The golden ratio is simple, like everything ingenious. Imagine a segment AB divided by point C. You just need to place point C so that you can create the equality CB/AC = AC/AB = 0.618. That is, the number obtained by dividing the smallest segment CB by the length of the middle segment AC must coincide with the number obtained by dividing the middle segment AC by the length of the large segment AB. This number will be 0.618. This is the golden, or, as they said in ancient times, the divine proportion - f(Greek "phi") Excellence index.
It is difficult to say exactly when and by whom it was noticed that following this proportion gives a feeling of harmony. But as soon as people began to create something with my own hands, then we intuitively tried to maintain this ratio. Buildings constructed taking into account f, always looked more harmonious compared to those in which the proportions of the golden section were violated. This has been repeatedly verified by all kinds of tests.
In geometry there are two objects that are inextricably linked with f: regular pentagon (pentagram) and logarithmic spiral. In a pentagram, each line, intersecting with a neighboring one, divides it in the golden proportion, and in a logarithmic spiral, the diameters of neighboring turns are related to each other in the same way as segments AC and CB on our straight line AB. But f works not only in geometry. It is believed that parts of any system (for example, protons and neutrons in the nucleus of an atom) can be among themselves in a proportion corresponding to the golden number. In this case, scientists believe, the system turns out to be optimal. True, scientific confirmation of the hypothesis requires more than a dozen years of research. Where f cannot be measured instrumental method, use the so-called Fibonacci number series, in which each subsequent number is the sum of the two previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. The peculiarity of this series is that when dividing any of its numbers by the next one, the result is as close as possible to 0.618. For example, let's take the numbers 2,3 and 5. 2/3 = 0.666, and 3/5 = 0.6. In essence, the same relationship is present here as between the components of our segment AB. Thus, if the measuring characteristics of some object or phenomenon can be entered into the Fibonacci number series, this means that the golden proportion is observed in their structure. And there are countless such objects and systems, and modern science is discovering more and more new ones. So the question is whether f the truly divine proportion on which our world rests is not at all rhetorical.
Golden ratio in nature
The golden proportion is also observed in nature, and already at the simplest levels. Take, for example, the protein molecules that make up the tissues of all living organisms. Molecules differ from each other in mass, which depends on the number of amino acids they contain. Not long ago it was found that the most common proteins are with masses of 31; 81.2; 140.6; 231; 319 thousand units. Scientists note that this series almost corresponds to the Fibonacci series - 3, 8,13, 21, 34 (here scientists do not take into account the decimal difference of these series).
Surely, with further research, a protein will be found whose mass will correlate with 5. Even the structure of the simplest gives this confidence - many viruses have a pentagonal structure. Tend to f and proportions of chemical elements. Plutonium is closest to it: the ratio of the number of protons in its nucleus to neutrons is 0.627. Farthest away is hydrogen. In turn, the number of atoms in chemical compounds is surprisingly often a multiple of the Fibonacci series numbers. This is especially true for uranium oxides and metal compounds.
If you cut an unopened bud of a tree, you will find two spirals there, directed in different directions. These are the beginnings of leaves. The ratio of the number of turns between these two spirals will always be 2/3, or 3/5, or 5/8, etc. That is, again according to Fibonacci. By the way, we see the same pattern in the arrangement of sunflower seeds and in the structure of cones coniferous trees. But let's get back to the leaves. When they open up, they will not lose their connection with f, since they will be located on the stem or branch in a logarithmic spiral. But that's not all. There is a concept of “leaf divergence angle” - this is the angle at which the leaves are relative to each other. Calculating this angle is not difficult. Imagine that a prism with a pentagonal base is inscribed in the stem. Now run a spiral down the stem. The points at which the spiral will touch the faces of the prism correspond to the points from which the leaves grow. Now draw a straight line up from the first leaf and see how many leaves will lie on this straight line. Their number in biology is denoted by the letter n (in our case it is two leaves). Now count the number of turns the spiral describes around the stem. The resulting number is called the leaf cycle and is denoted by the letter p (in our case it is 5). Now we multiply the maximum angle - 360 degrees by 2 (n) and divide by 5 (p). We get the desired angle of divergence of the leaves - 144 degrees. The ratio of n and p to each plant or tree is different, but they all do not fall outside the Fibonacci series: 1/2; 2/5; 3/8; 5/13, etc. Biologists have found that the angles formed according to these proportions tend to infinity to 137 degrees - the optimal divergence angle at which sunlight is evenly distributed over the branches and leaves. And in the leaves themselves we can notice the observance of the golden proportion, as well as in flowers - it is easiest to notice it in those that have the shape of a pentagram.
