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 the relief wooden board from the tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded.

The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for the construction of dynamic rectangles.

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, in particular, to the issues of the golden division.

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 “Principles” a geometric construction of the golden division is given. After Euclid, the study of golden division was carried out by Hypsicles (II century BC), Pappus (III century AD) and others. medieval Europe The golden division was introduced to the Arabic translations of Euclid’s Elements by the translator J. Campano from Navarre (III century). The secrets of the golden division were jealously guarded, 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 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. This is how it continues to this day.

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.” Albrecht Durer develops in detail the theory of proportions of the human body. He assigned an important place in his system of relationships to the golden section. 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).

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 Research”. Zeising considers the golden ratio 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 teaching on proportions “mathematical aesthetics.”

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 a Basic Morphological Law in Nature and Art.” In 1876, a small book was published in Russia outlining this work of Zeising.

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.

Science did not absorb art, but in those historical periods when mathematics and art came closer, this gave impetus to the development of both.

The concept of the golden ratio

Let's find out what the ancient Egyptian pyramids, Leonardo da Vinci's painting "Mona Lisa", a sunflower, a snail, a snowflake, a galaxy and human fingers have in common?

In mathematics, proportion (lat. proportio) is the equality of two ratios: a: b = c: d.

The golden ratio is 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.

Line segment AB can be divided into two parts by point C 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);
• in extreme and average terms in such a way that AB: AC = AC: BC.

The last one is the golden division.

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler. 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 as an infinite irrational fraction; if AB is taken as one, then AE = 0.618..., 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.

Construction of the second golden ratio. 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. Point E divides segment AD in the ratio 56:44.

The line of the second golden ratio of the rectangle is located midway between the golden ratio line and the middle line of the rectangle.

Pentagram

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

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 a pentagon divide each other into segments in 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, deposited at side, divides it in the golden ratio.

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: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

This series is known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the two previous ones, and the ratio of adjacent numbers in the series approaches the ratio of the golden division. Moreover, after the 13th number in the sequence, this division result becomes constant until the infinity of the series. It was this constant number of divisions that was called the Divine proportion in the Middle Ages, and is now called the golden section, the golden average, or the golden proportion. In algebra, this number is denoted by the Greek letter φ (phi).

So the Golden Ratio is 1:1.618

So, 21: 34 = 0.617, and 34: 55 = 0.618. This relationship is denoted by the symbol φ. This ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden proportion.

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. 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.

Golden rectangle and golden spiral

In geometry, a rectangle with a golden aspect ratio began to be called golden. Its long sides are in relation to the short ones - in a ratio of 1.168: 1.

The Golden Rectangle also has many amazing properties. By cutting a square from the golden rectangle, the side of which is equal to the smaller side of the rectangle, we again obtain a golden rectangle of smaller dimensions. This process can be continued indefinitely. As we continue to cut off squares, we will end up with smaller and smaller golden rectangles. Moreover, they will be located along a logarithmic spiral, having important V mathematical models natural objects. The pole of the spiral lies at the intersection of the diagonals of the initial rectangle and the first vertical one to be cut. Moreover, the diagonals of all subsequent decreasing golden rectangles lie on these diagonals. Of course, there is also the golden triangle.

When we look at a beautiful landscape, we are embraced by everything around us. Then we pay attention to details. A murmuring river or a majestic tree. We see a green field. We notice how the wind gently hugs him and shakes the grass from side to side. We can feel the aroma of nature and hear the singing of birds... Everything is harmonious, everything is interconnected and gives a feeling of peace, a sense of beauty. Perception proceeds in stages in slightly smaller fractions. Where will you sit on the bench: on the edge, in the middle, or anywhere? Most will answer that it is a little further from the middle. The approximate number for the proportion of the bench from your body to the edge would be 1.62. It’s the same in the cinema, in the library, everywhere. We instinctively create harmony and beauty, which I call the “Golden Ratio” all over the world.

Golden ratio in mathematics

Have you ever wondered whether it is possible to determine the measure of beauty? It turns out that from a mathematical point of view it is possible. Simple arithmetic gives the concept of absolute harmony, which is reflected in impeccable beauty, thanks to the principle of the Golden Ratio. The architectural structures of other Egypt and Babylon were the first to begin to correspond this principle. But Pythagoras was the first to formulate the principle. In mathematics, this is a division of a segment slightly more than half, or more precisely 1.628. This ratio is presented as φ =0.618= 5/8. A small segment = 0.382 = 3/8, and the entire segment is taken as one.

