Golden glow. Shkrudnev Fedor Dmitrievich - Golden ratio

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 operates in the forms of space and time.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science considers the golden ratio as “asymmetrical symmetry,” calling it in a broad sense a universal rule 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 in the golden section the divine trinity: 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 proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.

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. Goethe later noted nature’s attraction 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, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but the main indicator of the golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions of the female body - 8:5.

ART OF SPATIAL FORMS

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

Art critic F.V. Kovalev, having examined in detail the painting by Nikolai Ge “Alexander Sergeevich Pushkin in the village of Mikhailovskoye,” notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.

Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.

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

WORD, SOUND AND FILM

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

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

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

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

Golden ratio - mathematics

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) 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 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

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded.
The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for the construction of dynamic rectangles.

Rice. 7. Dynamic rectangles

Plato (427...347 BC) also knew about the golden division. His dialogue “Timaeus” is devoted to the mathematical and aesthetic views of the Pythagorean school and, in particular, to the issues of the golden division.
The façade of the ancient Greek temple of the Parthenon features golden proportions. During its excavations, compasses were discovered that were used by architects and sculptors of the ancient world. The Pompeian compass (museum in Naples) also contains the proportions of the golden division.

Rice. 8. Antique golden ratio compass

In the ancient literature that has come down to us, the golden division was first mentioned in Euclid’s Elements. In the 2nd book of the “Principles” the geometric construction of the golden division is given. After Euclid, the study of the golden division was carried out by Hypsicles (2nd century BC), Pappus (III century AD), and others. In medieval Europe, with the golden division We met through Arabic translations of Euclid’s Elements. The translator J. Campano from Navarre (III century) made comments on the translation. The secrets of the golden division were jealously guarded and kept in strict secrecy. They were known only to initiates.
During the Renaissance, interest in the golden division increased among scientists and artists due to its use in both geometry and art, especially in architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry.
Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “The Divine Proportion” was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity - God the son, God the father and God the holy spirit (it was implied that the small segment is the personification of God the son, the larger segment 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 endless 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 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

In subsequent centuries, the rule of the golden proportion turned into an academic canon, and when, over time, the struggle against academic routine began in art, in the heat of the struggle “they threw out the baby with the bathwater.” The golden ratio was “discovered” again in the middle of the 19th century. In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Studies”. What happened to Zeising was exactly what should inevitably happen to a researcher who considers a phenomenon as such, without connection with other phenomena. He absolutized the proportion of the golden section, declaring it universal for all phenomena of nature and art. Zeising had numerous followers, but there were also opponents who declared his doctrine of proportions to be “mathematical aesthetics.”

Rice. 10. Golden proportions in parts of the human body

Zeising did a tremendous job. He measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law. The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6. In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man. The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.

Rice. 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 edition does not mention a single work of painting.

At the end of the 19th – beginning of the 20th centuries. Many purely formalistic theories appeared about the use of the golden ratio in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

Fibonacci series

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci (son of Bonacci), is indirectly connected with the history of the golden ratio. He traveled a lot in the East, introduced Europe to Indian (Arabic) numerals. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time. One of the problems read “How many pairs of rabbits will be born from one pair in one year.” Reflecting on this topic, Fibonacci built the following series of numbers:

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division. So, 21: 34 = 0.617, and 34: 55 = 0.618. This 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.

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 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.

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

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

Everything that took on some form was formed, grew, strived to take a place in space and preserve itself. This desire is realized mainly in two options - growing upward or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly shorter than the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The idea of the golden ratio will be incomplete without talking about the spiral.

Rice. 12. Archimedes Spiral

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

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

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

Rice. 13. Chicory

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

Rice. 15. Bird's egg

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

Pierre Curie at the beginning of this century formulated a number of profound ideas about symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry of the environment.

The laws of “golden” symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms. These patterns, as indicated above, exist in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.

Golden ratio and symmetry

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

What do the Egyptian pyramids, Leonardo da Vinci's Mona Lisa, and the Twitter and Pepsi logos have in common?

