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

Story

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

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 dedicated 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. Antique golden ratio compass

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

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

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

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

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

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

Nature

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

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

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

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

Rice. Construction of a scale of golden proportion segments

Rice. Chicory

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

Rice. Viviparous lizard

Rice. bird egg

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

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

Human

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

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

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

The art of spatial forms

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

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

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

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

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

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

Golden ratio and symmetry

The golden ratio cannot be considered on its own, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulf (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.

The golden division is not a manifestation of asymmetry, something opposite to symmetry. According to modern concepts, the golden division is an asymmetrical symmetry. The science of symmetry includes such concepts as static And dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments and equal values. Dynamic symmetry is characterized by an increase in segments or their decrease, and it is expressed in the values of the golden section of an increasing or decreasing series.

Word, sound and film

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

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

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

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

We invite you to discuss the topic in our group -

Since ancient times, people have been concerned with the question of whether such elusive things as beauty and harmony are subject to any mathematical calculations. Of course, all the laws of beauty cannot be contained in a few formulas, but by studying mathematics, we can discover some components of beauty - the golden ratio. Our task is to find out what the golden ratio is and to establish where humanity has found the use of the golden ratio.

You probably noticed that we treat objects and phenomena of the surrounding reality differently. Be h decency, blah h Formality and disproportion are perceived by us as ugly and produce a repulsive impression. And objects and phenomena that are characterized by proportion, expediency and harmony are perceived as beautiful and evoke in us a feeling of admiration, joy, and lift our spirits.

In his activities, a person constantly encounters objects that are based on the golden ratio. There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit? In the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other relative to your body is approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you reproduced the “golden ratio”.

The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. Be h the final row after the decimal point is 1.6180339887... A strange, mysterious, inexplicable thing - this divine proportion mystically accompanies all living things. Inanimate nature does not know what the “golden ratio” is. But you will certainly see this proportion in the curves of sea shells, and in the shape of flowers, and in the appearance of beetles, and in the beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”. So what is the “golden ratio”? What is this perfect, divine combination? Maybe this is the law of beauty? Or is he still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The “Golden Ratio” is both. Only not separately, but simultaneously... And this is his true mystery, his great secret.

It is probably difficult to find a reliable measure for an objective assessment of beauty itself, and logic alone will not do it. However, the experience of those for whom the search for beauty was the very meaning of life, who made it their profession, will help here. These are, first of all, people of art, as we call them: artists, architects, sculptors, musicians, writers. But these are also people of exact sciences, primarily mathematicians.

Trusting the eye more than other sense organs, Man first learned to distinguish the 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, 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, a proportion is the equality of two ratios:

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 last one is the golden division (section).

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, in other words, 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.

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.

Dividing a straight line segment using the golden ratio. BC=1/2AB; 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 ratio are expressed without h the final 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 equal to 62, and the smaller part is 38 parts.

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

Solution to this equation:

The properties of the golden ratio have created a romantic aura of mystery and an almost mystical generation around this number. For example, in a regular five-pointed star, each segment is divided by the segment intersecting it in the proportion of the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is 1.618).

SECOND GOLDEN RATIO

This proportion is found in architecture.

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.

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 (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. 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 with 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 0 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 down on it three times a segment O of an arbitrary size, through the resulting point P we draw a perpendicular to the line AB, on the perpendicular to the right and left of point P we lay off segments O. We connect the resulting points d and d 1 with straight lines to point A. Segment dd 1 we put it on the line Ad 1, getting point C. It divided the line Ad 1 in the proportion of the golden section. Lines Ad 1 and dd 1 are used to construct a “golden” rectangle.

Construction of the golden triangle

HISTORY OF THE GOLDEN RATIO

Indeed, the proportions of the Cheops pyramid, temples, 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.

Dynamic rectangles

Plato also knew about the golden division. The Pythagorean Timaeus, in Plato’s dialogue of the same name, says: “It is impossible for two things to be perfectly united without a third, since a thing must appear between them that would hold them together. This can best be accomplished by proportion, for if three numbers have the property that the average is to the lesser as the greater is to the average, and, conversely, the lesser is to the average as the average is to the greater, then the latter and the first will be the average, and average - first and last. Thus, everything necessary will be the same, and since it will be the same, it will make up the whole.” Plato builds the earthly world using triangles of two types: isosceles and non-isosceles. He considers the most beautiful right triangle to be one in which the hypotenuse is twice as large as the smaller of the legs (such a rectangle is half of the equilateral, basic figure of the Babylonians, it has a ratio of 1: 3 1/2, which differs from the golden ratio by about 1/25, and is called Timerding "rival of the golden ratio"). Using triangles, Plato builds four regular polyhedra, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedra - the dodecahedron, all twelve of which are regular pentagons, claims to be a symbolic image of the celestial world.

ICOSAHEDRON AND DODECAHEDRON

The honor of discovering the dodecahedron (or, as was supposed, the Universe itself, this quintessence of the four elements, symbolized, respectively, by the tetrahedron, octahedron, icosahedron and cube) belongs to Hippasus, who later died in a shipwreck. This figure actually captures many relationships of the golden ratio, so the latter was given the main role in the heavenly world, which was what the Minorite brother Luca Pacioli later insisted on.

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

In the Middle Ages, the pentagram was demonized (as, indeed, much that was considered divine in ancient paganism) and found shelter in the occult sciences. However, the Renaissance again brings to light both the pentagram and the golden ratio. Thus, during that period of the establishment of humanism, a diagram describing the structure of the human body became widespread.

Leonardo da Vinci also repeatedly resorted to such a picture, essentially reproducing a pentagram. Her interpretation: the human body has divine perfection, because the proportions inherent in it are the same as in the main heavenly figure. 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 Duke Moreau, he came to Milan, where he gave lectures on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “On the Divine Proportion” (De divina proportione, 1497, published in Venice in 1509) 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. There is only one such proportion, and uniqueness is the highest property of God. It embodies the holy trinity. This proportion cannot be expressed in an accessible number, remains hidden and secret, and is called irrational by mathematicians themselves (in the same way, God cannot be defined or explained in words). God never changes and represents everything in everything and everything in each of its parts, so the golden ratio for any continuous and definite quantity (regardless of whether it is large or small) is the same, can neither be changed nor changed. otherwise perceived by reason. God called into existence heavenly virtue, otherwise called the fifth substance, with its help and four other simple bodies (four elements - earth, water, air, fire), and on their basis called into existence every other thing in nature; so our sacred proportion, according to Plato in the Timaeus, gives formal existence to the sky itself, for it is attributed the appearance of a body called the dodecahedron, which cannot be constructed without the golden ratio. These are Pacioli's arguments.

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

Construction of a scale of golden proportion segments

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.”

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 of 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 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.

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 concepts, the golden division is an asymmetrical symmetry. The science of symmetry includes such concepts as static and dynamic symmetry. Static symmetry characterizes peace and balance, while dynamic symmetry characterizes movement and growth. Thus, in nature, static symmetry is represented by the structure of crystals, and in art it characterizes peace, balance and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characterized by equal segments and equal values. Dynamic symmetry is characterized by an increase in segments or their decrease, and it is expressed in the values of the golden section of an increasing or decreasing series.