A:B=B:C and C:B=B:A

The principle of the golden ratio was used by great writers, architects, sculptors, musicians, people of art, and Christians who drew pictograms (five-pointed stars, etc.) with its elements in churches, fleeing from evil spirits, and people studying exact sciences, problem solver cybernetics.

Golden ratio in nature and phenomena.

Everything on earth takes shape, grows upward, to the side or in a spiral. Archimedes paid close attention to the latter and composed an equation. According to the Fibonacci series, there is a cone, a shell, a pineapple, a sunflower, a hurricane, a spider’s web, a DNA molecule, an egg, a dragonfly, a lizard...

Titirius proved that our entire Universe, space, galactic space - everything is planned based on the Golden Principle. One can read the highest beauty in absolutely everything living and non-living.

Golden ratio in man.

The bones are also designed by nature according to the proportion 5/8. This eliminates people’s reservations about “wide bones.” Most body parts in ratios apply to the equation. If all parts of the body obey the Golden Formula, then the external data will be very attractive and ideally proportioned.

The segment from the shoulders to the top of the head and its size = 1:1 .618
The segment from the navel to the top of the head and from the shoulders to the top of the head = 1:1 .618
The segment from the navel to the knees and from them to the feet = 1:1 .618
The segment from the chin to the extreme point of the upper lip and from it to the nose = 1:1 .618


All facial distances give a general idea of ​​the ideal proportions that attract the eye.
Fingers, palm, also obey the law. It should also be noted that the length of the spread arms with the torso is equal to the height of a person. Why, all organs, blood, molecules correspond to the Golden Formula. True harmony inside and outside our space.

Parameters from the physical side of surrounding factors.

Sound volume. Highest point sound, causing an uncomfortable feeling and pain in the ear = 130 decibels. This number can be divided by the proportion 1.618, then it turns out that the sound of a human scream will be = 80 decibels.
Using the same method, moving further, we get 50 decibels, which is typical for the normal volume of human speech. And the last sound that we get thanks to the formula is a pleasant whisper sound = 2.618.
Using this principle, it is possible to determine the optimal-comfortable, minimum and maximum numbers of temperature, pressure, and humidity. The simple arithmetic of harmony is embedded in our entire environment.

Golden ratio in art.

In architecture, the most famous buildings and structures are: Egyptian pyramids, Mayan pyramids in Mexico, Notre Dame de Paris, Greek Parthenon, Peter's Palace, and others.

In music: Arensky, Beethoven, Havan, Mozart, Chopin, Schubert, and others.

In painting: almost all the paintings of famous artists are painted according to the cross-section: the versatile Leonardo da Vinci and the inimitable Michelangelo, such relatives in writing as Shishkin and Surikov, the ideal of the purest art - the Spaniard Raphael, and the Italian Botticelli, who gave the ideal of female beauty, and many, many others.

In poetry: the ordered speech of Alexander Sergeevich Pushkin, especially “Eugene Onegin” and the poem “The Shoemaker”, the poetry of the wonderful Shota Rustaveli and Lermontov, and many other great masters of words.

In sculpture: a statue of Apollo Belvedere, Olympian Zeus, beautiful Athena and graceful Nefertiti, and other sculptures and statues.

Photography uses the “rule of thirds.” The principle is this: the composition is divided into 3 equal parts vertically and horizontally, key points are located either on intersection lines (horizon) or at intersection points (object). Thus the proportions are 3/8 and 5/8.
According to the Golden Ratio, there are many tricks that are worth examining in detail. I will describe them in detail in the next one.

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

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

This proportion is:

For example, in the correct five-pointed star, each segment is divided by a segment intersecting it in the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is equal 1.618

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

In 1855, the German researcher of the golden ratio, Professor Zeising, published his work "Aesthetic Research".
Zeising measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law.

Golden proportions in parts of the human body

The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6.

In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man.
The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, and poetic meters were studied.

Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they amounted to 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, 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. (or 1.618 , if you divide a larger number by a smaller one).

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 section.

Golden ratio in art

Back in 1925, art critic L.L. Sabaneev, having analyzed 1,770 musical works by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or by intonation structure, or by modal structure, which are in relation to each other golden ratio.

Moreover, the more talented the composer, the more golden sections are found in his works. In Arensky, Beethoven, Borodin, Haydn, Mozart, Scriabin, Chopin and Schubert, golden sections were found in 90% of all works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition.