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 the painting of the Sistine Chapel and in the paintings of Van Gogh. 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, orderly, balanced and respects 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 a harmonious combination of elements, then it won’t hurt to try to apply the principles of the golden ratio.

There are still many unsolved mysteries in the universe, some of which scientists have already been able to identify and describe. Fibonacci numbers and the golden ratio form the basis for unraveling the world around us, constructing its form and optimal visual perception by a person, with the help of which he can feel beauty and harmony.

Golden ratio

The principle of determining the dimensions of the golden ratio underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technology. The doctrine of the golden proportion was founded as a result of research by ancient scientists into the nature of numbers.

It is based on the theory of proportions and ratios of divisions of segments, which was made by the ancient philosopher and mathematician Pythagoras. He proved that when dividing a segment into two parts: X (smaller) and Y (larger), the ratio of the larger to the smaller will be equal to the ratio of their sum (the entire segment):

The result is an equation: x 2 - x - 1=0, which is solved as x=(1±√5)/2.

If we consider the ratio 1/x, then it is equal to 1,618…

Evidence of the use of the golden ratio by ancient thinkers is given in Euclid’s book “Elements,” written back in the 3rd century. BC, who applied this rule to construct regular pentagons. Among the Pythagoreans, this figure is considered sacred because it is both symmetrical and asymmetrical. The pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber abaci by Italian mathematician Leonardo of Pisa, who later became known as Fibonacci, was published in 1202. In it, the scientist for the first time cites the pattern of numbers, in a series of which each number is the sum of 2 previous digits. The Fibonacci number sequence is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.

The scientist also cited a number of patterns:

Any number from the series divided by the next one will be equal to a value that tends to 0.618. Moreover, the first Fibonacci numbers do not give such a number, but as we move from the beginning of the sequence, this ratio will become more and more accurate.
If you divide the number from the series by the previous one, the result will rush to 1.618.
One number divided by the next by one will show a value tending to 0.382.

The application of the connection and patterns of the golden section, the Fibonacci number (0.618) can be found not only in mathematics, but also in nature, history, architecture and construction, and in many other sciences.

Archimedes spiral and golden rectangle

Spirals, very common in nature, were studied by Archimedes, who even derived its equation. The shape of the spiral is based on the laws of the golden ratio. When unwinding it, a length is obtained to which proportions and Fibonacci numbers can be applied; the step increases evenly.

The parallel between Fibonacci numbers and the golden ratio can be seen by constructing a “golden rectangle” whose sides are proportional as 1.618:1. It is built by moving from a larger rectangle to smaller ones so that the lengths of the sides are equal to the numbers from the series. It can also be constructed in reverse order, starting with square “1”. When the corners of this rectangle are connected by lines at the center of their intersection, a Fibonacci or logarithmic spiral is obtained.

History of the use of golden proportions

Many ancient architectural monuments of Egypt were built using golden proportions: the famous pyramids of Cheops, etc. Architects of Ancient Greece widely used them in the construction of architectural objects such as temples, amphitheaters, and stadiums. For example, such proportions were used in the construction of the ancient temple of the Parthenon, (Athens) and other objects that became masterpieces of ancient architecture, demonstrating harmony based on mathematical patterns.

In later centuries, interest in the golden ratio subsided, and the patterns were forgotten, but it resumed again in the Renaissance with the book of the Franciscan monk L. Pacioli di Borgo “The Divine Proportion” (1509). It contained illustrations by Leonardo da Vinci, who established the new name “golden ratio”. 12 properties of the golden ratio were also scientifically proven, and the author talked about how it manifests itself in nature, in art and called it “the principle of building the world and nature.”

Vitruvian Man Leonardo

The drawing, which Leonardo da Vinci used to illustrate the book of Vitruvius in 1492, depicts a human figure in 2 positions with arms spread to the sides. The figure is inscribed in a circle and a square. This drawing is considered to be the canonical proportions of the human body (male), described by Leonardo based on studying them in the treatises of the Roman architect Vitruvius.