FIBONACCI SERIES

The name of the Italian mathematician monk Leonardo of Pisa, better known as Fibonacci, is indirectly connected with the history of the golden ratio. He traveled extensively in the East and introduced Arabic numerals to Europe. In 1202, his mathematical work “The Book of the Abacus” (counting board) was published, which collected all the problems known at that time.

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 the whole.

As shown in the bottom figure, the length of each finger joint is related to the length of the next joint by the proportion F. The same relationship appears in all fingers and toes. This connection is somehow unusual, because one finger is longer than the other without any visible pattern, but this is not accidental, just as everything in the human body is not accidental. The distances on the fingers, marked from A to B to C to D to E, are all related to each other by the proportion F, as are the phalanges of the fingers from F to G to H.

Take a look at this frog skeleton and see how each bone fits the F proportion pattern just like in the human body.

GENERALIZED GOLDEN RATIO

Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. 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 1, 2, 4, 8, discovered by him, are at first glance completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2=1+1; 4=2+2..., in the second - this is the sum of the two previous numbers 2=1+1, 3=2+1, 5=3+2... Is it possible to find a general mathematical formula from which the “binary » series, and Fibonacci series? Or maybe this formula will give us new numerical sets that have some new unique properties?

Indeed, let us define a numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider a number series, S+1, the first terms of which are ones, 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 we denote the nth term of this series by? S (n), then we get the general formula? S(n)=? S(n-1)+? S(n-S-1).

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

In general, the golden S-proportion is the positive root of the equation of the golden S-section x S+1 -x S -1=0.

It is easy to show that when S = 0 the segment is divided in half, and when S = 1 the familiar classical golden ratio is obtained.

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 from 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 the 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, in which an irrational number (which, recall, is the root of the golden ratio equation) is chosen as the fundamental basis of the beginning of notation; 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 FORM FORMATION IN NATURE

Everything that took on some form was formed, grew, sought to take a place in space and preserve itself. This desire is realized mainly in two ways: 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.

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and derived the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. 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.”

Mandelbrot series

The Golden Spiral is closely related to cycles. Modern chaos science studies simple cyclic operations with feedback and the fractal shapes they generate, previously unknown. The picture shows the famous Mandelbrot series - a page from the dictionary h limbs of individual patterns called Julian series. Some scientists associate the Mandelbrot series with the genetic code of cell nuclei. A consistent increase in sections reveals fractals that are amazing in their artistic complexity. And here, too, there are logarithmic spirals! This is all the more important since both the Mandelbrot series and the Julian series are not an invention of the human mind. They arise from the area of Plato's prototypes. As doctor R. Penrose said, “they are like Mount Everest.”

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.

The shoot makes a strong ejection into space, stops, releases a leaf, but this time 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 to be 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden proportion. In growth and conquest of space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

Chicory

In many butterflies, the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding its wings, the 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.

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

Viviparous lizard

In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.

Nature has carried out division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Of great interest is the study of the shapes of bird eggs. Their various forms fluctuate between two extreme types: one of them can be inscribed in a rectangle of the golden ratio, the other in a rectangle with a modulus of 1.272 (the root of the golden ratio)

Such shapes of bird eggs are not accidental, since it has now been established that the shape of eggs described by the golden ratio ratio corresponds to higher strength characteristics of the egg shell.

The tusks of elephants and extinct mammoths, the claws of lions, and the beaks of parrots are logarithmic in shape and resemble the shape of an axis that tends to turn into a spiral.

In living nature, forms based on “pentagonal” symmetry are widespread (starfish, sea urchins, flowers).

The golden ratio is present in the structure of all crystals, but most crystals are microscopically small, so we cannot see them with the naked eye. However, snowflakes, which are also water crystals, are quite visible to our eyes. All the exquisitely beautiful figures that form snowflakes, all axes, circles and geometric figures in snowflakes are also always, without exception, built according to the perfect clear formula of the golden ratio.

In the microcosm, three-dimensional logarithmic forms built according to golden proportions are ubiquitous. For example, many viruses have the three-dimensional geometric shape of an icosahedron. Perhaps the most famous of these viruses is the Adeno virus. The protein shell of the Adeno virus is formed from 252 units of protein cells arranged in a specific sequence. At each corner of the icosahedron there are 12 units of protein cells in the shape of a pentagonal prism, and spine-like structures extend from these corners.

Adeno virus

The golden ratio in the structure of viruses was first discovered in the 1950s. scientists from Birkbeck College London A. Klug and D. Kaspar. The Polyo virus was the first to display a logarithmic form. The form of this virus was found to be similar to that of the Rhino virus.

The question arises: how do viruses form such complex three-dimensional forms, the structure of which contains the golden ratio, which are quite difficult to construct even with our human mind? The discoverer of these forms of viruses, virologist A. Klug, gives the following comment: “Dr. Kaspar and I showed that for the spherical shell of the virus, the most optimal shape is symmetry such as the icosahedron shape. This order minimizes the number of connecting elements... Most of Buckminster Fuller's geodesic hemispherical cubes are built on a similar geometric principle. The installation of such cubes requires an extremely precise and detailed explanation diagram, while unconscious viruses themselves construct such a complex shell from elastic, flexible protein cellular units.”

Klug’s comment once again reminds us of an extremely obvious truth: in the structure of even a microscopic organism that scientists classify as “the most primitive form of life,” in this case a virus, there is a clear plan and an intelligent design implemented. This project is incomparable in its perfection and precision of execution to the most advanced architectural projects created by people. For example, projects created by the brilliant architect Buckminster Fuller.

Three-dimensional models of the dodecahedron and icosahedron are also present in the structure of the skeletons of single-celled marine microorganisms radiolarians (rayfish), the skeleton of which is made of silica.

Radiolarians form their bodies of very exquisite, unusual beauty. Their shape is a regular dodecahedron, and from each of its corners sprouts a pseudo-elongation-limb and other unusual shapes-growths.

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.

THE HUMAN BODY AND THE GOLDEN RATIO

All human bones are kept in proportion to the golden ratio. The proportions of the various parts of our body are a number very close to the golden ratio. If these proportions coincide with the golden ratio formula, then the person’s appearance or body is considered ideally proportioned.

Golden proportions in parts of the human body

If we take the navel point as the center of the human body, and the distance between a person’s foot and the navel point as a unit of measurement, then a person’s height is equivalent to the number 1.618.

the distance from shoulder level to the crown of the head and the size of the head is 1:1.618;
the distance from the navel point to the crown of the head and from shoulder level to the crown of the head is 1:1.618;
the distance of the navel point to the knees and from the knees to the feet is 1:1.618;
the distance from the tip of the chin to the tip of the upper lip and from the tip of the upper lip to the nostrils is 1:1.618;
the actual exact presence of the golden proportion in a person’s face is the ideal of beauty for the human gaze;
the distance from the tip of the chin to the upper line of the eyebrows and from the upper line of the eyebrows to the crown is 1:1.618;
face height/face width;
the central point of connection of the lips to the base of the nose/length of the nose;
face height/distance from the tip of the chin to the central point where the lips meet;
mouth width/nose width;
nose width/distance between nostrils;
distance between pupils/distance between eyebrows.