f didn't go around animal world. According to scientists, the presence of the golden proportion in the structure of the skeleton of living organisms solves a very important problem. This achieves the maximum possible strength of the frame with minimal possible weight, which, in turn, makes it possible to rationally distribute matter among parts of the body. This applies to almost all representatives of the fauna. Thus, starfish are perfect pentagons, and the shells of many mollusks are logarithmic spirals. The ratio of the length of a dragonfly's tail to its body is also equal f. And the mosquito is not simple: it has three pairs of legs, its abdomen is divided into eight segments, and on its head there are five antennae - the same Fibonacci series. The number of vertebrae in many animals, such as a whale or a horse, is 55. The number of ribs is 13, and the number of bones in the limbs is 89. And the limbs themselves have a three-part structure. The total number of bones of these animals, counting the teeth (of which there are 21 pairs) and the bones of the hearing aid, is 233 (Fibonacci number). Why be surprised when even an egg, from which, as many peoples believe, everything came, can be inscribed in a rectangle of the golden ratio - the length of such a rectangle is 1.618 times its width.
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Geometry is an exact and quite complex science, which at the same time is a kind of art. Lines, planes, proportions - all this helps to create many truly beautiful things. And oddly enough, this is based on geometry in its most varied forms. In this article we will look at one very unusual thing that is directly related to this. The golden ratio is exactly the geometric approach that will be discussed.
The shape of an object and its perception
People most often rely on the shape of an object in order to recognize it among millions of others. It is by its shape that we determine what kind of thing lies in front of us or stands in the distance. We first recognize people by the shape of their body and face. Therefore, we can confidently say that the shape itself, its size and appearance is one of the most important things in human perception.
For people, the form of anything is of interest for two main reasons: either it is dictated by vital necessity, or it is caused by aesthetic pleasure from beauty. The best visual perception and feeling of harmony and beauty most often comes when a person observes a form in the construction of which symmetry and a special ratio were used, which is called the golden ratio.
The concept of the golden ratio
So, the golden ratio is the golden ratio, which is also a harmonic division. To explain this more clearly, let's look at some features of the form. Namely: a form is something whole, and the whole, in turn, always consists of some parts. These parts most likely have different characteristics, at least different sizes. Well, such dimensions are always in a certain relationship, both among themselves and in relation to the whole.
This means, in other words, we can say that the golden ratio is a ratio of two quantities, which has its own formula. Using this ratio when creating a form helps to make it as beautiful and harmonious as possible for the human eye.
From the ancient history of the golden ratio
The golden ratio is often used in many different areas of life today. But the history of this concept goes back to ancient times, when sciences such as mathematics and philosophy were just emerging. As a scientific concept, the golden ratio came into use during the time of Pythagoras, namely in the 6th century BC. But even before that, knowledge about such a ratio was used in practice in Ancient Egypt and Babylon. A clear indication of this are the pyramids, for the construction of which exactly this golden proportion was used.
New period
The Renaissance brought new breath to harmonic division, especially thanks to Leonardo da Vinci. This ratio has increasingly begun to be used both in geometry and in art. Scientists and artists began to study the golden ratio more deeply and create books that examine this issue.
One of the most important historical works related to the golden ratio is a book by Luca Pancholi called The Divine Proportion. Historians suspect that the illustrations of this book were done by Leonardo himself before Vinci.
golden ratio
Mathematics gives a very clear definition of proportion, which says that it is the equality of two ratios. Mathematically, this can be expressed by the following equality: a: b = c: d, where a, b, c, d are some specific values.
If we consider the proportion of a segment divided into two parts, we can encounter only a few situations:
- The segment is divided into two absolutely even parts, which means AB:AC = AB:BC, if AB is the exact beginning and end of the segment, and C is the point that divides the segment into two equal parts.
