What is common in the Egyptian pyramids, paintings "Mona Lisa" Leonardo da Vinci and Twitter and Pepsi logos?

Let's not pull with the answer - all of them are created using the Rules of the Golden section. The gold cross section is the ratio of two values \u200b\u200bof A and B, which are not equal to each other. This proportion is often found in nature, the golden section rule is actively used in visual art and design - compositions created using the "divine proportion", well balanced and, called, pleasant to the eyes. But what exactly is golden cross section And is it possible to use it in modern disciplines, for example, in web design? Let's figure it out.

A little mathematics

Suppose we have a certain section of AB, separated by a point C. Cut length ratio: AC / BC \u003d BC / AB. That is, the segment is divided into unequal parts in such a way that most of the segment is the same share as a whole, an undeveloped segment, which smaller cut is greater.

Such an unequal separation is called a golden cross section. The gold cross section is indicated by the symbol φ. The value φ is 1.618 or 1.62. In general, if we say quite simple, this division of a segment or any other value in relation to 62% and 38%.

The "divine proportion" was known to people from ancient times, this rule used in the construction of Egyptian pyramids and parfenon, the golden section can be found in the painting of the Sistine Chapel and in the pictures of Van Gogh. The golden cross section is widely used and today - examples that are constantly in front of our eyes are Twitter and Pepsi logos.

The human brain is designed in such a way that he considers beautiful those images or objects in which an unequal ratio of parts can be detected. When we talk about someone that "it is proportionally complicated," we, without knowing it, we mean a golden cross section.

Golden section can be used to various geometric pieces. If you take a square and multiply one of its side by 1.618, then we will get a rectangle.

Now, if you put a square on this rectangle, we will be able to see the line of the golden section:

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

It is not yet clear where this crushing of geometric shapes will lead to us. Still a little bit and everything will become clear. If in each of the squares of the scheme to carry out a smooth line, equal to the quarter of the circle, then we will get a gold spiral.

This is an unusual spiral. It is sometimes called the Spiral Fibonacci, in honor of the scientist who explored the sequence in which each number is early as the sum of the two previous ones. The bottom line is that this is a mathematical ratio visually perceived by us as a spiral, it is literally everywhere - sunflowers, sea shells, spiral galaxies and typhoons - everywhere there is a golden spiral.

How can I use a golden cross section in design?

So, the theoretical part is completed, go to practice. Is it really a golden cross section in design? Yes, you can. For example, in web design. Given this rule, you can get the right ratio of composite elements of the layout. As a result, all parts of the design, up to the smallest, will be harmoniously combined with each other.

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

Sites with two columns occur very often and this is not by chance. Here, for example, the site is National Geographic. Two columns, the rule of the golden section. Good design, ordered, balanced and taking into account the requirements of the visual hierarchy.

One more example. Moodley Design Studio has developed a corporate identity for the Festival of Performing Art in Bregense. When the designers worked on the billboard, they unequivocally enjoyed the Golden section rule in order to correctly determine the size and location of all elements and as a result of obtaining an ideal composition.

The Lemon Graphic Agency, which created the visual image for Terkaya Wealth Management, also used the 1,618 ratio and the Golden Spiral. Three business card design elements fit perfectly into the scheme, as a result of which all parts are very well combined with each other.

But another interesting use of gold spirals. Before us again, the site is National Geographic. If you look at the design more closely, you can see that the page has another NG logo, only smaller, which is located closer to the center of the spiral.

Of course, it is not by chance - the designers knew perfectly well what they were doing. This is a great place to duplicate the logo, as our eye, considering the site, naturally shifts the center of the composition. So the subconscious works and this must be considered when working on the design.

Golden Circles

"Divine proportion" can be applied to any geometric figures, including circles. If you enter a circle in squares, the ratio between which is 1: 1.618, then we will get golden circles.

Here is the logo Pepsi. Everything is clear without words. And the ratio, and how a smooth arc of a white logo element was obtained.