It is enough just to bring your palm closer to you and look carefully at your index finger, and you will immediately find the formula of the golden ratio in it.

Each finger of our hand consists of three phalanges. The sum of the lengths of the first two phalanges of the finger in relation to the entire length of the finger gives the number of the golden ratio (with the exception of the thumb).

In addition, the ratio between the middle finger and little finger is also equal to the golden ratio.

A person has 2 hands, the fingers on each hand consist of 3 phalanges (except for the thumb). There are 5 fingers on each hand, that is, 10 in total, but with the exception of two two-phalanx thumbs, only 8 fingers are created according to the principle of the golden ratio. Whereas all these numbers 2, 3, 5 and 8 are Fibonacci sequence numbers.

Also worth noting is the fact that for most people, the distance between the ends of their outstretched arms is equal to their height.

The truths of the golden ratio are within us and in our space. The peculiarity of the bronchi that make up the human lungs lies in their asymmetry. The bronchi consist of two main airways, one of which (the left) is longer and the other (the right) is shorter. It was found that this asymmetry continues in the branches of the bronchi, in all the smaller respiratory tracts. Moreover, the ratio of the lengths of short and long bronchi is also the golden ratio and is equal to 1:1.618.

In the human inner ear there is an organ called Cochlea (“Snail”), which performs the function of transmitting sound vibration. This bony structure is filled with fluid and is also shaped like a snail, containing a stable logarithmic spiral shape =73 0 43".

Blood pressure changes as the heart works. It reaches its greatest value in the left ventricle of the heart at the moment of its compression (systole). In the arteries, during the systole of the ventricles of the heart, blood pressure reaches a maximum value equal to 115-125 mmHg in a young, healthy person. At the moment of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of maximum (systolic) to minimum (diastolic) pressure is on average 1.6, that is, close to the golden ratio.

If we take the average blood pressure in the aorta as a unit, then the systolic blood pressure in the aorta is 0.382, and the diastolic pressure is 0.618, that is, their ratio corresponds to the golden proportion. This means that the work of the heart in relation to time cycles and changes in blood pressure are optimized according to the same principle, the law of the golden proportion.

The DNA molecule consists of two vertically intertwined helices. The length of each of these spirals is 34 angstroms and the width is 21 angstroms. (1 angstrom is one hundred millionth of a centimeter).

The structure of the helix section of the DNA molecule

So, 21 and 34 are numbers following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the golden ratio 1:1.618.

GOLDEN RATIO IN SCULPTURE

Sculptural structures and monuments are erected to perpetuate significant events, to preserve in the memory of descendants the names of famous people, their exploits and deeds. It is known that even in ancient times the basis of sculpture was the theory of proportions. The relationships between the parts of the human body were associated with the golden ratio formula. The proportions of the “golden section” create the impression of harmony and beauty, which is why sculptors used them in their works. Sculptors claim that the waist divides the perfect human body in relation to the “golden ratio”. For example, the famous statue of Apollo Belvedere consists of parts divided according to golden ratios. The great ancient Greek sculptor Phidias often used the “golden ratio” in his works. The most famous of them were the statue of Olympian Zeus (which was considered one of the wonders of the world) and the Parthenon of Athens.

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.

GOLDEN RATIO IN ARCHITECTURE

In books about the “golden ratio” you can find the remark that in architecture, as in painting, everything depends on the position of the observer, and if some proportions in a building from one side seem to form the “golden ratio”, then from other points of view they will look different. The “Golden Ratio” gives the most relaxed ratio of the sizes of certain lengths.

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

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

The Parthenon has 8 columns on the short sides and 17 on the long sides. The projections are made entirely of squares of Pentilean marble. The nobility of the material from which the temple was built made it possible to limit the use of coloring, which is common in Greek architecture; it only emphasizes the details and forms a colored background (blue and red) for the sculpture. The ratio of the building's height to its length is 0.618. If we divide the Parthenon according to the “golden section”, we will get certain protrusions of the facade.

The “golden rectangles” can also be seen on the floor plan of the Parthenon.

We can see the golden ratio in the building of Notre Dame Cathedral (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.

For a long time it was believed that the architects of Ancient Rus' built everything “by eye”, without special mathematical calculations. However, the latest research has shown that Russian architects were well aware of mathematical proportions, as evidenced by the analysis of the geometry of ancient temples.

The famous Russian architect M. Kazakov widely used the “golden ratio” in his work. His talent was multifaceted, but it was revealed to a greater extent in the numerous completed projects of residential buildings and estates. For example, the “golden ratio” can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the Golitsyn Hospital was built in Moscow, which is currently called the First Clinical Hospital named after N.I. Pirogov.

Petrovsky Palace in Moscow. Built according to the design of M.F. Kazakova

Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect works of architecture by V. Bazhenov.

Pashkov House

The wonderful creation of V. Bazhenov has firmly entered the ensemble of the center of modern Moscow and enriched it. The exterior of the house has remained almost unchanged to this day, despite the fact that it was badly burned in 1812. During restoration, the building acquired more massive shapes. The internal layout of the building has not been preserved, which can only be seen in the drawing of the lower floor.

Many of the architect’s statements deserve attention today. About his favorite art, V. Bazhenov said: “Architecture has three main objects: beauty, tranquility and strength of the building... To achieve this, the knowledge of proportion, perspective, mechanics or physics in general serves as a guide, and the common leader of all of them is reason.”

GOLDEN RATIO IN MUSIC

Any piece of music has a temporal extension and is divided by certain “aesthetic milestones” into separate parts that attract attention and facilitate perception as a whole. These milestones can be the dynamic and intonation climaxes of a musical work. Separate time intervals of a musical work, connected by a “climax event,” as a rule, are in the Golden Ratio ratio.

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 related to each other in relation to the golden ratio. Moreover, the more talented the composer, the more golden ratios are found in his works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition. Sabaneev checked this result on all 27 Chopin etudes. He discovered 178 golden ratios in them. It turned out that not only large parts of the studies are divided by duration in relation to the golden ratio, but also parts of the studies inside are often divided in the same ratio.

Composer and scientist M.A. Marutaev counted the number of bars in the famous sonata “Appassionata” and found a number of interesting numerical relationships. In particular, in the development - the central structural unit of the sonata, where themes intensively develop and tones replace each other - there are two main sections. In the first - 43.25 measures, in the second - 26.75. The ratio 43.25:26.75=0.618:0.382=1.618 gives the golden ratio.

The largest number of works in which the Golden Ratio is present are by Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Chopin (92%), Schubert (91%).

If music is the harmonic ordering of sounds, then poetry is the harmonic ordering of speech. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry the sister of musical works. The golden ratio in poetry first of all manifests itself as the presence of a certain moment of the poem (culmination, semantic turning point, main idea of the work) in a line falling on the point of division of the total number of lines of the poem in the golden proportion. So, if a poem contains 100 lines, then the first point of the Golden Ratio falls on the 62nd line (62%), the second on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, including “Eugene Onegin”, are the finest correspondence to the golden proportion! Works by Shota Rustaveli and M.Yu. Lermontov are also built according to the principle of the Golden Section.