- The segment is divided into two unequal parts, which may be in the very different ratios among themselves, which means that here they are completely disproportionate.
- The segment is divided so that AB:AC = AC:BC.
As for the golden ratio, this is a proportional division of a segment into unequal parts, when the entire segment relates to the larger part, just as the larger part itself relates to the smaller one. There is another formulation: the smaller segment is related to the larger one, just as the larger one is to the entire segment. In mathematical terms, it looks like this: a:b = b:c or c:b = b:a. This is exactly what the golden ratio formula looks like.
Golden ratio in nature
The golden ratio, examples of which we will now consider, refers to incredible phenomena in nature. These are very beautiful examples of the fact that mathematics is not just numbers and formulas, but a science that has more than a real reflection in nature and our life in general.
For living organisms, one of the main tasks in life is growth. This desire to take one’s place in space, in fact, occurs in several forms - growing upward, almost horizontally spreading on the ground, or twisting in a spiral on some kind of support. And as incredible as it may be, many plants grow according to the golden ratio.
Another almost incredible fact is the relationships in the body of lizards. Their body looks quite pleasing to the human eye and this is possible due to the same golden ratio. To be more precise, the length of their tail relates to the length of the entire body as 62:38.
Interesting facts about the rules of the golden ratio
The golden ratio is a truly incredible concept, which means that throughout history we can come across many really interesting facts about this proportion. We present you some of them:
Golden ratio in the human body
In this section it is necessary to mention a very significant person, namely S. Zeizinga. This is a German researcher who has done a tremendous amount of work in the field of studying the golden ratio. He published a work entitled Aesthetic Studies. In his work he presented the golden ratio as absolute concept, which is universal for all phenomena both in nature and in art. Here we can recall the golden ratio of the pyramid along with the harmonious proportion of the human body and so on.
It was Zeising who was able to prove that the golden ratio, in fact, is the average statistical law for the human body. This was shown in practice, because during his work he had to measure a lot of human bodies. Historians believe that more than two thousand people took part in this experiment. According to Zeising's research, main indicator The golden ratio is the division of the body by the navel point. Thus, the male body with an average ratio of 13:8 is slightly closer to the golden ratio than the female body, where the golden ratio is 8:5. The golden ratio can also be observed in other parts of the body, such as the hand.
About the construction of the golden ratio
In fact, constructing the golden ratio is a simple matter. As we see, even ancient people coped with this quite easily. What can we say about modern knowledge and technologies of mankind. In this article we will not show how this can be done simply on a piece of paper and with a pencil in hand, but we will confidently declare that it is, in fact, possible. Moreover, this can be done in more than one way.
Since this is a fairly simple geometry, the golden ratio is quite simple to construct even at school. Therefore, information about this can be easily found in specialized books. By studying the golden ratio, 6th graders are fully able to understand the principles of its construction, which means that even children are smart enough to master such a task.
Golden ratio in mathematics
The first acquaintance with the golden ratio in practice begins with a simple division of a straight line segment in the same proportions. Most often this is done with the help of a ruler, compass and, of course, a pencil.
Segments of the golden proportion are expressed as an infinite irrational fraction AE = 0.618..., if AB is taken as one, BE = 0.382... In order to make these calculations more practical, very often they use not exact, but approximate values, namely - 0 .62 and .38. If the segment AB is taken as 100 parts, then its larger part will be equal to 62, and the smaller part will be equal to 38 parts, respectively.
The main property of the golden ratio can be expressed by the equation: x 2 -x-1=0. When solving, we get the following roots: x 1.2 =. Although mathematics is an exact and rigorous science, like its section - geometry, it is properties such as the laws of the golden section that cast mystery on this topic.
Harmony in art through the golden ratio
In order to summarize, let’s briefly consider what has already been discussed.
Basically, many pieces of art fall under the rule of the golden ratio, where a ratio close to 3/8 and 5/8 is observed. This is the rough formula of the golden ratio. The article has already mentioned a lot about examples of using the section, but we will look at it again through the prism of ancient and modern art. So, the most striking examples from ancient times:
As for the probably conscious use of proportion, starting from the time of Leonardo da Vinci, it came into use in almost all areas of life - from science to art. Even biology and medicine have proven that the golden ratio works even in living systems and organisms.