With the Twitter logo, everything is a bit more difficult, but it can be seen here that its design is based on the use of gold circles. It does not match the rule of "divine proportion", but for the most part all its elements fit into the scheme.

OUTPUT

As can be seen, despite the fact that the rule of the golden section is known from time immemorial, it is not at all over. Consequently, it can be used in design. It does not necessarily try to meet the scheme - the discipline design is inaccurate. But if you need to achieve a harmonious combination of elements, then try to apply the principles of the golden section does not prevent.

Cutting a square with a side and from a rectangle built on the principle of the golden section, we get a new, reduced rectangle with the same property

Golden. section (Golden proportion, division in extreme and middle terms, harmonic division, number of fidiy) - division of continuous magnitude to the part in such a respect, in which most of the way it relates to a smaller, as the whole value to the greater. For example, dividing segment AC into two parts in such a way that most of it AU refers to less Sun as the whole cut AC refers to AU (i.e. | AU| / |Sun| = |AC| / |AU|).

This proportion is made to denote by the Greek letter φ (the designation τ is also found). It is equal:

The formula "Golden Harmonies", which gives a pair of numbers satisfying the aforementioned proportion:

In the case of the number of parameter m. = 1.

In the antique literature, the division of a segment in extreme and middle respect (ἄκρος καὶ μέσος λόγος ) first meets in the "beginning of" Euclidea (approx. 300 BC), where it is used to build a correct pentagon.

C.aM The term "golden section" (it.goldener Schnitt.) was introduced by the German mathematician Martin Om in 1835.

Mathematical properties

Golden section in a five-pointed star

— irrational Algebraic number, positive solution of any of the following equations

it seems a chain fraction

for which suitable fractions are the relations of consecutive numbers of Fibonacci. In this way, .

In the right five-pointed star, each segment is divided by crossing it in a gold section (that is, the ratio of the blue segment to the green, as well as red to blue, as well as green to purple, equal).

Building a golden cross section

Here is another idea:

Geometric construction

Gold cross section cut AB can be built as follows: at point B. Restores perpendicular K. AB, put it on it segment BC.equal to half AB, on cut AC Decoration cut ADequal AC − CB., finally, on the segment AB Decoration cut AEequal AD. Then

Golden section and harmony

It is believed that the objects containing the Golden Section are perceived by people as the most harmonious. The proportions of the pyramid of Heops, temples, bas-reliefs, the objects of life and jewelry from the tomb of Tutankhamon allegedly indicate that Egyptian masters used the ratios of the golden section when creating them. Architect Le Corbusier "found" that in the relief from the Temple of the pharaoh of Network I in ABIDOS and in the relief, depicting the Pharaoh Ramses, the proportions of the figures correspond to the Golden section values. The architect hackira depicted on the relief of a wooden board from the tomb of his name, keeps in his hands the measuring instruments in which the proportions of the golden section are fixed. In the facade of the ancient Greek temple of Parfenon there are golden proportions. With its excavations, the circuses were found, which were used by architects and sculptors of the ancient world. In Pompeysky Circle (Museum in Naples), the proportions of gold division are also laid, and so on. Etc.

"Golden section" in art

Golden section and visual centers

Starting with Leonardo da Vinci, many artists consciously used the proportions of the Golden section.

It is known that Sergey Eisenstein artificially built a film of the battleship Potemkin according to the rules of the Golden section. He broke the ribbon for five parts. In the first three, the action unfolds on the ship. In the last two - in Odessa, where the uprising unfolds. This transition to the city occurs exactly at the point of the golden section. Yes, and in each part there is a fracture that happens on the law of the golden section. In the frame, scene, the episode occurs a certain leap in the development of the topic: the plot, mood. Eisenstein believed that, since such a transition is close to the point of the golden section, it is perceived as the most legitarily and natural.

Another example of using the Rules of Golden Section in Cinema is the location of the main components of the frame at high points - "visual centers." Four points are often used, located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.