Stradivari wrote that he used the golden ratio to determine the locations for f-shaped notches on the bodies of his famous violins.

GOLDEN RATIO IN POETRY

Research into poetic works from these positions is just beginning. And you need to start with the poetry of A.S. Pushkin. After all, his works are an example of the most outstanding creations of Russian culture, an example of the highest level of harmony. From the poetry of A.S. Pushkin, we will begin the search for the golden proportion - the measure of harmony and beauty.

Much in the structure of poetic works makes this art form similar to music. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry the sister of musical works. Each verse has its own musical form, its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden proportion will appear.

Let's start with the size of the poem, that is, the number of lines in it. It would seem that this parameter of the poem can change arbitrarily. However, it turned out that this was not the case. For example, N. Vasyutinsky’s analysis of the poems of A.S. Pushkina showed that the sizes of poems are distributed very unevenly; it turned out that Pushkin clearly prefers the sizes of 5, 8, 13, 21 and 34 lines (Fibonacci numbers).

Many researchers have noticed that poems are similar to pieces of music; they also have culminating points that divide the poem in proportion to the golden ratio. Consider, for example, the poem by A.S. Pushkin's "Shoemaker":

Let's analyze this parable. The poem consists of 13 lines. It has two semantic parts: the first in 8 lines and the second (the moral of the parable) in 5 lines (13, 8, 5 are Fibonacci numbers).

One of Pushkin’s last poems, “I do not value loud rights…” consists of 21 lines and there are two semantic parts in it: 13 and 8 lines:

I don’t value loud rights dearly,

Which makes more than one head spin.

I don't complain that the gods refused

It's my sweet fate to challenge taxes

Or prevent kings from fighting each other;

And it’s not enough for me to worry if the press is free

Fooling idiots, or sensitive censorship

In magazine plans, the joker is embarrassed.

All this, you see, is words, words, words.

Other, better rights are dear to me:

I need a different, better freedom:

Depend on the king, depend on the people -

Do we care? God be with them.

Don’t give a report, only to yourself

To serve and please; for power, for livery

Don’t bend your conscience, your thoughts, your neck;

To wander here and there at will,

Marveling at the divine beauty of nature,

And before the creations of art and inspiration

Trembling joyfully in the raptures of tenderness,

What happiness! That's right...

It is characteristic that the first part of this verse (13 lines), according to its semantic content, is divided into 8 and 5 lines, that is, the entire poem is structured according to the laws of the golden proportion.

The analysis of the novel “Eugene Onegin” made by N. Vasyutinsky is of undoubted interest. This novel consists of 8 chapters, each with an average of about 50 verses. The eighth chapter is the most perfect, most polished and emotionally rich. It has 51 verses. Together with Eugene’s letter to Tatiana (60 lines), this exactly corresponds to the Fibonacci number 55!

N. Vasyutinsky states: “The culmination of the chapter is Evgeny’s declaration of love for Tatyana - the line “To turn pale and fade away... this is bliss!” This line divides the entire eighth chapter into two parts: the first has 477 lines, and the second has 295 lines. Their ratio is 1.617! The finest correspondence to the value of the golden proportion! This is a great miracle of harmony accomplished by the genius of Pushkin!”

E. Rosenov analyzed many of the poetic works of M.Yu. Lermontov, Schiller, A.K. Tolstoy and also discovered the “golden ratio” in them.

Lermontov’s famous poem “Borodino” is divided into two parts: an introduction addressed to the narrator, occupying only one stanza (“Tell me, uncle, it’s not without reason...”), and the main part, representing an independent whole, which falls into two equal parts. The first of them describes, with increasing tension, the anticipation of the battle, the second describes the battle itself, with a gradual decrease in tension towards the end of the poem. The boundary between these parts is the culmination point of the work and falls exactly at the point of division by the golden section.

The main part of the poem consists of 13 seven-line lines, that is, 91 lines. Having divided it by the golden ratio (91:1.618=56.238), we are convinced that the division point is at the beginning of the 57th verse, where there is a short phrase: “Well, it was a day!” It is this phrase that represents the “culmination point of excited anticipation”, completing the first part of the poem (anticipation of the battle) and opening its second part (description of the battle).

Thus, the golden ratio plays a very meaningful role in poetry, highlighting the climax of the poem.

Many researchers of Shota Rustaveli’s poem “The Knight in the Skin of a Tiger” note the exceptional harmony and melody of his verse. These properties of the poem by the Georgian scientist, academician G.V. Tsereteli is attributed to the poet’s conscious use of the golden ratio both in the formation of the form of the poem and in the construction of its verses.

Rustaveli's poem consists of 1587 stanzas, each of which consists of four lines. Each line consists of 16 syllables and is divided into two equal parts of 8 syllables in each hemistich. All hemistiches are divided into two segments of two types: A - hemistich with equal segments and an even number of syllables (4+4); B is a hemistich with an asymmetrical division into two unequal parts (5+3 or 3+5). Thus, in the hemistich B the ratio is 3:5:8, which is an approximation to the golden proportion.

It has been established that in Rustaveli’s poem, out of 1587 stanzas, more than half (863) are constructed according to the principle of the golden ratio.

In our time, a new form of art was born - cinema, which absorbed the drama of action, painting, and music. It is legitimate to look for manifestations of the golden ratio in outstanding works of cinema. The first to do this was the creator of the world cinema masterpiece “Battleship Potemkin,” film director Sergei Eisenstein. In constructing this picture, he managed to embody the basic principle of harmony - the golden ratio. As Eisenstein himself notes, the red flag on the mast of the mutinous battleship (the climax of the film) flies at the point of the golden ratio, counted from the end of the film.

GOLDEN RATIO IN FONT AND HOUSEHOLD ITEMS

A special type of fine art of Ancient Greece should be highlighted in the production and painting of all kinds of vessels. In an elegant form, the proportions of the golden ratio are easily guessed.

In painting and sculpture of temples, and on household items, the ancient Egyptians most often depicted gods and pharaohs. The canons of depicting a person standing, walking, sitting, etc. were established. Artists were required to memorize individual forms and image patterns using tables and samples. The artists of Ancient Greece made special trips to Egypt to learn how to use the canon.

OPTIMAL PHYSICAL PARAMETERS OF THE EXTERNAL ENVIRONMENT

It is known that the maximum sound volume, which causes pain, is equal to 130 decibels. If we divide this interval by the golden ratio of 1.618, we get 80 decibels, which are typical for the volume of a human scream. If we now divide 80 decibels by the golden ratio, we get 50 decibels, which corresponds to the volume of human speech. Finally, if we divide 50 decibels by the square of the golden ratio 2.618, we get 20 decibels, which corresponds to a human whisper. Thus, all characteristic parameters of sound volume are interconnected through the golden proportion.

At a temperature of 18-20 0 C interval humidity 40-60% is considered optimal. The boundaries of the optimal humidity range can be obtained if the absolute humidity of 100% is divided twice by the golden ratio: 100/2.618 = 38.2% (lower limit); 100/1.618=61.8% (upper limit).

