Golden ratio. A New Look

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 relates to the larger, as the larger part relates 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 in the forms of space and time operates.

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 that reflects the structure and order of our world order.

Story.
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw the divine trinity in the golden section: the small segment personified the son, the large segment the father, and the whole the holy spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. 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 ratio : “It is arranged in such a way that the two Younger Members of This Infinite Proportion in the Sum give the Third Member, and any two Last Members, If Added, Give the Next Member, Moreover, the same Proportion is Preserved to Infinity.” Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden ratio in all its manifestations

Fibonacci numbers are a harmonic division, a measure of beauty. The golden ratio in nature, man, art, architecture, sculpture, design, mathematics, music https://psihologiyaotnoshenij.com/stati/zolotoe-sechenie-kak-eto-rabotaet

Leonardo da Vinci 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 golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it a second life in 1855. 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.
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.

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.
Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Later, Goethe noted the attraction of nature to spiral forms, calling the spiral the “Curve of Life.” 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, exploring 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 a Composition there is an Immutable Law, When in a Picture You Can’t Remove or Add Anything, You Can’t Even Put an Extra Point, This is Real Mathematics.” For a long time artists followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting can no longer be accomplished without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

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: on their list are the great pyramids of Giza, the cathedral Notre Dame of Paris, St. Basil's Cathedral, 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.

Word, sound and film.
Forms are temporary? The Go arts, 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 late period Pushkin's creativity 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 relationships 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 golden ratio rule, 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.

Golden ratio examples. How to get the golden ratio

So, the golden ratio is golden ratio, which is also a harmonic division. To explain this more clearly, let's look at some features of the form. Namely: a form is something whole, and the whole, in turn, always consists of some parts. These parts most likely have different characteristics, at least different sizes. Well, such dimensions are always in a certain relationship, both among themselves and in relation to the whole.

This means, in other words, we can say that the golden ratio is a ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.

There is a lot to be said for a spiral tattoo more meaning than it seems at first glance. Such a simple pattern is built according to the so-called golden ratio principle, which is found everywhere in nature. Moreover, this principle has been known since ancient times, which is confirmed by its presence at the base of the Egyptian pyramids.

Symbolism of spiral tattoos

In Ta-moko tattoos or in the same Celtic patterns, spirals are found very often, and this is not surprising. The absence of right angles in this figure symbolizes the connection with nature, which does not like right angles and always tries to smooth them out. A spiral tattoo means unity with nature; as a rule, calm, reasonable people make such a tattoo.

But this is only a general meaning; often people try to find out about the meaning of a spiral tattoo, actually confusing it with other tattoos. The spiral shell tattoo often misleads people; it has become quite popular lately. One has a completely different meaning, it suits closed people, loners, who have usually suffered some kind of shock and do not want to share about it, but in his honor they make such a tattoo.

A wave tattoo, which symbolizes love of the sea, or a black sun tattoo, the meaning of which we wrote in detail, is very similar to a spiral.

Often a spiral tattoo is made as a talisman, since it is a symbol of the cyclical nature of life; it conveys the energy of the world and existence. The spiral image can be applied to the shoulders, forearms, chest and back. The tattoo is more suitable for women, since another meaning of the tattoo is the feminine principle.

It is believed that Pythagoras was the first to introduce the concept of the golden ratio. The works of Euclid have survived to this day (he used the golden ratio to build regular pentagons, which is why such a pentagon is called “golden”), and the number of the golden ratio is named after the ancient Greek architect Phidias. That is, this is our number “phi” (denoted Greek letterφ), and it is equal to 1.6180339887498948482... Naturally, this value is rounded: φ = 1.618 or φ = 1.62, and in percentage terms the golden ratio looks like 62% and 38%.

What is unique about this proportion (and believe me, it exists)? Let's first try to figure it out using an example of a segment. So, we take a segment and divide it into unequal parts in such a way that its smaller part relates to the larger one, as the larger part relates to the whole. I understand, it’s not very clear yet what’s what, I’ll try to illustrate it more clearly using the example of segments:

So, we take a segment and divide it into two others, so that the smaller segment a relates to the larger segment b, just as the segment b relates to the whole, that is, the entire line (a + b). Mathematically it looks like this:

This rule works indefinitely; you can divide segments as long as you like. And, see how simple it is. The main thing is to understand once and that’s it.

But now let's take a closer look complex example, which comes across very often, since the golden ratio is also represented in the form of a golden rectangle (the aspect ratio of which is φ = 1.62). This is a very interesting rectangle: if we “cut off” a square from it, we will again get a golden rectangle. And so on endlessly. See:

But mathematics would not be mathematics if it did not have formulas. So, friends, now it will “hurt” a little. I hid the solution to the golden ratio under a spoiler; there are a lot of formulas, but I don’t want to leave the article without them.