It should be noted that in the examples described above, the approximate value of the "golden section" appeared: it is easy to make sure that no 3/2 5/3 is equal equal to the value of the golden cross section.

Russian architect Zholtovsky also used a golden cross section.

Criticism of the golden section

There are opinions that the significance of the golden section in art, architecture is exaggerated and is based on erroneous calculations.

When discussing the optimal relations of the parties of the rectangles (the size of the sheets of paper A0i multiples, the sizes of photoplastic (6: 9, 9:12) or photography frames (often 2: 3), the size of cinema and television screens - for example, 3: 4 or 9:16 ) Different options were tested. It turned out that most people do not perceive the golden Section as optimal and considers it the proportion "too elongated."

Number of readings: 8625

The gold cross section is a universal manifestation of structural harmony. It meets in nature, science, art - in everything, with which a person can come with. One day, having acquainted with the golden rule, humanity did not change him anymore.

Definition

The most capacious definition of the golden section states that a smaller part refers to a greater, as big - to all. Its approximate value is 1,6180339887. In a rounded percentage value of the proportion of parts of the whole will be correlated as 62% by 38%. This ratio acts in the forms of space and time.

Ancient saw in the golden section reflection of cosmic order, and Johann Kepler called him one of the treasures of geometry. Modern science considers the golden cross section as "asymmetric symmetry", calling it in a broad sense by a universal rule reflecting the structure and order of our world order.

HISTORY

The ancient Egyptians had an idea of \u200b\u200bthe golden proportions, they knew about them in Russia, but for the first time a scientific golden cross section explained the monk of Luke Pachet in the book "Divine proportion" (1509), the illustrations to which he allegedly made Leonardo da Vinci. Pachet saw divine trinity in the golden section: a small segment personified his son, a big father, and the whole - the Holy Spirit.

Directly with the Rule of the Golden section connected the name of the Italian mathematics Leonardo Fibonacci. As a result of a solution to one of the tasks, the scientist entered the sequence of numbers now known as a number of Fibonacci: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. At the ratio of this sequence to the golden proportion, the Kepler drew attention to: "It works so that two younger members of this endless proportion in the amount give the third dick, and any two of the last member, if they are folded, give the following member, and the same proportion is preserved to infinity " Now a series of fibonacci is an arithmetic basis for calculating the proportions of the golden section in all its manifestations.

Leonardo da Vinci also devoted a lot of time to study the features of the golden section, most likely, it is the term himself. Its drawings of the stereometric body formed by the correct pentagons prove that each of the rectangles obtained during the cross section gives the ratio of the parties in the gold division.

Over time, the Rule of the Golden section turned into an academic routine, and only the Adolf Ceasing philosopher in 1855 returned to his second life. He brought to the absolute of the proportion of the Golden section, making them versatile for all the phenomena of the surrounding world. However, his "mathematical aesthetics" caused a lot of criticism.

NATURE

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

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

More Archimedes, paying attention to the helix, brought the equation based on its form, which is now applied in the technique. Later Guete noted the nature of nature to spiral forms, calling the spiral of the "curve of life." Modern scientists were established that such manifestations of spiral forms in nature as a snail shell, sunflower seed location, web patterns, hurricane movement, structure of DNA and even the structure of galaxies enter into a series of fibonacci.

HUMAN

Fashion designers and designers of clothing All calculations are made based on the proportions of the golden section. Man is a universal form to verify the laws of the golden section. Of course, from nature, not all people have the proportions are ideal, which creates certain difficulties with the selection of clothing.

In the diary, Leonardo da Vinci has a picture inscribed in the circumference of a naked person, located in two positions superimposed on each other. Relying on the studies of the Roman architect Vitruvia, Leonardo was similar way to establish the proportions of the human body. Later, the French architect Le Corbusier, using the "Vitruvian man" Leonardo, created his own scale of "harmonic proportions", which influenced the aesthetics of the architecture of the XX century.