At air pressure 0.5 MPa, a person experiences unpleasant sensations, his physical and psychological activity worsens. At a pressure of 0.3-0.35 MPa, only short-term work is allowed, and at a pressure of 0.2 MPa, work is allowed for no more than 8 minutes. All these characteristic parameters are related to each other by the golden proportion: 0.5/1.618 = 0.31 MPa; 0.5/2.618=0.19 MPa.

Boundary parameters outside air temperature, within which the normal existence (and, most importantly, the origin has become possible) of a person is possible is the temperature range from 0 to + (57-58) 0 C. Obviously, there is no need to provide explanations for the first limit.

Let us divide the indicated range of positive temperatures by the golden section. In this case, we obtain two boundaries (both boundaries are temperatures characteristic of the human body): the first corresponds to the temperature, the second boundary corresponds to the maximum possible outside air temperature for the human body.

GOLDEN RATIO IN PAINTING

Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points, and they are located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.

This discovery was called the “golden ratio” of the painting by artists of that time.

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. His personality is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.”

He gained fame as an unsurpassed artist, a great scientist, a genius who anticipated many inventions that were not realized until the 20th century.

There is no doubt that Leonardo da Vinci was a great artist, this was already recognized by his contemporaries, but his personality and activities will remain shrouded in mystery, since he left to his descendants not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “about everything in the world."

He wrote from right to left in illegible handwriting and with his left hand. This is the most famous existing example of mirror writing.

The portrait of Monna Lisa (La Gioconda) has attracted the attention of researchers for many years, who discovered that the composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon. There are many versions about the history of this portrait. Here is one of them.

One day, Leonardo da Vinci received an order from the banker Francesco dele Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint the portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became lively and interesting.

FAIRY TALE. Once upon a time there lived one poor man, he had four sons: three were smart, and one of them was this and that. And then death came for the father. Before losing his life, he called his children to him and said: “My sons, I will soon die. As soon as you bury me, lock the hut and go to the ends of the world to find happiness for yourself. Let each of you learn something so that you can feed yourself.” The father died, and the sons dispersed around the world, agreeing to return to the clearing of their native grove three years later. The first brother came, who learned to carpenter, cut down a tree and hewed it, made a woman out of it, walked away a little and waited. The second brother returned, saw the wooden woman and, since he was a tailor, dressed her in one minute: like a skilled craftsman, he sewed beautiful silk clothes for her. The third son decorated the woman with gold and precious stones - after all, he was a jeweler. Finally, the fourth brother came. He did not know how to carpenter or sew, he only knew how to listen to what the earth, trees, grass, animals and birds were saying, he knew the movements of the celestial bodies and also knew how to sing wonderful songs. He sang a song that made the brothers hiding behind the bushes cry. With this song he revived the woman, she smiled and sighed. The brothers rushed to her and each shouted the same thing: “You must be my wife.” But the woman replied: “You created me - be my father. You dressed me, and you decorated me - be my brothers. And you, who breathed my soul into me and taught me to enjoy life, you are the only one I need for the rest of my life.”

Having finished the tale, Leonardo looked at Monna Lisa, her face lit up with light, her eyes shone. Then, as if awakening from a dream, she sighed, ran her hand over her face and without a word went to her place, folded her hands and assumed her usual pose. But the job was done - the artist awakened the indifferent statue; a smile of bliss, slowly disappearing from her face, remained in the corners of her mouth and trembled, giving her face an amazing, mysterious and slightly sly expression, like that of a person who has learned a secret and, carefully keeping it, cannot contain his triumph. Leonardo worked silently, afraid to miss this moment, this ray of sunshine that illuminated his boring model...

It is difficult to say what was noticed in this masterpiece of art, but everyone talked about Leonardo’s deep knowledge of the structure of the human body, thanks to which he was able to capture this seemingly mysterious smile. They talked about the expressiveness of individual parts of the picture and about the landscape, an unprecedented companion to the portrait. They talked about the naturalness of expression, the simplicity of the pose, the beauty of the hands. The artist did something unprecedented: the picture depicts air, it envelops the figure in a transparent haze. Despite the success, Leonardo was gloomy; the situation in Florence seemed painful to the artist; he got ready to go on the road. Reminders about the influx of orders did not help him.

The golden ratio in the painting by I.I. Shishkin "Pine Grove". In this famous painting by I.I. Shishkin clearly shows the motives of the golden ratio. A brightly sunlit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a sunlit hillock. It divides the right side of the picture horizontally according to the golden ratio. To the left of the main pine there are many pines - if you wish, you can successfully continue dividing the picture according to the golden ratio further.

Pine Grove

The presence in the picture of bright verticals and horizontals, dividing it in relation to the golden ratio, gives it a character of balance and calm in accordance with the artist’s intention. When the artist’s intention is different, if, say, he creates a picture with rapidly developing action, such a geometric composition scheme (with a predominance of verticals and horizontals) becomes unacceptable.

IN AND. Surikov. "Boyaryna Morozova"

Her role is given to the middle part of the picture. It is bound by the point of the highest rise and the point of the lowest decline of the plot of the picture: the rise of Morozova’s hand with the double-fingered sign of the cross as the highest point; a hand helplessly extended to the same noblewoman, but this time the hand of an old woman - a beggar wanderer, a hand from under which, along with the last hope of salvation, the end of the sledge slips out.

What about the “highest point”? At first glance, we have an apparent contradiction: after all, section A 1 B 1, spaced 0.618... from the right edge of the picture, does not pass through the hand, not even through the head or eye of the noblewoman, but ends up somewhere in front of the noblewoman’s mouth.

The golden ratio really cuts to the most important thing here. In him, and precisely in him, is Morozova’s greatest strength.

There is no painting more poetic than that of Botticelli Sandro, and the great Sandro has no painting more famous than his “Venus”. For Botticelli, his Venus is the embodiment of the idea of universal harmony of the “golden section” that dominates nature. The proportional analysis of Venus convinces us of this.

Venus

Raphael "The School of Athens". Raphael was not a mathematician, but, like many artists of that era, he had considerable knowledge of geometry. In the famous fresco “The School of Athens”, where in the temple of science there is a society of the great philosophers of antiquity, our attention is drawn to the group of Euclid, the greatest ancient Greek mathematician, analyzing a complex drawing.

The ingenious combination of two triangles is also constructed in accordance with the proportion of the golden ratio: it can be inscribed in a rectangle with an aspect ratio of 5/8. This drawing is surprisingly easy to insert into the top section of the architecture. The upper corner of the triangle rests on the keystone of the arch in the area closest to the viewer, the lower one on the vanishing point of the perspectives, and the side section indicates the proportions of the spatial gap between the two parts of the arches.

Golden spiral in Raphael's painting "Massacre of the Innocents". Unlike the golden ratio, the feeling of dynamics and excitement is manifested, perhaps, most strongly in another simple geometric figure - a spiral. The multi-figure composition, executed in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is precisely distinguished by the dynamism and drama of the plot. Raphael never brought his plan to completion, but his sketch was engraved by the unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the engraving “Massacre of the Innocents”.

Massacre of the innocents

If, in Raphael’s preparatory sketch, we mentally draw lines running from the semantic center of the composition - the point where the warrior’s fingers closed around the child’s ankle, along the figures of the child, the woman holding him close, the warrior with a raised sword, and then along the figures of the same group on the right side sketch (in the figure these lines are drawn in red), and then connect these pieces with a curved dotted line, then with very great accuracy a golden spiral is obtained. This can be checked by measuring the ratio of the lengths of the segments cut by a spiral on straight lines passing through the beginning of the curve.