Airbrushing is based on the same "pillars" as other forms of art.
Our entire world can be described by numbers. Many numbers play such a significant role in this description that they have their own names: Pi, exponent (e), etc. Among these “nominal” numbers there is something quite remarkable. At different times, mathematicians, artists, and architects called it the “golden number,” “divine number,” and “divine section.” The term “golden ratio” was coined by Claudius Ptolemy, and it became popular thanks to Leonardo Da Vinci, who used it extensively in his works. People of art have noticed that the proportions of forms that are especially pleasing to the eye for perception are based on the “golden ratio”.
So what is this number? The golden ratio is the number Phi (Phi) equal to 1.61803. The number is named after the great ancient Greek sculptor Phidias, who used it in his sculptures. How to clearly demonstrate the principle of the “golden ratio”? Let's give a simple example. If you build a rectangle, one side of which is 1.618 times longer than the other, then the resulting aspect ratio is the “golden ratio”. The most common "golden rectangles" in modern world- these are credit cards. The human body is considered beautiful, and its proportions are considered ideal, if the ratio between the smaller and larger parts of the body is equal to the ratio between the larger part and the whole, that is, equal to the number Phi.
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The most famous mathematical work of ancient science is Euclid's Elements. It was from the “Principles” that the geometric problem “on the division of a segment in extreme and mean ratio” came to us. Which is the “Golden Ratio” itself.
The essence of the task is this:
Let us divide the segment AB by point C in such a ratio that the larger part of the segment CB is related to the smaller part of the segment AC as the segment AB is to its larger part CB, i.e.
Let us denote proportion (1.1) by x. Then, taking into account that AB = AC + CB, proportion (1.1) can be written in the following form:
This gives us the following algebraic equation for calculating the required proportion x:
X*
= x + 1. (1.2)
x* - squared
From the “physical meaning” of proportion (1.1) it follows that the desired solution to equation (1.2) must be a positive number, from which it follows that the solution to the problem of dividing a segment in extreme and mean ratio is the positive root of equation (1.2), which we denote by , that is
The approximate value of the golden ratio is:
= 1,61803 39887 49894 84820 45868 34365 63811 77203…
GOLDEN GEOMETRIC FIGURES
Based on the above proportions in geometry, the following concepts of golden geometric figures are defined:
- golden rectangle (in which the ratio of the larger side to the smaller side is equal to the golden ratio);
- golden right triangle;
- golden ellipse;
- golden isosceles triangle.
A right triangle with sides 3:4:5 is called "perfect", "sacred" or "Egyptian".
The creators of the Egyptian pyramids chose as the “main geometric idea"for the pyramid of Cheops - a golden right-angled triangle, and for the pyramid of Khafre - a “sacred” triangle.
Pentagon (“pentagonon” - Greek), regular pentagon. If we draw all the diagonals in the pentagon, the result is a pentagonal star called a pentagram (“pentagrammon” - Greek: “pente” - five and “grammon” - line) or pentacle.
The pentagram, called the “witch’s foot” in popular belief, played a large role in all magical sciences and was considered as a means of protection against evil spirits.
Every eight years, the planet Venus describes an absolutely regular pentacle along the great circle of the celestial sphere.
The Pentagon building, the US military department, is shaped like a Pentagon.
The Pentagon and Pentacle include a number of remarkable figures that have been widely used in works of art. The so-called law of the golden cup, which was used by ancient sculptors and goldsmiths, is widely known in ancient art. The shaded portion of the pentagon gives a schematic representation of the golden cup.
Once upon a time in the Soviet Union there was a State Quality Mark, in which the motifs of the golden cup are clearly visible.
In living nature, forms based on pentagonal symmetry are widespread - starfish, sea urchins, flowers..
HARMONY OF THE GOLDEN RATIO
(short review art history)
The great works of Greek sculptors: Phidias, Polyctetus, Myron, Praxiteles have long been rightfully considered the standard of beauty of the human body, an example of a harmonious physique. In their creations, Greek masters used the principle of the golden proportion. One of the highest achievements of classical Greek art is the statue of Doryphoros, sculpted by Polyctetus in the 5th century BC. e. This statue is considered the best example for analyzing the proportions of the ideal human body, established by ancient Greek sculptors, and is directly related to the Golden Ratio. M=0.618…
The Venus de Milo, a statue of the goddess Aphrodite and the standard of female beauty, is one of the best monuments of Greek sculptural art.