Adolf Ceying, exploring the proportionality of a person, has done a colossal work. He measured about two thousand human bodies, as well as many ancient statues and led that the golden cross section expresses the average law. In a person, almost all parts of the body are subordinated to him, but the main indicator of the golden section is the division of the body of the Pup point.
As a result of measurements, the researcher found that the proportions of the male body 13: 8 closer to the golden section than the proportions of the female body - 8: 5.

Art of spatial shapes

The artist Vasily Surikov said "that there is an immutable law in the composition, when there is nothing to remove anything in the picture, neither add, even the extra point is impossible, this is a real mathematics." For a long time, artists followed this law intuitively, but after Leonardo Da Vinci, the process of creating a picturesque web can no longer be without solving geometric tasks. For example, Albrecht Durer to determine the points of the golden section used the proportional circuit invented.

Art historian F.V. Kovlev, examining in detail the picture of Nikolai Ge Alexander Sergeevich Pushkin in the village of Mikhailovsky, notes that every detail of the canvas, whether it is a fireplace, a shelf, chair or poet himself, strictly inscribed in golden proportions.

The investigors of the golden section without tired and measured the masterpieces of architecture, claiming that they became such because they were created on gold canons: in their list, the Great Pyramids Giza, the Cathedral of the Parisian Mother of God, the Temple of Basil Blessed, Parfenon.

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

Word, sound and film

Forms temporarily? The arts in their own way demonstrate to us the principle of gold division. Literary critics, for example, noticed that the most popular number of lines in the poems of the late period of Pushkin's creativity corresponds to the Fibonacci row - 5, 8, 13, 21, 34.

There is a Rule of the Golden section and in separately taken works of Russian classics. So the climax of the "peak lady" is the dramatic scene of Herman and Countess, ending with the death of the latter. In the lead of 853 lines, and the climax accounts for 535 row (853: 535 \u003d 1.6) - this is the point of the golden section.

The Soviet musicologist E.K. Tyunov notes the striking accuracy of the ratios of the golden section in the strict and free forms of the works of Johann Sebastian Baha, which corresponds to a thoughtful, concentrated, technically verified style of the master. This is true in relation to the outstanding creations of other composers, where the most bright or unexpected musical solution is usually accounted for on the golden section.

Film director Sergei Eisenstein Scenario of his film "Bramenos Potemkin" consciously agreed with the Rule of Golden Section, dividing the tape for five parts. In the first three sections, the action unfolds on the ship, and in the last two - in Odessa. Go to scenes in the city and there golden mean Film.

Modern web design includes 2 features that should be clearly observed: aesthetics and the right area of \u200b\u200bapplication. If you follow these concepts, web design can be considered successful.

As for the aesthetics, it means that draws one or another object image, we use a variety of different manipulations: creating a mesh, layout, use typographical methods to obtain a good item structural structure. It is important to maintain a feeling of harmony, order and visual equilibrium with any graphic processing. This will help us with a "golden section" and "Rule of Three".

Surely, earlier you heard about these concepts. And perhaps, you have an idea, in what specific projects they can be used. The "Golden section" and "Rule of Three" applies to change the image and present it in the best form than it is actually. Such technologies help improve even the most primitive picture.

Let's stay in more detail on these features and find out what areas of web design can be applied.

What is a "golden section" and how did it appear?

This term at first glance may be incomprehensible. Why is the "golden"? What to apply this technology? Today, still remains a mystery, who invented the "Golden section", from where it originated. However, it is known that the technology has been used for 2400 years. It should also be noted that the golden section is used in various industries of science: in astronomy, mathematics, architecture, music, painting and many others.

The gold cross section is formed by the example of a simple mathematical equation that shows the relationship. In the simplest mathematical form, this attitude is as follows:

As you can see, it is a unique equation that shares the relationship between the two dimensions and the proportions of the lines. In decimal calculus B divided by A is 1.618033 ... if a\u003e b. In the example below, it is assumed that B is equal to 5. Then the equation will look like this:

Perhaps earlier you have already heard of Fibonacci sequence. How does it really work? For example, there are a number of numbers in which any specified number is created by adding the two previous ones. Starting with 0, then the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... etc.:

The written expression is presented in the formula form: xn \u003d xn-1 + xn-2.