The center of the body as an equidistant point from the end of the arms and legs is the navel, the length of the arms is equal to the height of the person, the maximum width of the shoulders = 1/8 of the height, the distance from the top of the chest to the hair = 1/7, from the top of the chest to the top of the head = 1/6 etc.

Since then, the drawing has been used as a symbol showing the internal symmetry of the human body.

Leonardo used the term “Golden Ratio” to designate proportional relationships in the human figure. For example, the distance from the waist to the feet is related to the same distance from the navel to the top of the head in the same way as height is to the first length (from the waist down). This calculation is done similarly to the ratio of segments when calculating the golden proportion and tends to 1.618.

All these harmonious proportions are often used by artists to create beautiful and impressive works.

Research on the golden ratio in the 16th to 19th centuries

Using the golden ratio and Fibonacci numbers, research on the issue of proportions has been going on for centuries. In parallel with Leonardo da Vinci, the German artist Albrecht Durer also worked on developing the theory of correct proportions of the human body. For this purpose, he even created a special compass.

In the 16th century The question of the connection between the Fibonacci number and the golden ratio was devoted to the work of astronomer I. Kepler, who first applied these rules to botany.

A new “discovery” awaited the golden ratio in the 19th century. with the publication of the “Aesthetic Investigation” of the German scientist Professor Zeisig. He raised these proportions to absolutes and declared that they are universal for all natural phenomena. He conducted studies of a huge number of people, or rather their bodily proportions (about 2 thousand), based on the results of which conclusions were drawn about statistically confirmed patterns in the ratios of various parts of the body: the length of the shoulders, forearms, hands, fingers, etc.

Objects of art (vases, architectural structures), musical tones, and sizes when writing poems were also studied - Zeisig displayed all this through the lengths of segments and numbers, and he also introduced the term “mathematical aesthetics.” After receiving the results, it turned out that the Fibonacci series was obtained.

Fibonacci number and the golden ratio in nature

In the plant and animal world there is a tendency towards morphology in the form of symmetry, which is observed in the direction of growth and movement. Division into symmetrical parts in which golden proportions are observed - this pattern is inherent in many plants and animals.

The nature around us can be described using Fibonacci numbers, for example:

the arrangement of leaves or branches of any plants, as well as distances, correspond to a series of given numbers 1, 1, 2, 3, 5, 8, 13 and so on;
sunflower seeds (scales on cones, pineapple cells), arranged in two rows along twisted spirals in different directions;
the ratio of the length of the tail and the entire body of the lizard;
the shape of an egg, if you draw a line through its wide part;
ratio of finger sizes on a person's hand.

And, of course, the most interesting shapes include spiraling snail shells, patterns on spider webs, the movement of wind inside a hurricane, the double helix in DNA and the structure of galaxies - all of which involve the Fibonacci sequence.

Use of the golden ratio in art

Researchers searching for examples of the use of the golden ratio in art study in detail various architectural objects and works of art. There are famous sculptural works, the creators of which adhered to golden proportions - statues of Olympian Zeus, Apollo Belvedere and

One of Leonardo da Vinci’s creations, “Portrait of the Mona Lisa,” has been the subject of research by scientists for many years. They discovered that the composition of the work consists entirely of “golden triangles” united together into a regular pentagon-star. All of da Vinci’s works are evidence of how deep his knowledge was in the structure and proportions of the human body, thanks to which he was able to capture the incredibly mysterious smile of Mona Lisa.

Golden ratio in architecture

As an example, scientists examined architectural masterpieces created according to the rules of the “golden ratio”: Egyptian pyramids, Pantheon, Parthenon, Notre Dame de Paris Cathedral, St. Basil's Cathedral, etc.

The Parthenon - one of the most beautiful buildings in Ancient Greece (5th century BC) - has 8 columns and 17 on different sides, the ratio of its height to the length of the sides is 0.618. The protrusions on its facades are made according to the “golden ratio” (photo below).