Leonardo Da Vinci used the Golden Ratio proportions in many of his most famous works, most notably The Last Supper and the famous La Gioconda.
Researchers of the painting “La Gioconda” discovered that the compositional structure of the painting is based on two golden triangles, their bases facing each other. Harmonic analysis of the picture shows that the pupil of the left eye, through which the vertical axis of the canvas passes, is located at the intersection of two bisectors of the upper golden triangle, which, on the one hand, bisect the angles at the base of the golden triangle, and on the other hand, at the points of intersection with the hips of the golden triangle triangles divide them in proportion to the Golden Ratio. Thus, Leonardo Da Vinci used in his painting not only the principle of symmetry, but also the Golden Ratio.
The painting “The Holy Family” by Michelangelo is recognized as one of the masterpieces of Western European art of the Renaissance. Harmonic analysis showed that the composition of the painting is based on a pentacle.
The proportions of the statue of David (by Michelangelo) are based on the Golden Ratio.
A striking example of Baroque architecture, the Smolny Cathedral in St. Petersburg makes an indelible impression. The Golden Ratio is also seen in its basic proportions.
In the famous painting “Ship Grove” by Ivan Shishkin, motifs of the Golden Ratio are visible. A brightly sunlit pine tree (standing in the foreground) divides the picture horizontally with the Golden Ratio. To the right of the pine tree is a hill lit by the sun. He divides the picture vertically with the Golden Ratio. To the left of the main pine tree there are many pine trees - you can continue dividing the Golden Ratio horizontally on the left side of the picture. 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.
Construction of the UN headquarters in New York was completed in 1943. The building then attracted everyone's attention not only as a public building created using the latest architectural means, but also as the first example of the use of a continuous solar modulating screen on one of the facades. This building also displays the Golden Ratio motifs. In the composition of the building, three golden rectangles placed on top of each other clearly stand out, which are its main architectural idea.
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. In the musical works of various composers, not just one Golden Ratio is usually stated, but a whole series of similar sections. Largest quantity works in which the Golden Ratio is present in Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), 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 primarily manifests itself as the presence of a certain moment in the poem (culmination, semantic turning point, main idea product) in the line at the division point total number lines of a poem in 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" - 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.
One of the modern forms of art is cinema, which incorporates dramaturgy of action, painting, and music. It is right 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.
For many millennia, the Golden Ratio has been the object of admiration and worship of outstanding scientists and thinkers: Pythagoras, Plato, Euclid, Luca Pacioli, Johannes Kepler, Pavel Florensky...
Currently, the Golden Ratio is a source of new fruitful ideas in mathematics and theoretical physics, biology and botany, economics and computer science...
The material is based on the book “The Da Vinci Code and Fibonacci Series” by A. Stakhov, A. Sluchenkova, I. Shcherbakov, published in 2007, publishing house “Peter”.
Geometry has two treasures: one of them is the Pythagorean theorem, and the other is the division of a segment in the mean and extreme ratio. The first can be compared to a measure of gold; the second one looks more like a precious stone.
I. Kepler
Did you know that when going to school or work, listening to music, doing housework, relaxing on vacation at sea or signing business contracts, we constantly come across examples of the golden ratio. Plants, animals, dishes and even some letters are built according to the principle of the golden ratio. The golden ratio has even been found in the DNA molecule.
I would like to introduce you closer to this incredible, in my opinion, phenomenon and tell you specifically where and how we encounter it and how we use it.
It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, an ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them. The French architect Le Corbusier found that in the relief from the temple of Pharaoh Seti I in Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the values of the golden division. The architect Khesira, depicted on a relief of a wooden board from a tomb named after him, holds in his hands measuring instruments in which the proportions of the golden division are recorded. The Greeks were skilled geometers. They even taught arithmetic to their children using geometric figures. The Pythagorean square and the diagonal of this square were the basis for the construction of dynamic rectangles.