GOLDEN RATIO AND IMAGE PERCEPTION

The ability of the human visual analyzer to identify objects constructed using the golden ratio algorithm as beautiful, attractive and harmonious has been known for a long time. The golden ratio gives the feeling of the most perfect whole. The format of many books follows the golden ratio. It is chosen for windows, paintings and envelopes, stamps, business cards. A person may not know anything about the number F, but in the structure of objects, as well as in the sequence of events, he subconsciously finds elements of the golden proportion.

Studies have been conducted in which subjects were asked to select and copy rectangles of various proportions. There were three rectangles to choose from: a square (40:40 mm), a “golden ratio” rectangle with an aspect ratio of 1:1.62 (31:50 mm) and a rectangle with elongated proportions 1:2.31 (26:60 mm).

When choosing rectangles in the normal state, in 1/2 of the cases preference is given to the square. The right hemisphere prefers the golden ratio and rejects the elongated rectangle. On the contrary, the left hemisphere gravitates towards elongated proportions and rejects the golden ratio.

When copying these rectangles, the following was observed: when the right hemisphere was active, the proportions in the copies were maintained most accurately; when the left hemisphere was active, the proportions of all rectangles were distorted, the rectangles were elongated (the square was drawn as a rectangle with an aspect ratio of 1:1.2; the proportions of the elongated rectangle increased sharply and reached 1:2.8). The proportions of the “golden” rectangle were most distorted; its proportions in copies became the proportions of a rectangle 1:2.08.

When drawing your own pictures, proportions close to the golden ratio and elongated ones prevail. On average, the proportions are 1:2, with the right hemisphere giving preference to the proportions of the golden section, the left hemisphere moving away from the proportions of the golden section and drawing out the pattern.

Now draw some rectangles, measure their sides and find the aspect ratio. Which hemisphere is dominant for you?

GOLDEN RATIO IN PHOTOGRAPHY

An example of the use of the golden ratio in photography is the placement of key components of the frame at points that are located 3/8 and 5/8 from the edges of the frame. This can be illustrated with the following example: a photograph of a cat, which is located in an arbitrary place in the frame.

Now let’s conditionally divide the frame into segments, in proportion to 1.62 total lengths from each side of the frame. At the intersection of the segments there will be the main “visual centers” in which it is worth placing the necessary key elements of the image. Let's move our cat to the points of the “visual centers”.

GOLDEN RATIO AND SPACE

From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, with the help of this series, found a pattern and order in the distances between the planets of the solar system.

However, one case that seemed to contradict the law: there was no planet between Mars and Jupiter. Focused observation of this part of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century. The Fibonacci series is widely used: it is used to represent the architectonics of living beings, man-made structures, and the structure of Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

The two Golden Spirals of the galaxy are compatible with the Star of David.

Notice the stars emerging from the galaxy in a white spiral. Exactly 180 0 from one of the spirals another unfolding spiral emerges... For a long time, astronomers simply believed that everything that is there is what we see; if something is visible, then it exists. They were either completely unaware of the invisible part of Reality, or they did not consider it important. But the invisible side of our Reality is actually much larger than the visible side and is probably more important... In other words, the visible part of Reality is much less than one percent of the whole - almost nothing. In fact, our real home is the invisible universe...

In the Universe, all galaxies known to mankind and all the bodies in them exist in the form of a spiral, corresponding to the formula of the golden ratio. The golden ratio lies in the spiral of our galaxy

CONCLUSION

Nature, understood as the whole world in the diversity of its forms, consists, as it were, of two parts: living and inanimate nature. Creations of inanimate nature are characterized by high stability and low variability, judging on the scale of human life. A person is born, lives, ages, dies, but the granite mountains remain the same and the planets revolve around the Sun the same way as in the time of Pythagoras.

The world of living nature appears to us completely different - mobile, changeable and surprisingly diverse. Life shows us a fantastic carnival of diversity and uniqueness of creative combinations! The world of inanimate nature is, first of all, a world of symmetry, giving his creations stability and beauty. The natural world is, first of all, a world of harmony, in which the “law of the golden ratio” operates.

In the modern world, science is of particular importance due to the increasing impact of humans on nature. Important tasks at the present stage are the search for new ways of coexistence between man and nature, the study of philosophical, social, economic, educational and other problems facing society.

This work examined the influence of the properties of the “golden section” on living and non-living nature, on the historical course of development of the history of mankind and the planet as a whole. Analyzing all of the above, you can once again marvel at the enormity of the process of understanding the world, the discovery of its ever new patterns and conclude: the principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. It can be expected that the laws of development of various natural systems, the laws of growth, are not very diverse and can be traced in a wide variety of formations. This is where the unity of nature is manifested. The idea of such unity, based on the manifestation of the same patterns in heterogeneous natural phenomena, has retained its relevance from Pythagoras to the present day.

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.

Every person who comes across the geometry of objects in space is well acquainted with the golden section method. It is used in art, interior design and architecture. Even in the last century, the golden ratio turned out to be so popular that now many supporters of the mystical vision of the world have given it a different name - the universal harmonic rule. The features of this method are worth considering in more detail. This will help you find out why he is interested in several fields of activity at once - art, architecture, design.

The essence of universal proportion

The principle of the golden ratio is just a relationship between numbers. However, many are biased towards it, attributing some mystical powers to this phenomenon. The reason lies in the unusual properties of the rule:

Many living objects have proportions of the torso and limbs that are close to the golden ratio.
Dependencies of 1.62 or 0.63 determine size ratios only for living beings. Objects related to inanimate nature very rarely correspond to the meaning of the harmonic rule.
The golden proportions of the body structure of living beings are an essential condition for the survival of many biological species.

The golden ratio can be found in the structure of the bodies of various animals, tree trunks and bush roots. Proponents of the universality of this principle are trying to prove that its meaning is vital for representatives of the living world.

You can explain the golden ratio method using the image of a chicken egg. The ratio of segments from points of the shell equally distant from the center of gravity is equal to the golden ratio. The most important indicator of an egg for the survival of birds is its shape, and not the strength of the shell.

Important! The golden ratio is calculated based on measurements of many living objects.

Origin of the golden ratio

The universal rule was known to the mathematicians of Ancient Greece. It was used by Pythagoras and Euclid. In the famous architectural masterpiece - the Cheops pyramid, the ratio of the dimensions of the main part and the length of the sides, as well as bas-reliefs and decorative details, correspond to the harmonic rule.

The golden section method was adopted not only by architects, but also by artists. The mystery of harmonic proportion was considered one of the greatest mysteries.

The first to document universal geometric proportion was the Franciscan monk Luca Pacioli. His abilities in mathematics were brilliant. The golden ratio received wide recognition after the publication of the results of Zeising's research on the golden ratio. He studied the proportions of the human body, ancient sculptures, and plants.