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

Golden ratio in architecture, sculpture, painting

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

Visible in the pictures whole line patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

On the floor plan of the Parthenon you can also see the “golden rectangles”:

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

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

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

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. Let's look closely at the painting "La Gioconda". The composition of the portrait is based on “golden triangles”.

Golden ratio in fonts and household items

Golden ratio in nature

Biological studies have shown that, starting with viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems.

It was found that the Fibonacci number series characterizes structural organization many living systems. For example, the helical leaf arrangement on a branch forms a fraction (number of revolutions on the stem/number of leaves in a cycle, eg 2/5; 3/8; 5/13), corresponding to the Fibonacci series.

The “golden” proportion of five-petal flowers of apple, pear and many other plants is well known. The carriers of the genetic code - DNA and RNA molecules - have a double helix structure; its dimensions almost completely correspond to the numbers of the Fibonacci series.

Goethe emphasized nature's tendency toward spirality.

The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. A frightened herd of reindeer scatters in a spiral.

Goethe called the spiral the “curve of life.” The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc.

Flowers and seeds of sunflowers, chamomiles, scales in pineapple fruits, conifer cones are “packed” in logarithmic (“golden”) spirals, curling towards each other, and the numbers of “right” and “left” spirals are always related to each other, like neighboring numbers Fibonacci.

Consider a chicory shoot. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time 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.

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

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

What do they have in common? Egyptian pyramids, the Mona Lisa painting by Leonardo da Vinci and the Twitter and Pepsi logos?

Let’s not delay the answer - they were all created using the golden ratio rule. The golden ratio is the ratio of two quantities a and b, which are not equal to each other. This proportion is often found in nature, and the rule of the golden ratio is also actively used in fine arts and design - compositions created using the “divine proportion” are well balanced and, as they say, pleasing to the eye. But what exactly is the golden ratio and can it be used in modern disciplines, for example, in web design? Let's figure it out.

A LITTLE MATH

Let's say we have a certain segment AB, divided in two by point C. The ratio of the lengths of the segments is: AC/BC = BC/AB. That is, a segment is divided into unequal parts in such a way that the larger part of the segment makes up the same share in the whole, undivided segment as the smaller segment makes up in the larger one.

This unequal division is called the golden ratio. The golden ratio is designated by the symbol φ. The value of φ is 1.618 or 1.62. In general, to put it very simply, this is the division of a segment or any other value in the ratio of 62% and 38%.

“Divine proportion” has been known to people since ancient times; this rule was used in the construction of the Egyptian pyramids and the Parthenon; the golden ratio can be found in paintings Sistine Chapel and in Van Gogh's paintings. The golden ratio is still widely used today - examples that are constantly before our eyes are the Twitter and Pepsi logos.

The human brain is designed in such a way that it considers as beautiful those images or objects in which an unequal proportion of parts can be detected. When we say about someone that “he is well-proportioned,” we unknowingly mean the golden ratio.

The golden ratio can be applied to various geometric shapes. If we take a square and multiply one side by 1.618, we get a rectangle.

Now, if we superimpose a square on this rectangle, we can see the golden ratio line:

If we continue to use this proportion and break the rectangle into smaller parts, we get this picture:

It is not yet clear where this fragmentation of geometric figures will lead us. A little more and everything will become clear. If we draw a smooth line equal to a quarter of a circle in each of the squares of the diagram, then we will get a Golden Spiral.

This is an unusual spiral. It is also sometimes called the Fibonacci spiral, in honor of the scientist who studied the sequence in which each number is early to the sum of the two previous ones. The point is that this mathematical relationship, which we visually perceive as a spiral, is found literally everywhere - sunflowers, sea shells, spiral galaxies and typhoons - there is a golden spiral everywhere.

HOW CAN YOU USE THE GOLDEN RATIO IN DESIGN?

So, the theoretical part is over, let's move on to practice. Is it really possible to use the golden ratio in design? Yes, you can. For example, in web design. Taking this rule into account, you can obtain the correct ratio of the compositional elements of the layout. As a result, all parts of the design, down to the smallest ones, will be harmoniously combined with each other.

If we take a typical layout with a width of 960 pixels and apply the golden ratio to it, we will get this picture. The ratio between the parts is the already known 1:1.618. The result is a two-column layout, with a harmonious combination of two elements.

Sites with two columns are very common and this is far from accidental. Here, for example, is the National Geographic website. Two columns, golden ratio rule. Good design, ordered, balanced and taking into account the requirements of visual hierarchy.