The principle of the golden ratio. Successful creation or the rule of the golden ratio

Capturing the moment - this is precisely the moment of creation of an artist or photographer. In addition to inspiration, the master must follow strictly defined rules, which include: contrast, placement, balance, the rule of thirds and many others. But the rule of the golden section, also known as the rule of thirds, is still recognized as a priority.

Just something complicated

If we present the basis of the golden ratio rule in a simplified form, then in fact it is the division of the reproduced moment into nine equal parts (three vertically by three horizontally). For the first time, Leonardo da Vinci specifically introduced it, arranging all his compositions in this peculiar grid. It was he who practically confirmed that the key elements of the image should be concentrated at the points of intersection of vertical and horizontal lines.

The rule of the golden ratio in photography is subject to certain correction. In addition to the nine-segment grid, it is recommended to use so-called triangles. The principle of their construction is based on the rule of thirds. To do this, a diagonal is drawn from the extreme upper point to the lower one, and from the opposite upper point a ray is drawn, dividing the already existing diagonal at one of the internal intersection points of the grid. The key element of the composition should be displayed in the average size of the resulting triangles. It’s worth making a remark here: the given diagram for constructing triangles reflects only their principle, and, therefore, it makes sense to experiment with the given instructions.

How to use grid and triangles?

The golden ratio rule in photography operates according to certain standards depending on what is depicted in it.

Horizon factor. According to the rule of thirds, it should be placed along horizontal lines. Moreover, if the captured object is above the horizon, then the factor passes through the bottom line, and vice versa.

Location of the main object. The classic arrangement is one in which the central element is located at one of the intersection points. If the photographer selects two objects, then they should be diagonal or at parallel points.

Using triangles. The golden section rule in the case under consideration deviates from the canons, but only slightly. The object does not have to be located at the intersection point, but is as close as possible to it in the middle triangle.

Direction. This principle of shooting is used in dynamic photography and consists in the fact that two-thirds of the image space should remain in front of the moving object. This will provide the effect of moving forward and indicating the target. Otherwise, the photo may remain misunderstood.

Correction of the golden ratio rule

Despite the fact that the rule of thirds is considered classic in the existing theory of composition, more and more photographers are inclined to abandon it. Their motivation is simple: analysis of paintings by famous artists shows that the rule of the golden ratio does not hold true. One can argue with this statement.

Let's consider the well-known Mona Lisa, which opponents of using the rule of thirds cite as an example (forgetting that da Vinci himself was at the origins of its practical use). Their argument is that the master did not consider it necessary to arrange the key elements of the picture at the points of intersection, as required by the classical image. But they overlook the factor of horizontal lines, according to which the head and torso of the person depicted are positioned in such a way that the silhouette as a whole does not “hurt the eye.” Moreover, in this work to a greater extent a spiral is used, which is mostly forgotten by photography theorists. And so it is possible to refute statements regarding almost every creation cited as an example.

The golden ratio rule can be used or abandoned if you want to emphasize the disharmony of the composition. However, it is impossible to say that it is not key in the formation of an art object.

Golden ratio in architecture. How to get the golden ratio

The golden ratio is most easily represented as the ratio of two parts of one object 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 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:

The classic division of one of the sides of a right triangle and the construction of perpendiculars and secant arcs. To do this, from one end of the segment it is necessary to restore a perpendicular with a height of ½ its length and construct right triangle, as in the diagram.
If we plot the height of the perpendicular on the hypotenuse, then with a radius equal to the remaining segment, the base is cut into two segments with lengths proportional to the golden ratio;
Using the method of constructing the pentagram of Dürer, the brilliant German graphic artist and geometer. Today we know Dürer's method of the golden section as a method of constructing a star or pentagram inscribed in a circle in which there are at least four segments of harmonious proportion;
In architecture and construction, the golden ratio is often used in an improved form. In this case, the division of a right triangle is used not along the leg, but along the hypotenuse, as a diagram.

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.

Many people dream of an ideal appearance, but not everyone has a clear idea of what proportions can be considered harmonious. The formula for the golden ratio of the face is inextricably linked with the number 1.618 and other ratios. Thus, the proportions of beauty can be described as follows:

the ratio of the height and width of the face should be 1.618;
if you divide the length of the mouth and the width of the wings of the nose, you get 1.618;
when dividing the distances between the pupils and eyebrows, again, the result is 1.618;
the length of the eyes should match the distance between them, as well as the width of the nose;
the areas of the face from the hairline to the eyebrows, from the bridge of the nose to the tip of the nose, and the lower part to the chin should be equal;
if you draw from the pupils vertical lines to the corners of the lips, you get three sections equal in width.