The sequence is closely related to the gold proportion, because if you take any two consecutive numbers and divide to the previous one, the fraction will be very close to the golden proportion. As the value of the number increases, the fraction becomes even closer to the golden proportion. For example, 8/5 1.6, 34/21 is 1.619 and so on.

"Golden Spiral". Rectangle

So, you probably met such equations. But for what designers use geometry in their projects? Why do you need overlay figures? The scheme is called "Spiral Fibonacci." It is actually quite simple and is the most optimal for many geometric shapes. The spiral is created using a quarter of circles, which are drawn inside the square array based on Fibonacci sequence.

The diagram below shows a sample:

It turns out that each subsequent radius is greater than the previous one close to the golden section. The resulting helix is \u200b\u200bused in many areas, more often in drawing and architecture, but it can also be observed in natural phenomena.

"Rule three"

This rule is one of the options for gold spirals and is often used when cropping photos, video. Used to trim the frames and give them aesthetic species. To apply the "rule of three", it is necessary to divide the image into 9 equal parts. Conduct 2 horizontal lines and 2 vertical. It is important to arrange them smoothly. The meaning is to align the focus with the extreme left vertical separator. The horizon or the "gathering point" should be at a level with a horizontal separator.

Application "Golden Spiral"

As it was already noted, the Fibonacci sequence is closely related to the gold proportion. The use of the golden section is performed using the painted spiral. The image shows an example of using this method. So, we see a rectangle, the basis of which extends from the right wrist of a woman to her left elbow.

The rectangle expands vertically until it reaches the macushkin. If we draw squares inside the golden rectangle, all important parts of the woman are on the edges of the inner squares: its chin, eyes and lips. Leonardo da Vinci used a set of times a golden cross section in his works. Below examples of gold spirals in nature and space.

Application in web design

Many designers make a mistake, believing that by usual division or multiplication by 1.61 ... You can get a harmonious proportion. It is far enough, it is just the basis of the process. It is impossible to simply use one or another number and get a magic proportion. However, there are certain ways that help get a golden cross section. Some artists tend to think that the theory of golden section is a myth. We give another example, how the golden section works. Take the site prototype and consider the use of the golden section on it.

It looks pretty simple, right? Yes, in fact it is. The design is based on a 960-pixel grid. Design is presented using a golden proportion. If you use 1 gold spiral that covers 960px, it can be seen as a headline, the Logo ITD.

We move our spiral below and rely on its size

A certain cascade of spirals is obtained in which the basic design elements are inscribed in rectangles with a gold proportion.

The grid based on the golden section has a number of proportional relations inside it, which are made clearly proportionally inside the rectangle. At the bottom of this article, I attached the PSD file that my example contains, you can try to use it in your project for experimenting with a golden section.

People for a long time worried about the question whether such elusive things are subject to beauty and harmony by any mathematical calculations. Of course, all the laws of beauty cannot be accommodated in several formulas, but, studying mathematics, we can open some of the well-known - golden cross section. Our task is to know what kind of golden section and establish - where humanity has found the use of a golden section.

You probably paid attention to that we are not the same in the subjects and phenomena of the surrounding reality. B. z.decency, Be. z.form, disproportion is perceived by us as ugly and produce repulsive impression. And objects and phenomena, which are peculiar to measure, feasibility and harmony are perceived as beautiful and cause us a feeling of admiration, joy, raise the mood.

A person in its activities is constantly facing objects that are based on a golden cross section. There are things that cannot be explained. Here you come to an empty bench and sit on it. Where will you sit down? In the middle? Or maybe from the very edge? No, most likely, not something else. You will sit down so that the attitude of one piece of the bench to another relative to your body will be approximately 1.62. A simple thing, absolutely instinctive ... sitting on the bench, you played the "golden cross section".