How to calculate the golden ratio

An explanation based on the lengths of the segments will help you understand what the golden ratio is. For example, inside a large one there are several small ones. Then the lengths of small segments are related to the total length of the large segment as 0.62. This definition helps to figure out how many parts a certain line can be divided into so that it corresponds to the harmonic rule. Another advantage of using this method is that you can find out what the ratio of the largest segment to the length of the entire object should be. This ratio is 1.62.

Such data can be represented as proportions of measured objects. At first they were sought out, selected empirically. However, now the exact relationships are known, so building an object in accordance with them will not be difficult. The golden ratio is found in the following ways:

Construct a right triangle. Break one of its sides, and then draw perpendiculars with secant arcs. When carrying out calculations, you should construct a perpendicular from one end of the segment equal to ½ of its length. Then a right triangle is completed. If you mark a point on the hypotenuse that shows the length of the perpendicular segment, then a radius equal to the remaining part of the line will cut the base into two halves. The resulting lines will relate to each other according to the golden ratio.
Universal geometric values are also obtained in another way - by building the Dürer pentagram. She is a star that is placed in a circle. It contains 4 segments, the lengths of which correspond to the golden ratio rule.
In architecture, harmonic proportion is used in a modified form. To do this, the right triangle should be divided along the hypotenuse.

Important! When compared to the classic concept of the golden ratio method, the version for architects has a ratio of 44:56.

If, in the traditional interpretation of the harmonic rule for graphics, it was calculated as 37:63, then for architectural structures 44:56 was more often used. This is due to the need to construct high-rise buildings.

The secret of the golden ratio

If in the case of living objects the golden ratio, manifested in the proportions of the body of people and animals, can be explained by the need to adapt to the environment, then the use of the rule of optimal proportions in the 12th century for building houses was new.

The Parthenon, preserved from the times of Ancient Greece, was built using the golden ratio method. Many castles of nobles of the Middle Ages were created with parameters corresponding to the harmonic rule.

Golden ratio in architecture

Many buildings from antiquity that have survived to this day confirm that architects from the Middle Ages were familiar with the harmonic rule. The desire to maintain harmonious proportion in the construction of churches, significant public buildings, and residences of royalty is very noticeable.

For example, Notre Dame Cathedral was built in such a way that many of its sections correspond to the golden ratio rule. You can find many works of architecture from the 18th century that were built in accordance with this rule. The rule was also applied by many Russian architects. Among them was M. Kazakov, who created projects for estates and residential buildings. He designed the Senate building and the Golitsyn hospital.

Naturally, houses with such a ratio of parts were built even before the discovery of the golden ratio rule. For example, such buildings include the Church of the Intercession on the Nerl. The beauty of the building becomes even more mysterious if we consider that the building of the Pokrovsk Church was erected in the 18th century. However, the building acquired its modern appearance after restoration.

In writings about the golden ratio it is mentioned that in architecture the perception of objects depends on who is observing. The proportions formed using the golden ratio give the most relaxed relationship between the parts of the structure relative to each other.

A striking representative of a number of buildings that comply with the universal rule is the architectural monument Parthenon, erected in the fifth century BC. e. The Parthenon is built with eight columns on the smaller facades and seventeen on the larger ones. The temple was built from noble marble. Thanks to this, the use of coloring is limited. The height of the building refers to its length 0.618. If you divide the Parthenon according to the proportions of the golden section, you will get certain protrusions of the facade.

All these structures have one similarity - a harmonious combination of forms and excellent quality of construction. This is explained by the use of the harmonic rule.

The importance of the golden ratio for humans

The architecture of ancient buildings and medieval houses is quite interesting for modern designers. This is due to the following reasons:

Thanks to the original design of houses, you can avoid annoying cliches. Each such building is an architectural masterpiece.
Mass application of the rules for decorating sculptures and statues.
By maintaining harmonious proportions, the eye is drawn to more important details.

Important! When creating a building project and creating an external appearance, medieval architects used universal proportions, based on the laws of human perception.

Today, psychologists have come to the conclusion that the principle of the golden ratio is nothing more than a human reaction to a certain ratio of sizes and shapes. In one experiment, a group of subjects were asked to bend a sheet of paper so that the sides had optimal proportions. In 85 out of 100 results, people bent the sheet almost exactly according to the harmonic rule.

According to modern scientists, the indicators of the golden section belong more to the sphere of psychology than to characterize the laws of the physical world. This explains why hoaxers are showing such interest in him. However, when constructing objects according to this rule, a person perceives them more comfortably.

Using the Golden Ratio in Design

The principles of using universal proportions are increasingly used in the construction of private houses. Particular attention is paid to maintaining optimal design proportions. Much attention is paid to the correct distribution of attention within the house.

The modern interpretation of the golden ratio no longer refers only to the rules of geometry and shape. Today, not only the dimensions of façade details, the area of rooms or the lengths of gables, but also the color palette used to create the interior are subject to the principle of harmonious proportions.

It is much easier to build a harmonious structure on a modular basis. Many departments and rooms in this case are constructed as separate blocks. They are designed in strict accordance with the harmonic rule. Constructing a building as a set of individual modules is much easier than creating a single box.

Many companies involved in the construction of country houses follow the harmonic rule when creating a project. This helps give clients the impression that the building's design has been carefully designed. Such houses are usually described as the most harmonious and comfortable to use. With the optimal choice of room areas, residents psychologically feel calm.

If the house is built without taking into account harmonious proportions, you can create a layout that, in terms of the ratio of wall sizes, will be close to 1:1.61. To do this, additional partitions are installed in the rooms, or furniture is rearranged.

Similarly, the dimensions of doors and windows are changed so that the opening has a width whose value is 1.61 times less than the height.

It is more difficult to choose color solutions. In this case, you can observe the simplified value of the golden ratio - 2/3. The main color background should occupy 60% of the room space. The shade takes up 30% of the room. The remaining surface area is painted with tones close to each other, enhancing the perception of the selected color.

The internal walls of the rooms are divided by a horizontal strip. It is placed 70 cm from the floor. The height of the furniture should be in a harmonious relationship with the height of the walls. This rule also applies to length distribution. For example, a sofa should have dimensions that are no less than 2/3 of the length of the partition. The area of the room occupied by pieces of furniture should also have a certain meaning. It relates to the total area of the entire room as 1:1.61.

The golden ratio is difficult to apply in practice due to the presence of only one number. That is why. I design harmonious buildings using a series of Fibonacci numbers. This ensures a variety of options for shapes and proportions of structural parts. The Fibonacci number series is also called the golden number. All values strictly correspond to a certain mathematical relationship.

In addition to the Fibonacci series, another design method is used in modern architecture - the principle laid down by the French architect Le Corbusier. When choosing this method, the starting unit of measurement is the height of the home owner. Based on this indicator, the dimensions of the building and internal premises are calculated. Thanks to this approach, the house is not only harmonious, but also acquires individuality.

Any interior will take on a more complete look if you use cornices in it. When using universal proportions, you can calculate its size. The optimal values are 22.5, 14 and 8.5 cm. The cornice should be installed according to the rules of the golden ratio. The small side of the decorative element should relate to the larger one as it relates to the added values of the two sides. If the large side is 14 cm, then the small side should be 8.5 cm.