You need to understand that in nature the coincidence of all parameters is quite rare. But there's nothing wrong with that. This does not mean at all that persons who do not comply perfect proportions, can be called ugly or unpretty. On the contrary, it is “defects” that sometimes give a face an unforgettable charm.

The golden ratio in the composition of drawings in paint.net
Mathematically, the “Golden Ratio” can be described as follows: the ratio of the whole to its larger part must be equal to the ratio of the larger part to the smaller. Let us illustrate with the example of a segment.

In our case, the entire segment B is divided into two parts - larger A and smaller B. Then, if B / A is equal to A / B, the division of the segment will be carried out according to the principle called the “Golden Section”.
Not exactly accurate, but close to the Golden Ratio, for example a ratio of 2/3 or 5/8. Numbers in such ratios are often called “golden”.
Why do we need this information for drawing in paint.net? The Golden Ratio is important for composition. It is believed that objects containing the “golden ratio” are perceived by people as the most harmonious. It was in similar ratios that famous artists chose the sizes of hosts for their paintings.
Let's consider a simplified version of constructing the “Golden Ratio” for the composition of a drawing, or the “Rule of Thirds”. The rule of thirds is that we mentally divide the frame into three parts horizontally and vertically, and at the intersection points of imaginary lines, we place the key and important details of our drawing or photo collage.

The principle of the "golden ratio" can be applied when cropping an image. So, for example, a frame formed according to the “golden ratio” rule from a large photograph may look like this.

Golden ratio in music. Golden section method in musical works

The “golden ratio” is a rather mathematical concept and its study is a task of science. This is the division of a certain quantity into two parts in such a ratio that the larger part will be related to the smaller one as the whole is to the larger one. This attitude turns out to be equal to the transcendental number Ф=1.6180339... with amazing properties.

The golden section method is a search for function values on a given interval. This method is based on the principle of dividing a segment in the so-called golden ratio. Most widespread it was obtained for searching for extreme values when solving optimization problems. In addition to mathematics, the golden section method is used in a variety of fields, from architecture, art to astronomy. For example, the famous Soviet director Sergei Eisenstein used it in his film “Battleship Potemkin,” and Leonardo da Vinci used it when he wrote the famous “La Gioconda.”

The golden ratio method is also used in music. It turned out that this golden proportion occurs very often in musical works. At the beginning of the 20th century, at a meeting of the Moscow Music Circle, a message was made containing information about the application of the golden ratio in music. The message was listened to with great interest by members of the musical circle, composers S. Rachmaninov, S. Taneyev, R. Gliere and others. Report by musicologist E.K. Rosenov “The Law of the Golden Ratio in Music and Poetry” marked the beginning of research into mathematical patterns associated with the golden ratio in music. He analyzed the musical works of Mozart, Bach, Beethoven, Wagner, Chopin, Glinka and other composers and showed that this “divine proportion” was present in their works.

The climax of many musical works is not located in the center, but is slightly shifted towards the end of the work in a ratio of 62:38 - this is the point of the golden proportion. Doctor of Art History, Professor L. Mazel noticed, while studying the eight-bar melodies of Chopin, Beethoven, Scriabin, that in many works of these composers the climax, as a rule, falls on the weak beat of the fifth, that is, at the point of the golden ratio - 5/8. L. Mazel believed that almost every composer who adheres to the harmonic style can find a similar musical structure: five bars of ascent and three bars of descent. This suggests that the golden section method was actively used by composers, either consciously or unconsciously. Probably, this structural arrangement of climactic moments gives a musical work a harmonic sound and emotional coloring.

A serious study of musical works for the manifestation of the golden proportion in them was undertaken by composer and musicologist L. Sabaneev. He studied about two thousand works of different composers and came to the conclusion that in approximately 75% of cases the golden ratio was present in a musical work at least once. The most a large number of works in which the golden proportion occurs, he noted in such composers as Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%) , Schubert (91%). He studied Chopin's etudes most closely and came to the conclusion that the golden ratio was determined in 24 out of 27 etudes. Only in three of Chopin's etudes was the golden ratio not found. Sometimes the structure of a musical work included both symmetry and the golden ratio. For example, many of Beethoven's works are divided into symmetrical parts, and in each of them the golden ratio appears.

So, we can say that the presence of the golden ratio in a piece of music is one of the criteria for the harmony of a musical composition.

