowed the author to put forward the hypothesis that gold S-sections are numerical invariants of self-organizing systems. Once confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

Using golden codes S-proportions can be expressed by any real number as a sum of powers 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, with S> 0 turn out to be irrational numbers. Thus, new number systems with irrational bases seem to put the historically established hierarchy of relations between rational and irrational numbers “from head to foot.” The fact is that natural numbers were first “discovered”; then their ratios are rational numbers. And only later - after the discovery of incommensurable segments by the Pythagoreans - irrational numbers were born. For example, in decimal, quinary, binary and other classical positional number systems, natural numbers were chosen as a kind of fundamental principle - 10, 5, 2 - from which, according to certain rules, all other natural numbers, as well as rational and irrational numbers, were constructed.

A kind of alternative to the existing methods of notation is a new, irrational system, as a fundamental principle, the beginning of which is an irrational number (which, recall, is the root of the golden ratio equation); other real numbers are already expressed through it.

In such a number system, any natural number can always be represented as finite - and not infinite, as previously thought! - the sum of the degrees of any of the gold S-proportions. This is one of the reasons why “irrational” arithmetic, having amazing mathematical simplicity and elegance, seems to have absorbed the best qualities of classical binary and “Fibonacci” arithmetic.

Principles of formation in nature

Everything that took on some form was formed, grew, strived to take a place in space and preserve itself. This desire is realized mainly in two options - growing upward 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 shorter than the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The idea of the golden ratio will be incomplete without talking about the spiral.

Rice. 12. Archimedes spiral

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and came up with an equation for the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

Goethe also emphasized the tendency of nature towards spirality. The helical and spiral arrangement of leaves on tree branches was noticed a long time ago. The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that the Fibonacci series manifests itself in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, and pine cones, and therefore, the law of the golden ratio manifests itself. The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral the “curve of life.”

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there.

Rice. 13. Chicory

The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

Rice. 14. Viviparous lizard

At first glance, the lizard has proportions that are pleasant to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38.

In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding 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 division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Rice. 15. bird egg

The great Goethe, a poet, naturalist and artist (he drew and painted in watercolors), 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 this century formulated a number of profound ideas about 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 cosmic systems, in the gene structures of living organisms. These patterns, as indicated above, 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.

Golden ratio and symmetry

The golden ratio cannot be considered on its own, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulf (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 ideas, the golden division is asymmetrical symmetry. The science of symmetry includes such concepts as static And dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, 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 and equal values. 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.

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.

Definition

The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time. The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “asymmetrical symmetry”, calling it in a broad sense a universal rule reflecting the structure and order of our world order.

Story

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, 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 Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusien 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 a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded.

Plato(427...347 BC) also knew about the golden division. His dialogue “Timaeus” is dedicated to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division.

Rice. Antique golden ratio compass

In the ancient literature that has come down to us, the golden division was first mentioned in the “Elements” Euclid. In the 2nd book of the Elements, a geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd century BC), Pappus (3rd century AD), and others. In medieval Europe, they became acquainted with the golden division through Arabic translations of Euclid’s Elements. The translator J. Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates.

The concept of golden proportions was also known in Rus', but for the first time the golden ratio was scientifically explained monk Luca Pacioli in the book “The Divine Proportion” (1509), the illustrations of which were supposedly made by Leonardo da Vinci. Pacioli saw in the golden section the divine trinity: the small segment personified the Son, the large segment the Father, and the whole the Holy Spirit. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry.

Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of Duke Moreau, he came to Milan, where he gave lectures on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time.

The name of the Italian mathematician is directly associated with the golden ratio rule Leonardo Fibonacci. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations.

Leonardo da Vinci He also devoted a lot of time to studying the features of the golden ratio; most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in the golden division.

Over time, the rule of the golden ratio turned into an academic routine, and only the philosopher Adolf Zeising in 1855 he gave it a second life. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his “mathematical aesthetics” caused a lot of criticism.

Nature

16th century astronomer Johannes Kepler called the golden ratio one of the treasures of geometry. He was the first to draw attention to the importance of the golden proportion for botany (plant growth and their structure).

The construction of a series of segments of the golden proportion can be done both in the direction of increase (increasing series) and in the direction of decrease (descending series).

Rice. Construction of a scale of golden proportion segments

Rice. Chicory

Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.

Rice. Viviparous lizard

Rice. bird egg

The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting forms is spiral twisting.

More Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted the attraction of nature to spiral forms, calling spiral of "life curve". Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.

Human

Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.

In Leonardo da Vinci's diary there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo’s “Vitruvian Man,” created his own scale of “harmonic proportions,” which influenced the aesthetics of 20th-century architecture. Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but the main indicator of the golden ratio is the division of the body by the navel point.