What is the golden ratio, application of the golden ratio in mathematics.
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole a: b = b: c or c: b = b: a.
This proportion can be constructed as follows:
From point B we restore a perpendicular equal to half AB. The resulting point C is connected by a line to point A. On the resulting line we lay off a segment BC ending in point D. The segment AD is transferred to the line AB. The resulting point E divides the segment AB in the golden proportion.
The properties of the golden ratio are described by the equation: x*x – x – 1 = 0.
Solution to this equation:
In nature, a second golden ratio was also discovered, which follows from the main section and gives another ratio of 44:56. This proportion has been discovered in architecture, and also occurs when constructing compositions of images of an elongated horizontal format.
We divide this segment AB in the proportion of the golden section. From point C we restore the perpendicular CD. Using radius AB we find point D, then connect it with a line to point A. Divide the right angle ACD in half. From point C we draw a line to the intersection with AD. Let's call the resulting point the letter E, which divides the segment AD in the ratio 44:56.
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.
If the square AEFD is isolated from the golden rectangle ABCD, then the remaining part EBCF turns out to be a new golden rectangle, which can again be divided into the square GHCF and the smaller golden rectangle EBHG. By repeating this procedure many times, we will obtain an infinite sequence of squares and golden rectangles, which ultimately converge to point O. Note that such an endless repetition of the same geometric figures, that is, a square and a golden rectangle, gives us an unconscious aesthetic sense of rhythm and harmony. It is believed that this very circumstance is the reason that many rectangular-shaped objects that a person deals with (matchboxes, lighters, books, suitcases) often have the shape of a golden rectangle. For example, we use credit cards extensively in our Everyday life, but we don’t pay attention to the fact that in many cases credit cards have the shape of a golden rectangle.
Golden rectangle and credit card
Pentagram and Pentagon
If we draw all the diagonals in the pentagram, the result will be the well-known pentagonal star. It has been proven that the points of intersection of diagonals in a pentagram are always points of the golden ratio of diagonals. In this case, these points form a new pentagram FGHKL. In a new pentagram, diagonals can be drawn, the intersection of which forms another pentagram, and this process can be continued indefinitely. Thus, the pentagram ABCDE seems to consist of an infinite number of pentagrams, which are each time formed by the points of intersection of the diagonals. This endless repetition of the same geometric figure creates a sense of rhythm and harmony that is unconsciously recorded by our minds. The pentagram was especially admired by the Pythagoreans and was considered their main identification sign. The building of the US military department is shaped like a pentagram and is called the “Pentagon”, which means a regular pentagon.
So, I told you what the golden ratio is, and now, since my report is devoted to the application of the golden ratio, I will now talk about it.
The rabbit problem. Fibonacci numbers.
THE RABBIT PROBLEM
Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that after a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second month after his birth.
It is clear that if we consider the first pair of rabbits to be newborns, then in the second month we will still have one pair; for the 3rd month - 1+1=2; on the 4th month - 2 + 1 = 3 pairs (because of the two existing pairs, only one pair produces offspring); on the 5th month - 3+2=5 pairs (only 2 pairs born on the 3rd month will give birth to offspring on the 5th month); on the 6th month - 5 + 3 = 8 pairs (because only those couples born in the 4th month will produce offspring), etc.
From this problem followed the discovery of a certain series of sequence of natural numbers, each member of which, starting from the third, is equal to the sum of the two previous members: Uk = 1,1,2,3,5,8,13,21,34,55,89,144,233,377,. ,This sequence is called the Fibonacci Sequence, and its members are called Fibonacci numbers. The ratio of the next member of the series to the previous one tends to the golden ratio
In algebra, it is commonly denoted by the Greek letter phi.
The golden ratio has not bypassed humans either.
The golden ratio is the basis for constructing harmonious forms, as it is the absolute law of shape formation in nature, of which we are a part. The laws of harmony are numerical laws.
When modeling an ordinary person, we most likely do not take a ruler and a calculator to calculate the golden proportions. We simply intuitively feel these forms, because the forms of a human being come across our eyes more often than anything else, but when creating a model of an unusual creature, plant, structure, we should use knowledge of geometry and the golden ratio so that the result of the work can be looked at without disgust, although if it is the feeling of disgust that you are seeking, then you know what you have to do.