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 the 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

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 with the help of geometric shapes. The Pythagorean square and the diagonal of this square were the basis for the construction of dynamic rectangles.

Rice. 7. Dynamic rectangles

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” a geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (II century BC), Pappus (III century AD) and others. medieval Europe we got acquainted with the golden division through Arabic translations Euclid's "Beginnings". 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 remains 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. Dividing the body by the navel point - the most important indicator 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 and architectural structures were examined different eras, plants, animals, bird eggs, musical tones, poetic meters. 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 publication does not mention a single work of painting.

IN late XIX- early 20th century 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, equal to the sum two previous ones 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 number sets, possessing some new unique properties?

Indeed, let's ask 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 get general formulaφ 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! In such cases, mathematicians 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 gold 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 Pythagoreans discovered incommensurable segments - 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 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! - sum of 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. Collaboration Botanists and mathematicians 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. Frightened herd reindeer spirals away. 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 pleasing 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 our century formulated a series deep ideas 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.

They say that “divine proportion” is inherent in nature, and in many things around us. You can find it in flowers, beehives, sea shells, and even our bodies.

This divine ratio, also known as the golden ratio, divine ratio, or golden ratio can be applied to various forms of art and learning. Scientists say that the closer an object is to the golden ratio, the better the human brain perceives it.

Since this ratio was discovered, many artists and architects have used it in their works. You can find the golden ratio in several Renaissance masterpieces, architecture, painting, and more. The result is a beautiful and aesthetically pleasing masterpiece.

Few people know what the secret of the golden ratio is, which is so pleasing to our eyes. Many believe that the fact that it appears everywhere and is a “universal” proportion forces us to accept it as something logical, harmonious and organic. In other words, it simply “feels” what we need.

So what is the golden ratio?

The golden ratio, also known as “phi” in Greek, is a mathematical constant. It can be expressed by the equation a/b=a+b/a=1.618033987, where a is greater than b. This can also be explained by the Fibonacci sequence, another divine proportion. The Fibonacci sequence starts with 1 (some say 0) and adds the previous number to it to get the next one (i.e. 1, 1, 2, 3, 5, 8, 13, 21...)

If you try to find the quotient of two subsequent Fibonacci numbers (i.e. 8/5 or 5/3), the result is very close to the golden ratio of 1.6 or phi.

The golden spiral is created using a golden rectangle. If you have a rectangle of squares 1, 1, 2, 3, 5 and 8 respectively as shown in the picture above, you can start building the golden rectangle. By using the side of the square as the radius, you create an arc that touches the points of the square diagonally. Repeat this procedure with each square in the golden triangle and you will end up with a golden spiral.

Where can we see it in nature?

The Golden Ratio and Fibonacci Sequence can be found in flower petals. For most flowers, the number of petals is reduced to two, three, five or more, which is similar to the golden ratio. For example, lilies have 3 petals, buttercups have 5, chicory flowers have 21, and daisies have 34. Flower seeds probably also follow the golden ratio. For example, sunflower seeds germinate from the center and grow toward the outside, filling the seed head. They are usually spiral-shaped and resemble a golden spiral. Moreover, the number of seeds is usually reduced to Fibonacci numbers.

Hands and fingers are also an example of the golden ratio. Look closer! The base of the palm and the tip of the finger are divided into parts (bones). The ratio of one part in comparison to another is always 1.618! Even the forearms and hands are in the same ratio. And fingers, and face, and the list goes on...

Application in art and architecture

The Parthenon in Greece is said to have been built using golden proportions. It is believed that the dimensional ratios of height, width, columns, distance between pillars, and even the size of the portico are close to the golden ratio. This is possible because the building looks proportionally perfect and has been like this since ancient times.

Leonardo Da Vinci was also a fan of the golden ratio (and many other curiosities, in fact!). The wondrous beauty of the Mona Lisa may be due to the fact that her face and body represent the golden ratio, just like the real ones. human faces in life. In addition, the numbers in the painting “The Last Supper” by Leonardo Da Vinci are arranged in the order that is used in the golden ratio. If you draw golden rectangles on the canvas, Jesus will be right in the central lobe.

Application in logo design

It's no surprise that you can also find the use of the golden ratio in many modern projects, particularly design. For now, let's focus on how this can be used in logo design. First, let's look at some of the world's most famous brands that have used the golden ratio to perfect their logos.

Apparently Apple used circles from Fibonacci numbers, joining and cutting the shapes to create the Apple logo. It is unknown whether this was done intentionally or not. However, the result is a perfect and visually aesthetic logo design.

The Toyota logo uses the ratio of a and b, forming a grid in which three rings are formed. Notice how this logo uses rectangles instead of circles to create the golden ratio.