One of the scientists who came up with and successfully applied an improvement to the modular system of proportions for architectural objects (the so-called “modulor”) was the French architect Le Corbusier. The modulator is based on a measuring system associated with the conditional division into parts of the human body.

Russian architect M. Kazakov, who built several residential buildings in Moscow, as well as the Senate building in the Kremlin and the Golitsyn hospital (now the 1st Clinical named after N. I. Pirogov), was one of the architects who used the laws in design and construction about the golden ratio.

Applying proportions in design

In clothing design, all fashion designers create new images and models taking into account the proportions of the human body and the rules of the golden ratio, although by nature not all people have ideal proportions.

When planning landscape design and creating three-dimensional park compositions with the help of plants (trees and shrubs), fountains and small architectural objects, the laws of “divine proportions” can also be applied. After all, the composition of the park should be focused on creating an impression on the visitor, who will be able to freely navigate it and find the compositional center.

All elements of the park are in such proportions as to create an impression of harmony and perfection with the help of geometric structure, relative position, illumination and light.

Application of the golden ratio in cybernetics and technology

The laws of the golden section and Fibonacci numbers also appear in energy transitions, in processes occurring with elementary particles that make up chemical compounds, in space systems, and in the genetic structure of DNA.

Similar processes occur in the human body, manifesting itself in the biorhythms of his life, in the action of organs, for example, the brain or vision.

Algorithms and patterns of golden proportions are widely used in modern cybernetics and computer science. One of the simple tasks that novice programmers are given to solve is to write a formula and determine the sum of Fibonacci numbers up to a certain number using programming languages.

Modern research into the theory of the golden ratio

Since the mid-20th century, interest in the problems and influence of the laws of golden proportions on human life has increased sharply, and from many scientists of various professions: mathematicians, ethnic researchers, biologists, philosophers, medical workers, economists, musicians, etc.

In the United States, the magazine The Fibonacci Quarterly began publishing in the 1970s, where works on this topic were published. Works appear in the press in which the generalized rules of the golden ratio and the Fibonacci series are used in various fields of knowledge. For example, for information coding, chemical research, biological research, etc.

All this confirms the conclusions of ancient and modern scientists that the golden proportion is multilaterally related to fundamental issues of science and is manifested in the symmetry of many creations and phenomena of the world around us.

Geometry is an exact and quite complex science, which at the same time is a kind of art. Lines, planes, proportions - all this helps to create many truly beautiful things. And oddly enough, this is based on geometry in its most varied forms. In this article we will look at one very unusual thing that is directly related to this. The golden ratio is exactly the geometric approach that will be discussed.

The shape of an object and its perception

People most often rely on the shape of an object in order to recognize it among millions of others. It is by its shape that we determine what kind of thing lies in front of us or stands in the distance. We first recognize people by the shape of their body and face. Therefore, we can confidently say that the shape itself, its size and appearance is one of the most important things in human perception.

For people, the form of anything is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and feeling of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special ratio were used, which is called the golden ratio.

The concept of the golden ratio

So, the golden ratio is the 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.

From the ancient history of the golden ratio

The golden ratio is often used in many different areas of life today. But the history of this concept goes back to ancient times, when sciences such as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the 6th century BC. But even before that, knowledge about such a ratio was used in practice in Ancient Egypt and Babylon. A clear indication of this are the pyramids, for the construction of which exactly this golden proportion was used.

New period

The Renaissance brought new breath to harmonic division, especially thanks to Leonardo da Vinci. This ratio has increasingly begun to be used both in geometry and in art. Scientists and artists began to study the golden ratio more deeply and create books that examine this issue.

One of the most important historical works related to the golden ratio is a book by Luca Pancholi called The Divine Proportion. Historians suspect that the illustrations of this book were done by Leonardo himself before Vinci.

golden ratio

Mathematics gives a very clear definition of proportion, which says that it is the equality of two ratios. Mathematically, this can be expressed by the following equality: a: b = c: d, where a, b, c, d are some specific values.