As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

The art of spatial forms

The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real mathematics.” For a long time, artists have followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Durer To determine the points of the golden section, he used the proportional compass he invented.

Art critic F.V. Kovalev, having examined in detail Nikolai Ge’s painting “Alexander Sergeevich Pushkin in the village of Mikhailovskoye,” notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair, or the poet himself, is strictly inscribed in golden proportions. Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.

And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling in the viewer.

Goethe, a poet, naturalist and artist (he drew and painted in watercolors), dreamed of creating a unified doctrine of the form, formation and transformation of organic bodies. It was he who introduced the term into scientific use morphology.

Golden ratio and symmetry

The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the golden division is an asymmetrical symmetry. The science of symmetry includes such concepts as static And dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, 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 and equal values. 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.

Word, sound and film

The forms of temporary art in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin’s work corresponds to the Fibonacci series - 5, 8, 13, 21, 34.

The rule of the golden section also applies in individual works of the Russian classic. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.

Soviet musicologist E.K. Rosenov notes the amazing accuracy of the ratios of the golden section in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point.

Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

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The Golden Ratio is a simple principle that can help make a design visually pleasing. In this article we will explain in detail how and why to use it.

A common mathematical proportion in nature, called the Golden Ratio, or Golden Mean, is based on the Fibonacci Sequence (which you most likely heard about in school, or read about in Dan Brown's book "The Da Vinci Code"), and implies an aspect ratio of 1 :1.61.

This ratio is often found in our lives (shells, pineapples, flowers, etc.) and therefore is perceived by a person as something natural and pleasing to the eye.

→ The golden ratio is the relationship between two numbers in the Fibonacci sequence
→ Plotting this sequence to scale produces the spirals that can be seen in nature.

It is believed that the Golden Ratio has been used by mankind in art and design for more than 4 thousand years, and perhaps even more, according to scientists who claim that the ancient Egyptians used this principle when building the pyramids.

Famous examples

As we have already said, the Golden Ratio can be seen throughout the history of art and architecture. Here are some examples that only confirm the validity of using this principle:

Architecture: Parthenon

In ancient Greek architecture, the Golden Ratio was used to calculate the ideal proportion between the height and width of a building, the dimensions of a portico, and even the distance between columns. Subsequently, this principle was inherited by the architecture of neoclassicism.

Art: last supper

For artists, composition is the foundation. Leonardo da Vinci, like many other artists, was guided by the principle of the Golden Ratio: in the Last Supper, for example, the figures of the disciples are located in the lower two-thirds (the larger of the two parts of the Golden Ratio), and Jesus is placed exactly in the center between two rectangles.

Web design: Twitter redesign in 2010

Twitter creative director Doug Bowman posted a screenshot on his Flickr account explaining the use of the Golden Ratio principle for the 2010 redesign. “Anyone interested in #NewTwitter proportions, know that everything was done for a reason,” he said.

Apple iCloud

The iCloud service icon is also not a random sketch. As Takamasa Matsumoto explained in his blog (original Japanese version), everything is built on the mathematics of the Golden Ratio, the anatomy of which can be seen in the picture on the right.

How to construct the Golden Ratio?

The construction is quite simple, and starts with the main square:

Draw a square. This will form the length of the “short side” of the rectangle.

Divide the square in half with a vertical line so that you get two rectangles.

In one rectangle, draw a line by joining opposite corners.

Expand this line horizontally as shown in the figure.

Create another rectangle using the horizontal line you drew in the previous steps as a guide. Ready!

"Golden" instruments

If drawing and measuring is not your favorite activity, leave all the “grunt work” to tools that are designed specifically for this. With the help of the 4 editors below you can easily find the Golden Ratio!

The GoldenRATIO application helps you develop websites, interfaces and layouts in accordance with the Golden Ratio. It's available on the Mac App Store for $2.99, and has a built-in calculator with visual feedback, and a handy Favorites feature that stores settings for recurring tasks. Compatible with Adobe Photoshop.

This calculator will help you create the perfect typography for your website according to the principles of the Golden Ratio. Just enter the font size, content width in the field on the site, and click “Set my type”!

This is a simple and free application for Mac and PC. Just enter a number and it will calculate the proportion for it according to the Golden Ratio rule.

A convenient program that will relieve you of the need for calculations and drawing grids. It makes finding ideal proportions easier than ever! Works with all graphic editors, including Photoshop. Despite the fact that the tool is paid - $49, it is possible to test the trial version for 30 days.

20.05.2017

The golden ratio is something every designer should know about. We'll explain what it is and how you can use it.

There is a general mathematical relationship found in nature that can be used in design to create pleasing, natural-looking compositions. It is called the Golden Ratio or the Greek letter “phi”. If you are an illustrator, art director or graphic designer, you should definitely use the Golden Ratio in every project.