The Pepsi logo is created by two intersecting circles, one larger than the other. As shown in the picture above, the larger circle is proportional to the smaller circle - you guessed it! Their latest non-emboss logo is simple, effective and beautiful!

Besides Toyota and Apple, the logos of several other companies such as BP, iCloud, Twitter, and Grupo Boticario are also believed to have used the golden ratio. And we all know how famous these logos are - all because the image immediately springs to mind!

Here's how you can apply it in your projects

Sketch a golden rectangle as shown above in yellow. This can be achieved by constructing squares with height and width from numbers belonging to the golden ratio. Start with one block and place another next to it. And place another square, whose area is equal to those two, above them. You will automatically receive a side of 3 blocks. After building this 3 block structure, you will end up with a side of 5 quads from which you can make another (5 block area) box. This can go on as long as you like until you find the size you need!

The rectangle can move in any direction. Select small rectangles and use each one to assemble a layout that will serve as a logo design grid.

If the logo is more rounded, then you will need a circular version of the golden rectangle. You can achieve this by drawing circles proportional to the Fibonacci numbers. Create a golden rectangle using only circles (this means the largest circle will have a diameter of 8, and the smaller circle will have a diameter of 5, and so on). Now separate these circles and place them so that you can form the basic outline for your logo. Here's an example of the Twitter logo:

Note: You don't have to draw all the golden ratio circles or rectangles. You can also use the same size more than once.

How to use it in text design

It's easier than designing a logo. A simple rule for applying the golden ratio in text is that subsequent larger or smaller text must conform to Phi. Let's look at this example:

If my font size is 11, then the subtitle should be written in a larger font. I multiply the text font by the golden ratio number to get larger number(11*1.6=17). This means the subtitle should be written in font size 17. And now the title or title. I’ll multiply the subtitle by the proportion and get 27 (1*1.6=27). Like this! Your text is now proportional to the golden ratio.

How to apply it in web design

But here it’s a little more complicated. You can stay true to the golden ratio even in web design. If you are an experienced web designer, you have already guessed where and how it can be applied. Yes, we can effectively use the golden ratio and apply it to our web page grids and UI layouts.

Take total number grid pixels for width or height and use it to build a golden rectangle. Divide the largest width or length to get smaller numbers. This can be the width or height of your main content. What's left could be the sidebar (or bottombar if you applied it to height). Now continue using the golden rectangle to further apply it to windows, buttons, panels, images and text. You can also build a full mesh based on small versions of the golden rectangle placed both horizontally and vertically to create smaller interface objects that are proportional to the golden rectangle. To get the proportions you can use this calculator.

Spiral

You can also use the golden spiral to determine where to place content on your site. If your home page loads with graphic content, such as an online store website or photography blog, you can use the golden spiral method that many artists use in their work. The idea is to place the most valuable content in the center of the spiral.

Content with grouped material can also be placed using a golden rectangle. This means that the closer the spiral moves to the central squares (to one square block), the “dense” the contents there.

You can use this technique to indicate the placement of your header, images, menus, toolbar, search box, and other elements. Twitter is famous not only for its use of the golden rectangle in its logo design, but also for its use in web design. How? Through the use of the golden rectangle, or in other words the golden spiral concept, in the users' profile page.

But this won't be easy to do on CMS platforms, where the content author determines the layout instead of the web designer. The Golden Ratio is suitable for WordPress and other blog designs. This is probably because a sidebar is almost always present in a blog design, which fits nicely into the golden rectangle.

Easier way

Very often, designers skip complex mathematics and apply the so-called “rule of thirds”. It can be achieved by dividing the area into three equal parts horizontally and vertically. The result is nine equal parts. The intersection line can be used as the focal point of the form and design. You can place a key theme or main elements on one or all of the focal points. Photographers also use this concept for posters.

The closer the rectangles are to the ratio 1:1.6, the more pleasant the picture is perceived by the human brain (since it is closer to the golden ratio).

Golden ratio - mathematics

Golden ratio - harmonic proportion

In mathematics, proportion (lat. proportio) is the equality of two ratios: a: b = c: d.
A straight line segment AB can be divided into two parts in the following ways:
into two equal parts – AB: AC = AB: BC;
into two unequal parts in any respect (such parts do not form proportions);
thus, when AB: AC = AC: BC.
The latter is the golden division or the division of a segment in extreme and average ratio.
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole

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

Rice. 1. Geometric image of the golden ratio

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

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

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

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

The properties of the golden ratio are described by the equation:
x2 – 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.