You can add coziness to the room by dividing wall surfaces using plaster mirrors. If the wall is divided by a border, the height of the cornice strip should be subtracted from the remaining larger part of the wall. To create a mirror of optimal length, the same distance should be set back from the curb and cornice.

Conclusion

Houses built according to the golden ratio principle are indeed very comfortable. However, the price of building such buildings is quite high, since the cost of building materials increases by 70% due to the atypical sizes. This approach is not new at all, since most houses of the last century were created based on the parameters of the owners.

Thanks to the use of the golden ratio method in construction and design, buildings are not only comfortable, but also durable. They look harmonious and attractive. The interior is also designed according to universal proportions. This allows you to use space wisely.

In such rooms a person feels as comfortable as possible. You can build a house using the golden ratio principle yourself. The main thing is to calculate the loads on the building elements and choose the right materials.

The golden ratio method is used in interior design, placing decorative elements of certain sizes in the room. This allows you to give the room coziness. Color solutions are also chosen in accordance with universal harmonious proportions.

A person distinguishes objects around him by their shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the shape. The form, the construction of which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a feeling of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.

Golden ratio - harmonic proportion

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

Straight segment AB can be divided into two parts in the following ways:

into two equal parts - AB : AC = AB : Sun;

into two unequal parts in any respect (such parts do not form proportions);

thus, when AB : AC = AC : Sun.

The latter is the golden division or division of a segment in extreme and average ratio.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole

a : b = b : c or With : b = b : A.

Rice. 1. Geometric image of the golden ratio

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

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

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

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

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

x 2 - x - 1 = 0.

Solution to this equation:

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

Second golden ratio

The Bulgarian magazine "Fatherland" (No. 10, 1983) published an article by Tsvetan Tsekov-Karandash "On the second golden section", which follows from the main section and gives another ratio of 44: 56.

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

Rice. 3. Construction of the second golden ratio

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

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

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

Golden Triangle

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

Rice. 5. Construction of a regular pentagon and pentagram

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O- center of the circle, A- a point on a circle and E- the middle of the segment OA. Perpendicular to radius OA, restored at the point ABOUT, intersects the circle at the point D. Using a compass, plot a segment on the diameter C.E. = ED. The side length of a regular pentagon inscribed in a circle is DC. Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio.

Rice. 6. Construction of the golden triangle

We carry out a direct AB. From point A lay a segment on it three times ABOUT arbitrary value, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside the segments ABOUT. Received points d And d 1 connect with straight lines to a point A. Line segment dd put 1 on the line Ad 1, getting a point WITH. She split the line Ad 1 in proportion to the golden ratio. Lines Ad 1 and dd 1 is used to construct a “golden” rectangle.

History of the golden ratio

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

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.

Rice. 8. Antique golden ratio compass

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

During the Renaissance, interest in the golden division increased among scientists and artists due to its use in both geometry and art, especially in architecture. Leonardo da Vinci, an artist and scientist, saw that Italian artists had a lot of empirical experience, but little knowledge . He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Franceschi, who wrote two books, one of which was called “On Perspective in Painting.” He is considered the creator of descriptive geometry.

Luca Pacioli perfectly understood the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked in Milan at the Moro court at that time. In 1509, Luca Pacioli’s book “The Divine Proportion” was published in Venice with brilliantly executed illustrations, which is why it is believed that they were made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. Among the many advantages of the golden proportion, the monk Luca Pacioli did not fail to name its “divine essence” as an expression of the divine trinity - God the son, God the father and God the holy spirit (it was implied that the small segment is the personification of God the son, the larger segment - God the father, and the entire segment - God of the Holy Spirit).

Leonardo da Vinci also paid a lot of attention to the study of the golden division. He made sections of a stereometric body formed by regular pentagons, and each time he obtained rectangles with aspect ratios in the golden division. That's why he gave this division the name golden ratio. So it still remains as the most popular.

At the same time, in the north of Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches the introduction to the first version of the treatise on proportions. Dürer writes. “It is necessary that someone who knows how to do something should teach it to others who need it. This is what I set out to do.”

Judging by one of Dürer's letters, he met with Luca Pacioli while in Italy. Albrecht Durer develops in detail the theory of proportions of the human body. Dürer assigned an important place in his system of relationships to the golden section. A person's height is divided in golden proportions by the line of the belt, as well as by a line drawn through the tips of the middle fingers of the lowered hands, the lower part of the face by the mouth, etc. Dürer's proportional compass is well known.

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

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

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

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

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

Rice. eleven. Golden proportions in the human figure

Fibonacci series

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

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

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

Generalized golden ratio

The Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden division.

Scientists continued to actively develop the theory of Fibonacci numbers and the golden ratio. Yu. Matiyasevich solves Hilbert's 10th problem using Fibonacci numbers. Elegant methods are emerging for solving a number of cybernetic problems (search theory, games, programming) using Fibonacci numbers and the golden ratio. In the USA, even the Mathematical Fibonacci Association is being created, which has been publishing a special journal since 1963.

One of the achievements in this field is the discovery of generalized Fibonacci numbers and generalized golden ratios.

The Fibonacci series (1, 1, 2, 3, 5, 8) and the “binary” series of weights discovered by him 1, 2, 4, 8, 16... at first glance are completely different. But the algorithms for their construction are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 = 1 + 1; 4 = 2 + 2..., in the second - this is the sum of the two previous numbers 2 = 1 + 1, 3 = 2 + 1, 5 = 3 + 2.... Is it possible to find a general mathematical formula from which we obtain “ binary series and Fibonacci series? Or maybe this formula will give us new numerical sets that have some new unique properties?

Indeed, let us set the numerical parameter S, which can take any values: 0, 1, 2, 3, 4, 5... Consider a number series, S+ 1 of the first terms of which are units, and each of the subsequent ones is equal to the sum of two terms of the previous one and separated from the previous one by S steps. If n We denote the th term of this series by φ S ( n), then we obtain the general formula φ S ( n) = φ S ( n- 1) + φ S ( n - S - 1).

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

Overall golden S-proportion is the positive root of the golden equation S-sections x S+1 - x S - 1 = 0.

It is easy to show that when S= 0, the segment is divided in half, and when S= 1 - the familiar classical golden ratio.

Relations between neighbors S- Fibonacci numbers coincide with absolute mathematical accuracy in the limit with gold S-proportions! Mathematicians in such cases say that gold S-sections are numerical invariants S-Fibonacci numbers.

Facts confirming the existence of gold S-sections in nature, cites the Belarusian scientist E.M. Soroko in the book “Structural Harmony of Systems” (Minsk, “Science and Technology”, 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermal stable, hard, wear-resistant, resistant to oxidation, etc.) only if the specific gravities of the original components are related to each other by one of gold S-proportions. This allowed the author to put forward the hypothesis that gold S-sections are numerical invariants of self-organizing systems. Once confirmed experimentally, this hypothesis may be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

Using golden codes S-proportions can be expressed by any real number as a sum of powers of gold S-proportions with integer coefficients.

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

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

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

Principles of formation in nature

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

Rice. 12. Archimedes spiral

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

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

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

Rice. 13. Chicory

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

Rice. 14. Viviparous lizard

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

Nature has carried out division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.

Rice. 15. bird egg

Golden ratio and 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.