One more example. Design studio Moodley has developed a corporate identity for the Bregenz performing arts festival. When the designers worked on the event poster, they clearly used the golden ratio rule in order to correctly determine the size and location of all elements and, as a result, obtain the ideal composition.

Lemon Graphic, who created the visual identity for Terkaya Wealth Management, also used a 1:1.618 ratio and a golden spiral. The three elements of the business card design fit perfectly into the scheme, resulting in all the parts coming together very well

Here's another interesting use of the golden spiral. Before us again is the National Geographic website. If you look at the design more closely, you can see that there is another NG logo on the page, only a smaller one, which is located closer to the center of the spiral.

Of course, this is not accidental - the designers knew very well what they were doing. This is a great place to duplicate a logo, as our eye naturally moves toward the center of the composition when viewing a site. This is how the subconscious works and this must be taken into account when working on design.

GOLDEN CIRCLES

“Divine proportion” can be applied to any geometric shapes, including circles. If we inscribe a circle in squares, the ratio between which is 1:1.618, then we get golden circles.

Here is the Pepsi logo. Everything is clear without words. Both the ratio and the way the smooth arc of the white logo element was achieved.

With the Twitter logo, things are a little more complicated, but here too you can see that its design is based on the use of golden circles. It doesn't follow the "divine proportion" rule a little, but for the most part all of its elements fit into the scheme.

CONCLUSION

As you can see, despite the fact that the golden ratio rule has been known since time immemorial, it is not at all outdated. Therefore, it can be used in design. It is not necessary to try your best to fit into the scheme - design is an imprecise discipline. But if you need to achieve harmonious combination elements, then it wouldn’t hurt to try to apply the principles of the golden ratio.

04/18/2011 A. F. Afanasyev Updated 06/16/12

Dimensions and proportions are one of the main tasks in the search for an artistic image of any work of plastic art. It is clear that the issue of size is decided taking into account the room where it will be located and the objects surrounding it.

Speaking about proportions (the ratio of dimensional values), we take them into account in the format of a flat image (painting, marquetry), in the ratios overall dimensions(length, height, width) of a three-dimensional object, in the ratio of two objects of the same ensemble that are different in height or length, in the ratio of the sizes of two clearly prominent parts of the same object, etc.

In the classics of fine art for many centuries, a technique for constructing proportions has been traced, called the golden section, or the golden number (this term was introduced by Leonardo da Vinci). The principle of the golden ratio, or dynamic symmetry, is that “the ratio between two parts of a single whole is equal to the ratio of its larger part to the whole” (or, accordingly, the whole to the larger part). Mathematically this is

the number is expressed as - 1 ± 2?5 - which gives 1.6180339... or 0.6180339... In art, 1.62 is taken as the golden number, i.e. an approximate expression of the ratio larger size in proportion to its smaller value.
From approximate to more accurate, this relationship can be expressed: etc., where: 5+3=8, 8+5=13, etc. Or: 2,2:3,3:5,5:8 ,8, etc., where 2.2+3.3-5.5, etc.

Graphically, the golden ratio can be expressed by the ratio of segments obtained by various constructions. More convenient, in our opinion, is the construction shown in Fig. 169: if you add its short side to the diagonal of a half-square, you get a value in the ratio of the golden number to its long side.

Rice. 169. Geometric construction of a rectangle in the golden ratio 1.62: 1. Golden number 1.62 in relation to segments (a and b)

Rice. 170. Graphic construction of the golden ratio function 1.12: 1

Proportion of two golden ratios

creates a visual feeling of harmony and balance. There is another harmonious ratio of two adjacent quantities, expressed by the number 1.12. It is a function of the golden number: if you take the difference between two values ​​of the golden ratio, divide it also in the golden ratio and add each fraction to the smaller value of the original golden ratio, you get a ratio of 1.12 (Fig. 170). In this relation, for example, the middle element (shelf) is drawn in the letters H, R, Z, etc. in some fonts, the proportions of height and width are taken for wide letters, this relation is also found in nature.

The golden number is observed in proportions harmoniously developed person(Fig. 171): the length of the head divides in the golden ratio the distance from the waist to the crown; the kneecap also divides the distance from the waist to the sole of the feet; the tip of the middle finger of an outstretched hand divides the entire height of a person in the golden proportion; The ratio of the phalanges of the fingers is also a golden number. The same phenomenon is observed in other structures of nature: in the spirals of mollusks, in the corollas of flowers, etc.