The division is carried out as follows. Segment AB is divided in proportion to the golden ratio. From point C, a perpendicular CD is restored. The radius AB is point D, which is connected by a line to point A. Right angle ACD is divided in half. A line is drawn from point C to the intersection with line AD. The point Divides the segment AD in the ratio 56:44.

Rice. 3. Construction of the second golden ratio

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

The figure 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 the 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 be the center of the circle, A a point on the circle, and E the midpoint of segment OA. The perpendicular to the radius OA, restored at point O, intersects the circle at point D. Using a compass, plot the segment CE = ED on the diameter. The side length of a regular pentagon inscribed in a circle is equal to DC. We plot the segments DC on the circle and 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.

We draw straight AB. From point A we lay a segment of arbitrary size on it three times, through the resulting point P we draw a perpendicular to line AB, on the perpendicular to the right and left of point P we lay segments O. We connect the resulting points d and d1 with straight lines to point A. We lay segment dd1 on line Ad1 , obtaining point C. She divided the line Ad1 in proportion to the golden ratio. Lines Ad1 and dd1 are used to construct a “golden” rectangle.

Rice. 6. Construction of the golden triangle

History of the golden ratio

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 is the god of 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. Therefore, 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 we put aside segment m on a straight line of arbitrary length, we put aside segment M next to it. 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 segments of the golden ratio

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. 11. 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 publication 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

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

Generalized golden ratio

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 it 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 both the “binary” series and the Fibonacci series are obtained? 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 at S = 0 the segment is divided in half, and at S = 1 the familiar classical golden ratio results.

The ratios of neighboring Fibonacci S-numbers coincide with absolute mathematical accuracy in the limit with the golden S-proportions! Mathematicians in such cases say that the golden S-ratios are numerical invariants of the Fibonacci S-numbers.

Facts confirming the existence of golden S-sections in nature are given by 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 golden S-proportions. This allowed the author to put forward the hypothesis that the golden 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 S-proportion codes, you can express any real number as a sum of powers of golden S-proportions with integer coefficients.

The fundamental difference between this method of encoding numbers is that the bases of the new codes, which are the golden S-proportions, turn out to be irrational numbers when S> 0. 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 Pythagoreans discovered incommensurable segments - 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.

In such a number system, any natural number can always be represented as finite - and not infinite, as previously thought! – the sum of powers of any of the golden 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

Rice. 12. Archimedes Spiral

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. 15. Bird's egg

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.

Golden ratio and symmetry

The golden ratio cannot be considered on its own, separately, without connection with symmetry. The great Russian crystallographer G.V. Wolf (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.

Examples of the golden ratio in architecture can be found everywhere, as long as you can see it. Even a schoolchild can figure this out. In 2013, 10th grade student Elena Sivakova conducted her own research on buildings from the 19th and 20th centuries. Let's see how she did it and learn to see and identify it in architectural structures in 5 minutes. After reading the article, there will be no questions left about what it is and whether its unusual properties can be used in your life.

7+ examples of the golden ratio in Russian architecture

Saint Petersburg

Building historical center St. Petersburg were built in different styles, such as Baroque, Empire, Eclectic, Neo-Baroque, Neo-Gothic. Do they obey the golden rule?

Saint Isaac's Cathedral

The court architect of Alexander I, Auguste Montferrand, built this cathedral from 1819 to 1858. Late style, in which the features of neo-Renaissance and eclecticism are already manifested. Elena asked the question: “What is the reason for the harmony of a rather bulky building?”

The first row is determined by the width of the building, which is taken to be 400 units. and represents the following numbers: 400, 247, 153, 94, 58...

If we divide 400 by the number ≈1.618, we get approximately 247; repeat the action with the following number: 247: 1.618≈153.

And this is how we find all the numbers. Now let's look at the drawing. The main part with the columns fits into a rectangle with sides 400 and 247. Since the sides are in the ratio Ф≈1.618, they form a Golden Rectangle.

The next row is represented by the height of the building: 370, 228, 140, 87, 53, 33, 20, 12. These dimensions are embedded in smaller details. Vertically, St. Isaac's Cathedral is divided by the Golden Ratio at the base of the dome, which makes the relationship between the main part and the dome harmonious.

The third row of sizes begins with 113, and is the width of the base of the main dome: 113, 69, 42, 26, 16. The numbers in this row are found in the sizes of windows, in the heights of columns and other parts of the cathedral.

Golden rectangular and isosceles triangle and take place in the building of St. Isaac's Cathedral, as can be seen from the picture.

Kunstkamera

On the University embankment of Vasilievsky Island there is the Kunstkamera building, founded in 1718 under the leadership of the German architect Georg Mattarnovi: Petrovsky Baroque, two 3-story buildings and a complex multi-tiered domed tower.