In any case, knowledge of the laws of nature (numerical laws) helps us achieve the desired result as quickly as possible.
The German professor Zeising did a great job in the mid-18th century: he measured more than 2000 bodies and suggested that the golden ratio expresses the average statistical law: dividing a body by the navel point is one of the main indicators 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.
in young children (about a year old) the proportion is 1:1.
Recently, our contemporary, American surgeon Stephen Marquart, created, using the principle of the “golden ratio,” a geometric mask that can serve as a standard for a beautiful face. To find out whether a face matches the ideal, just copy the mask onto transparent film and overlay it on a photograph of the appropriate size.
So, dividing the segment between the crown and the Adam's apple in relation to the “golden section”, we get a point lying on the line of the eyebrows (B). With further golden division of the resulting parts, we obtain sequentially the tip of the nose (C), the end of the chin (D).
Golden ratio in the human ear.
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 = 73º 43'.
Since the golden ratio has touched a person, I will say that it is present even in the structure of the DNA molecule.
All information about the physiological characteristics of living beings is stored in a microscopic DNA molecule, the structure of which also contains 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). 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.
Each of us, at least once in our lives, has been to the sea and held a spiral-shaped shell in our hands. Well, here it is: such a 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.
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.
Golden ratio in painting and photography.
In photography
When we want to take a beautiful photo, we often notice that we do not know how to mentally arrange objects so that they later look in the finished photograph in in the best possible way. The golden ratio rule can help us with this. Using horizontal and vertical lines, we mentally divide the viewfinder into nine identical sectors. The four central points of intersection of horizontal and vertical lines will be key for us.
Practical use of the Golden Ratio rule when composing a frame.
Below are various options for grids created on the basis according to the “Zloty section” rule, for various compositional options. In order to understand the principles, you need to experiment on your own, try and combine the grids with your photographs. Basic meshes look like this:
Here is a photo of a cat, which is located in a random place in the frame.
Now let’s conditionally divide the frame into segments, in the proportion of 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".
This is what the composition looks like now. Isn't it much better?
In order to understand the essence of the golden ratio, try to take a few photographs yourself of a person sitting on garden bench. Make sure that the most harmonious photo will be one in which the person is sitting not in the center or on the edge, but at a point corresponding to the golden ratio (dividing the bench in a ratio of approximately 2:3).
In painting
Masters Ancient Greece, who knew how to consciously use the golden proportion, which, in essence, is very simple, skillfully applied its harmonic values in all types of art and achieved such perfection in the structure of forms expressing their social ideals, which is rarely found in the practice of world art. All ancient culture passed under the sign of the golden proportion. They knew this proportion in Ancient Egypt. I will show this using the example of such painters as: Raphael, Leonardo da Vinci, Botticelli, Shishkin.
In Raphael's preparatory sketch, red lines are drawn 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 his sword raised, and then along the figures of the same group on the right side sketch. If you naturally connect these pieces with a curved dotted line, then you get very accurate results. golden spiral! This can be checked by measuring the ratio of the lengths of the segments cut by a spiral on straight lines that pass through the beginning of the curve. "Massacre of the Innocents" Raphael
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 corner touches the vanishing point of the perspectives, and the side section indicates the proportions of the spatial gap between the two parts of the arches.
LEONARDO da VINCI
The portrait of Mona Lisa (La Gioconda) by Leonardo da Vinci is attractive because the composition of the picture is built on “golden triangles”, more precisely on triangles that are pieces of a regular star-shaped pentagon.