Rice. 172. Golden proportions of a carved geranium (pelargonium) leaf. Construction: 1) Using a scale graph (see Fig. 171) do we build? ABC,	Rice. 173. Five-petalled and three-petaled grape leaves. The length to width ratio is 1.12. The golden ratio is expressed

In Fig. 172 and 173 show the construction of a pattern of a geranium (pelargonium) leaf and a grape leaf in the proportions of golden numbers 1.62 and 1.12. In a geranium leaf, the construction is based on two triangles: ABC and CEF, where the ratio of the height and base of each of them is expressed by the numbers 0.62 and 1.62, and the distances between the three pairs of the most distant points of the leaf are equal: AB=CE=SF. The construction is indicated in the drawing. The design of such a leaf is typical of geraniums, which have similar carved leaves.

The generalized sycamore leaf (Fig. 173) has the same proportions as the grape leaf, in the ratio of 1.12, but the greater proportion of the grape leaf is its length, and that of the plane tree leaf is its width. The sycamore leaf has three proportional sizes in a ratio of 1.62. Such a correspondence in architecture is called a triad (for four proportions - tetrad and further: pectad, hexode).

In Fig. 174 shows a method for constructing a maple leaf in the proportions of the golden ratio. With a width to length ratio of 1.12, it has several proportions with the number 1.62. The construction is based on two trapezoids, in which the ratio of the height and length of the base is expressed by a golden number. The construction is shown in the drawing, and options for the shape of a maple leaf are also given.

In works of fine art, an artist or sculptor, consciously or subconsciously, trusting his trained eye, often applies the ratio of sizes in the golden ratio. Thus, while working on a copy of the head of Christ (according to Michelangelo), the author of this book noticed that adjacent curls in strands of hair in their size reflect the ratio of the golden ratio, and in their shape - the Archimedean spiral, the involute. The reader can see for himself that in a number of paintings by classical artists the central figure is located from the sides of the format at distances forming the proportion of the golden ratio (for example, the placement of the head both vertically and horizontally in V. Borovikovsky’s portrait of M. I. Lopukhina; position along the vertical center of the head in the portrait of A. S. Pushkin by O. Kiprensky and others). The same thing can sometimes be seen with the placement of the horizon line (F. Vasiliev: “Wet Meadow”, I. Levitan: “March”, “Evening Bells”).

Of course, this rule is not always a solution to the problem of composition, and it should not replace the intuition of rhythm and proportions in the artist’s work. It is known, for example, that some artists used the ratios of “musical numbers” for their compositions: thirds, fourths, fifths (2:3, 3:4, etc.). Art historians, not without reason, note that the design of any classical architectural monument or sculpture, if desired, can be adjusted to any number ratio. Our task in this case, and especially the task of a beginning artist or woodcarver, is to learn to build a deliberate composition of his work not according to random relationships, but according to harmonious proportions, proven by practice. These harmonious proportions must be able to be identified and emphasized by the design and shape of the product.

As an example of finding a harmonious proportion, consider determining the size of the frame for the work shown in Fig. 175. The format of the image placed in it is set in the proportion of the golden ratio. External dimensions frames with the same width of its sides will not give the golden proportion. Therefore, the ratio of its length and width (ЗЗ0X220) is taken to be slightly less than the golden number, i.e. equal to 1.5, and the width of the transverse links is correspondingly increased compared to the side sides. This made it possible to arrive at the dimensions of the frame in the light (for the painting), giving the proportions of the golden ratio. The ratio of the width of the lower link of the frame to the width of its upper link is adjusted to another golden number, i.e. 1.12. Also, the ratio of the width of the lower link to the width of the side link (94:63) is close to 1.5 (in the figure - the option on the left).

Now we’ll do an experiment: we’ll increase the long side of the frame to 366 mm due to the width of the lower link (it will be 130 mm) (in the picture - the option on the right), which will bring not only the ratio closer but also to the gold
number 1.62 instead of 1.12. The result is a new composition that can be used in some other product, but for the frame there is a desire to make it shorter. Cover its lower part with a ruler so much that the eye “accepts” the resulting proportion, and we will get its length of 330 mm, i.e. we will approach the original version.

So, analyzing various options(there may be others besides the two discussed), the master settles on the only possible solution from his point of view.

It is best to apply the principle of the golden ratio in search of the desired composition using a simple device, circuit diagram the design of which is shown in Fig. 176. Two rulers of this device can, rotating around hinge B, form an arbitrary angle. If, for any angle solution, we divide the distance AC in the golden section with a point K and mount two more rulers: KM\\BC and KE\\AB with hinges at points K, E and M, then for any solution AC this distance will be divided by point K in relation to the golden ratio.