The study begins with the main quantities: the height and length of the building, from which the golden row is built. Length - 450 units, then 277, 170, 105, 65, 40, 24. These dimensions can be seen in height and width different levels towers, length of hulls. The tower part itself is inscribed in a golden isosceles triangle from base to top. The golden ratio is seen to a greater extent in this main element, which is correct from an architectural point of view. Conclusion: the basis of the Kunstkamera obeys the golden rule and maintains compositional harmony.

The new golden row begins with the height of the building: 211, 130, 80, 49, 30. Looking at the dimensions of the drawing, it becomes clear that the choice of a three-story type of buildings is due to the proportionality with the tower.

Trading house "Esders and Scheyfals" at the intersection of Moika and Gorokhovaya

Built in 1907 according to the design of Vladimir Aleksandrovich Lipsky and Konstantin Nikolaevich de Rochefort (Rochefort). In 1905, the Belgian S. Esders and the Dutchman N. Scheyfals submitted a petition for permission to build a five-story building with a dome and a spire on a corner tower for their trading house instead of the old one.

From a building length of 671 units. the series of the Golden Ratio begins, observed in sizes: 671, 414, 256, 158, 98, 60, 37, 23. We pay attention to the main element - the spire. We make sure that the compositional solution is completed with a harmonious combination of height values.

Built in 1941 according to the design of Noah Abramovich Trotsky. Building Soviet period viewed as a creative interpretation. The central portico with fourteen columns completes the sculptural ensemble on the theme of the construction of socialism and the coat of arms of the Russian Soviet Federative Socialist Republic.

Five-story buildings are located symmetrically on the sides. The length of the House reaches 1472 units, from which, by dividing by the number F, a number of sizes of building elements are obtained: 1472, 909, 562, 34, 214, 132, 81, 50 (Appendix 21): height of the structure, height of the entrance, etc.

The top of the Golden Isosceles Triangle coincides with the top of the building, and its sides pass through the top points of the main entrance. A right-angled golden triangle is formed by the vertices at the top of the building and at the end of the inside of the side wing. Proportionality is obvious, although it does not have much compositional significance.

Moscow

Moscow State University on Sparrow Hills

A team led by B.M. Iofan worked on his project, who was later removed from the post of chief architect. An example of post-war Soviet architecture, built from 1949 to 1953.

B.M. Iofan proposed a composition of five components with a central tower. During the years of construction it was the tallest building in Europe.

The length of the building is 1472 units. and begins the series: 909, 562, 347, 214, 132, 81, 50. Mainly height dimensions are subject to the golden ratio. Another series follows from the width of the tower: 538, 332, 205, 126, which is visible in latitudinal dimensions.

The golden right triangle with its hypotenuse runs through the corner of the building and covers the extensions.

Thus, in all the buildings studied, the student discovered the Golden Ratio, which preserves harmony.

5 additional examples

To simplify the task of finding the ZS, you can take rational fractions 3/2; 5/3; 8/5; 13/8; 21/13; 34/21; 55/34; 89/55; and so on. The pattern is clear: 3+2 =5; 5+3=8; 8+5=13... Or even simpler. Make yourself a compass to determine the proportion according to the instructions in the video. It will take about 10 minutes. They will also tell and show you how to use this compass to determine the proportionality of elements.

Using this method, we find the golden proportion of the Russian architect Matvey Kazakov in the Kremlin Senate building, and in all other works: the Prechistensky Palace in Moscow, the Noble Assembly, the Golitsyn Hospital (named after Pirogov) ...

Created by another great architect Vasily Ivanovich Bazhenov, the Pashkov House in Moscow (Russian State Library) is considered one of the examples of perfect architectural monuments, in which it is easy to identify the AP.

The terrible symbol of Paris and the golden ratio

When the metal Eiffel Tower was being assembled in Paris, many French people were indignant. Critics wrote about it as “the ugliness of the city,” “the disgrace of Paris,” “a skinny pyramid of metal stairs.” Among them were Emile Zola, Dumas Jr., Guy de Maupassant. Now this most visited monument is the pride of Parisians. Could this be due to the “divine” proportion?

It is also observed in the most famous French cathedral, Notre Dame de Paris.

The whole truth about the ancient builders

Did great architects intuitively or consciously build buildings with these proportions in mind? Ancient mathematicians knew about the golden ratio since the time of Pythagoras. More and more evidence is being found of its use in architectural proportions. However, not a single ancient record can be found with a direct recommendation to use the “divine proportion”. Nor does Vitruvius (1st century BC), who wrote “Ten Books on Architecture,” in which he discussed proportionality, among other things. Strange fact, isn't it?