In this article, we'll explain how to use it and also share some great tools for further inspiration and learning.

Closely related to the Fibonacci Sequence, which you may remember from math class or Dan Brown's Da Vinci Code, the Golden Ratio describes a perfectly symmetrical relationship between two proportions.

Approximately equal to the ratio 1:1.61, the Golden Ratio can be illustrated as the Golden Rectangle: a large rectangle containing a square (in which the sides are equal to the length of the shortest side of the rectangle) and a smaller rectangle.

If you remove the square from the rectangle, you will be left with another, small Golden Rectangle. This process can continue indefinitely, just like Fibonacci numbers, which work in reverse. (Adding a square with sides equal to the length of the longest side of the rectangle gets you closer to the Golden Rectangle and the Golden Ratio.)

Golden Ratio in action

The Golden Ratio is believed to have been used for around 4,000 years in art and design. However, many people agree that the construction of the Egyptian Pyramids also used this principle.

In more modern times, this rule can be seen in the music, art and design around us. By using a similar working methodology, you can bring the same design features into your work. Let's take a look at some inspiring examples.

Greek architecture

In ancient Greek architecture, the Golden Ratio was used to determine the pleasing spatial relationship between the width of a building and its height, the size of the portico, and even the position of the columns supporting the structure.

The result is a perfectly proportional structure. The Neoclassical architecture movement also used these principles.

last supper

Leonardo Da Vinci, like many other artists of yesteryear, often used the Golden Ratio to create pleasing compositions.

In the Last Supper, the figures are located in the lower two-thirds (the larger of the two parts of the Golden Ratio), and Jesus is perfectly sketched between the golden rectangles.

Golden ratio in nature

There are many examples of the Golden Ratio in nature - you can find them around you. Flowers, seashells, pineapples and even honeycombs show the same ratio.

How to calculate the Golden Ratio

The calculation of the Golden Ratio is quite simple, and starts with a simple square:

01. Draw a square

It forms the length of the short side of the rectangle.

02. Divide the square

Divide the square in half using a vertical line, creating two rectangles.

03. Draw a diagonal

In one of the rectangles, draw a line from one corner to the opposite.

04. Turn

Rotate this line so that it lies horizontal to the first rectangle.

05. Create a new rectangle

Create a rectangle using a new horizontal line and the first rectangle.

How to use the Golden Ratio

Using this principle is easier than you think. There are a couple of quick tricks you can use in your layouts, or take a little more time and fully flesh out the concept.

Fast way

If you've ever encountered the Rule of Thirds, you'll be familiar with the idea of dividing a space into equal thirds vertically and horizontally, with where lines intersect to create natural points for objects.

The photographer places the key subject on one of these intersecting lines to create a pleasing composition. This principle can also be used in your page layout and poster designs.

The rule of thirds can be applied to any shape, but if you apply it to a rectangle with approximately 1:1.6 proportions, you will end up very close to the golden rectangle, which will make the composition more pleasing to the eye.

Full implementation

If you want to fully implement the Golden Ratio in your design, then simply arrange the main content and sidebar (in web design) in a ratio of 1: 1.61.

You can round the values down or up: if the content area is 640px and the sidebar is 400px, then this markup is quite suitable for the Golden Ratio.

Of course, you can also divide the content and sidebar areas into the same relationship, and the relationship between the web page's header, content area, footer, and navigation can also be designed using the same principle.

Useful tools

Here are some tools to help you use the Golden Ratio in design and create proportional designs.

GoldenRATIO is an application for creating website designs, interfaces and templates suitable for the Golden Ratio. Available on the Mac App Store for $2.99. Includes a visual Golden Ratio calculator.

The application also has a “Favorites” function, which saves settings for recurring tasks and a “Click-thru” mod that allows you to minimize the application in Photoshop.

This Golden Ratio calculator from Pearsonified helps you create the perfect typography for your website. Enter the font size, container width in the field, and click the button Set my type! If you need to optimize the number of letters per line, you can additionally enter a CPL value.

This simple, useful, and free app is available for Mac and PC. Enter any number and the application will calculate the second digit according to the Golden Ratio principle.

This application allows you to design with golden proportions, saving a lot of time on calculations.

You can change shapes and sizes to focus on your project. A permanent license costs $49, but you can download a free version for a month.

Golden Section Training

Here are some useful tutorials on the Golden Ratio (English):

In this Digital Arts tutorial, Roberto Marras shows how to use the Golden Ratio in your artistic work.

Tutorial from Tuts+ showing how to use the golden principles in web design projects.

A tutorial from Smashing Magazine about proportions and the rule of thirds.