“The Last Supper” is Leonardo’s most mature and complete work. In this painting, the master avoids everything that could obscure the main course of the action he depicts; he achieves a rare convincingness of the compositional solution. In the center he places the figure of Christ, highlighting it with the opening of the door. He deliberately moves the apostles away from Christ in order to further emphasize his place in the composition. Finally, for the same purpose, he forces all perspective lines to converge at a point directly above the head of Christ. Leonardo divides his students into four symmetrical groups, full of life and movement. He makes the table small, and the refectory - strict and simple. This gives him the opportunity to focus the viewer’s attention on figures with enormous plastic power. All these techniques reflect the deep purposefulness of the creative plan, in which everything is weighed and taken into account. "
Botticelli - "Birth of Venus"
The painting does not depict the birth of the goddess itself, but the moment that followed, when she, driven by the breath of the geniuses of the air, reaches the shore, where she is met by one of the graces. According to the ancient Greek poet Hesiod (Theogony, 188-200), Venus was born from the sea - from the foam produced by the genitals of castrated Uranus (SATURN), thrown into the water by Cronus. She floats to the shore in an open shell, driven by a soft breeze, and finally lands in Paphos (Cyprus) - one of the main places of veneration and cult in antiquity. Her Greek name Aphrodite possibly comes from aphros, which means "foam".
Near the island of Cythera, Aphrodite, daughter of Uranus, was born from the snow-white foam of sea waves. A light, caressing breeze brought her to the island of Cyprus. There the young Oras surrounded the goddess of love who emerged from the sea waves. They dressed her in gold-woven clothing and crowned her with a wreath of fragrant flowers. Wherever Aphrodite stepped, flowers grew magnificently. The whole air was full of fragrance. Eros and Himerot led the wondrous goddess to Olympus. The gods greeted her loudly. Since then, golden Aphrodite, forever young, the most beautiful of goddesses, has always lived among the gods of Olympus.
In this famous painting by I. I. Shishkin, the motifs of the golden ratio are clearly visible. 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 tree there are many pines - if you wish, you can successfully continue dividing the picture according to the golden ratio further.
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.
Golden ratio in architecture
Architecture is the ability of our consciousness to consolidate the feeling of an era in material forms. Le Corbusier
One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).
The figure shows a number of patterns associated with the golden ratio.
On the floor plan of the Parthenon you can also see the “golden rectangles”:
In the proportions of the Notre Dame Cathedral building in Paris we also see the golden proportion.
M. Kazakov used the “golden ratio” quite widely 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.
Many ancient sculptors used the rule of the golden proportion when constructing their works.
Consider this using the example of the statue of Apollo Belvedere: the umbilical line divides the height of the depicted person in relation to the golden ratio.
And a few more examples to prove that we observe the golden ratio in sculpture.
Doryphorus of Polykleitos and his harmonic analysis
Venus de Milo and its harmonic analysis
Michelangelo's David
6. Golden ratio in living nature
Everything in the world is connected to a single beginning:
In the movement of the waves - a Shakespearean sonnet,
In the symmetry of a flower are the foundations of the universe,
And in the singing of birds there is a symphony of planets.
Living nature in its development strived for the most harmonious organization, the criterion of which is the golden proportion, manifesting itself at a variety of levels - from atomic combinations to the structure of the bodies of higher animals.
Flowers and seeds of sunflower, chamomile, scales in pineapple fruits, conifer cones“packed” in logarithmic spirals, curling towards each other. Moreover, the numbers of “right” and “left” spirals always relate to each other, like neighboring Fibonacci numbers.
In the formulas for leaf arrangement (phyllotaxis) of many plants there are Fibonacci numbers arranged strictly regularly - through one, for example, hazel -1/3, oak, cherry - 2/5, sea buckthorn -5/13
Consider a chicory shoot. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again.
If the first emission is taken as 100 units, then the second is equal to 62 units, the third – 38, the fourth – 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.
Many butterflies and other insects have not avoided collisions with this remarkable, in my opinion, phenomenon of the golden ratio. The ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden proportion. Folding its wings, the moth forms a regular equilateral triangle. But as soon as she spreads her wings, you will see the same principle of dividing the body by 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.
Snowflakes are water crystals that are visible to our naked eye. They are incredibly beautiful and different in shape, but all their components are geometric shapes, and also, without exception, built on the principle of the golden proportion.
The golden ratio has even affected poetry and music.
In poetry
In the structure of each poem we cannot help but notice certain patterns, and, consequently, there are the golden proportion and Fibonacci numbers. Every second poem by A. S. Pushkin contains an example (pattern) of the golden ratio. And a sample (pattern) of mirror symmetry is in every third. One of the two patterns is found in two out of three poems (524 or 66%), and both patterns are found in every fifth poem (150 or 19%).
The main functions of the golden section in Pushkin’s works are:
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