Maybe all the above studies are an adjustment to a known result? It's not that hard to choose from a variety architectural elements those that confirm the hypothesis, since no one requires absolute accuracy. A logical question to ponder is, “What if the Greeks DIDN’T use the golden ratio?”

As a matter of fact, for Luca Pacioli, who wrote the work “Divine Proportion” in 1509, its applied significance was not so important. It was important to substantiate its mystical nature. And they began to use it consciously only from the moment the book was published.

The Mystery of the Architecture of Ancient Greece

Beautiful and harmonious objects always comply with the GS rule, and when analyzing quantities, this proportionality is determined. Art historians carefully studied the Greek Parthenon, erected in honor of the victory over the Persians - the temple of the goddess Athena. The ratio of the length of the temple to the width gives the golden number with a small error. If you subtract 14 cm from the length of the structure and add it to the width, you get a complete coincidence with the mathematical value. The façade of the building narrows slightly towards the top and deviates from its rectangular shape. Taking into account visual perception, the builders did this deliberately. Therefore, it is not entirely correct to consider it a rectangle of the golden ratio. But the proportions are respected, so it is logical to assume that the architects Iktin and Kallikrates deliberately incorporated the rule into the project?

Myths and strange facts about the pyramid

The Pyramid of Cheops was also built taking this condition into account. Without going into the mathematical proof of the presence of the golden formula, we will only say that it contains a right-angled golden triangle, the sides of which are the height and half of the side of the base of the building. Anything surprising?

But then the question arises about the level of ancient Egyptian mathematics. It turns out that the Pythagorean theorem was known to them two thousand years before the birth of the scientist himself. Attention is drawn to the fact that the heirs of Cheops built their pyramids with different proportions. Why?

It has been established that pyramid-shaped structures with terrestrial structures have a phenomenal effect on those in them: plants grow better, metals become stronger, water remains fresh for a long time. Scientists have been working with these mysteries for many years, but the mystery remains.

It is noticed that the pyramid brings the structure of space into a harmonious state. Everything that falls within the action zone is also organized in a similar way: the psycho-emotional state of people improves, radiation harmful to humans decreases, and geopathogenic zones disappear. The Internet claims that if the size of the figure doubles, then the influence of the pyramid increases a hundred times.

How can you build a “Golden” house for yourself?

The correct distribution of energies inside the house, harmonious designs in combination with the ecology and safety of building materials encourage modern architects and designers to use the principles and concepts of the Golden Ratio. This increases the estimate and creates the impression of a deep study of the project. The cost increases by 60-80%.

For talented artists and architects, the rule remains intuitive during creative process. However, some of them consciously implement this provision.

In nature, such proportionality is found everywhere. Anyone who feels the harmony of space will create a proportional building without any special effort.

For example, our ancestors built mansions proportional to a person. Height and length were measured in fathoms, cubits, arshins, spans. Does anyone object that the golden proportion is observed in the human body? The length of the arm from the tips of the fingers to the armpit is related to the distance from the same point to the elbow as this value is to the size of the palm.

The famous French architect Le Corbusier used the owner's height as a starting unit to calculate the parameters of the future house and interior. All his works are truly individual and harmonious.

5 ways to follow the rule in the interior

In a house built without taking proportions into account, rooms can be rearranged so that the proportions match.
Sometimes it is enough to rearrange the furniture or make an additional partition.
The height and length of windows and doors changes in the same way.
In color design, obtaining a simplified ratio is achieved by using 60% of the primary color, 30% as a shading color, and the remaining 10% as tones that enhance perception.
The height and length of the furniture should be commensurate with the height of the ceilings and the width of the partitions.

The application of this norm in, as an architecturally designed space, is combined with the concepts of self-organization, recursion, asymmetry, and beauty.

About the golden ratio in simple words

What is it? Segments of the golden proportion are expressed as an infinite irrational fraction, the decimal value of which is approximately equal to the number Ф≈1.618 or Ф≈1.62. In other words: if we take the whole and divide it into two parts so that one of them is 62% and the other is 38%, we get the Golden Proportion.

Golden rectangle: when we divide the length of the larger side by the length of the smaller one and get the number F. When dividing the smaller side by the larger one, we get the inverse value φ ≈ 0.618.

Golden isosceles triangle: if the ratio of the size of one side and the size of the base is the golden number Ф; angle between equal sides equal to 36°.

Kepler's golden right triangle combines the Pythagorean theorem and the GS: the ratio of the squares of its sides is 1.618.

Watch an educational video on the topic