If we consider the proportion of a segment divided into two parts, we can encounter only a few situations:

The segment is divided into two absolutely even parts, which means AB:AC = AB:BC, if AB is the exact beginning and end of the segment, and C is the point that divides the segment into two equal parts.
The segment is divided into two unequal parts, which can be in very different proportions to each other, which means that here they are completely disproportionate.
The segment is divided so that AB:AC = AC:BC.

As for the golden ratio, this is a proportional division of a segment into unequal parts, when the entire segment relates to the larger part, just as the larger part itself relates to the smaller one. There is another formulation: the smaller segment is related to the larger one, just as the larger one is to the entire segment. In mathematical terms, it looks like this: a:b = b:c or c:b = b:a. This is exactly what the golden ratio formula looks like.

Golden ratio in nature

The golden ratio, examples of which we will now consider, refers to incredible phenomena in nature. These are very beautiful examples of the fact that mathematics is not just numbers and formulas, but a science that has more than a real reflection in nature and our life in general.

For living organisms, one of the main tasks in life is growth. This desire to take one’s place in space, in fact, occurs in several forms - growing upward, almost horizontally spreading on the ground, or twisting in a spiral on some kind of support. And as incredible as it may be, many plants grow according to the golden ratio.

Another almost incredible fact is the relationships in the body of lizards. Their body looks quite pleasing to the human eye and this is possible due to the same golden ratio. To be more precise, the length of their tail relates to the length of the entire body as 62:38.

Interesting facts about the rules of the golden ratio

The golden ratio is a truly incredible concept, which means that throughout history we can come across many really interesting facts about this proportion. We present you some of them:

Golden ratio in the human body

In this section it is necessary to mention a very significant person, namely S. Zeizinga. This is a German researcher who has done a tremendous amount of work in the field of studying the golden ratio. He published a work entitled Aesthetic Studies. In his work, he presented the golden ratio as an absolute concept that is universal for all phenomena both in nature and in art. Here we can recall the golden ratio of the pyramid along with the harmonious proportion of the human body and so on.

It was Zeising who was able to prove that the golden ratio, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experiment. According to Zeising's research, the main indicator of the golden ratio is the division of the body by the navel point. Thus, the male body with an average ratio of 13:8 is slightly closer to the golden ratio than the female body, where the golden ratio is 8:5. The golden ratio can also be observed in other parts of the body, such as the hand.

About the construction of the golden ratio

In fact, constructing the golden ratio is a simple matter. As we see, even ancient people coped with this quite easily. What can we say about modern knowledge and technologies of mankind. In this article we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we will confidently declare that it is, in fact, possible. Moreover, this can be done in more than one way.

Since this is a fairly simple geometry, the golden ratio is quite simple to construct even at school. Therefore, information about this can be easily found in specialized books. By studying the golden ratio, 6th graders are fully able to understand the principles of its construction, which means that even children are smart enough to master such a task.

Golden ratio in mathematics

The first acquaintance with the golden ratio in practice begins with a simple division of a straight line segment in the same proportions. Most often this is done using a ruler, compass and, of course, a pencil.

Segments of the golden proportion are expressed as an infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... In order to make these calculations more practical, very often they use not exact, but approximate values, namely - 0 .62 and .38. If the segment AB is taken as 100 parts, then its larger part will be equal to 62, and the smaller part will be equal to 38 parts, respectively.

The main property of the golden ratio can be expressed by the equation: x 2 -x-1=0. When solving, we get the following roots: x 1.2 =. Although mathematics is an exact and rigorous science, like its section - geometry, it is properties such as the laws of the golden section that cast mystery on this topic.

Harmony in art through the golden ratio

In order to summarize, let’s briefly consider what has already been discussed.

Basically, many pieces of art fall under the rule of the golden ratio, where a ratio close to 3/8 and 5/8 is observed. This is the rough formula of the golden ratio. The article has already mentioned a lot about examples of using the section, but we will look at it again through the prism of ancient and modern art. So, the most striking examples from ancient times:

As for the probably conscious use of proportion, starting from the time of Leonardo da Vinci, it came into use in almost all areas of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.