Ancient Egypt and Babylon, in India and China, knew about the golden section, in India and China. The Great Pythagoras created a secret school, where the mystical essence of the "golden section" was studied. Euclid applied him, creating his geometry, and fidium is its immortal sculptures. Plato told that the Universe was arranged according to the "Golden section". Aristotle found the compliance of the "golden section" with an ethical law. Leonardo da Vinci and Michelangelo will preach the highest harmony of the "golden section" and Michelangelo, because the beauty and "golden cross section" are the same. And Christian mystics will draw on the walls of their monasteries of the Pentagram "Golden section", fleeing the devil. At the same time, scientists - from Pachet to Einstein - will search, but they will not find its exact meaning. B. z.the final series of comma is 1,6180339887 ... Strange, mysterious, inexplicable thing - this divine proportion is mystically accompanied by everything alive. Non-fat nature does not know what the Golden section is. But you will certainly see this proportion and in the bends of sea shells, and in the form of flowers, and in the guise of beetles, and in a beautiful human body. All the living and everything is beautiful - everything obeys the divine law, the name of which is the "golden cross section". So what is the "golden section"? What is this perfect, divine combination? Maybe this is the law of beauty? Or is he still a mystical mystery? Scientific phenomenon or ethical principle? The answer is unknown so far. More precisely - no, known. "Golden section" is both both, and third. Only not separately, but at the same time ... and this is his genuine mystery, his great mystery.

Probably it is difficult to find a reliable measure for an objective assessment of the beauty itself, and one logic will not cost here. However, the experience of those for whom the search for beauty was the meaning of life, who did it with their profession himself. This is, above all, people of art, as we call them: artists, architects, sculptors, musicians, writers. But these are the people of the exact sciences, above all, are mathematics.

Trusting the eye more than other senses, a person first and foremost studied to distinguish between his objects in shape. Interest in the form of a subject can be dictated by the vital necessity, and may be caused by the beauty of the form. The form, the basis of the construction of which lies the combination of symmetry and the golden section, promotes the best spectacular perception and the appearance of the feeling of beauty and harmony. The integer always consists of parts, part of different values \u200b\u200bare in a certain relation to each other and to the whole. 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.

Golden section - harmonic proportion

In mathematics, the equality of two relations is called the proportion:

Cut the straight AB can be divided into two parts in the following ways:

• into two equal parts - AV: AS \u003d AV: Sun;
• on two unequal parts in any respect (such parts of the proportion are not formed);
• thus, when AV: AC \u003d AC: Sun.

Last and there is a golden division (section).

Golden section - this is a proportional division of a segment to unequal parts, in which the entire segment belongs to most of the way, as most of them belongs to the smaller, in other words, a smaller cut is so relate to more as greater than

a: B \u003d B: C or C: B \u003d B: a.

Geometric image of golden proportion

A practical acquaintance with the golden section starts from the division of a straight line in a gold proportion with a circulation and a ruler.

The division of the segment is straight along the golden section. Bc \u003d 1 / 2ab; CD \u003d BC.

From the point in restoring perpendicular equal to half AB. The resulting point with the line is connected to the point A. On the resulting line, the segment of the aircraft is postponed, ending with a point D. Segment AD is transferred to direct AV. The resulting point E divides the segment AB in the ratio of the gold proportion.

Segments of gold proportions are expressed be z.the ultimate fraction of AE \u003d 0.618 ..., if AV adopt a unit, be \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment is taken for 100 parts, then most of the segment is 62, and less than 38 parts.

The properties of the gold cross section are described by the equation:

Solution of this equation:

The properties of the Golden section created a romantic halo of mysteriousness around this number and almost a mystical generation. For example, in the correct five-pointed star, each segment is divided into its segment in the proportion of the golden section (i.e., the ratio of the blue segment to the green, red to blue, green to purple, is equal to 1.618).

Second golden cross section

Such a proportion is found in architecture.

Building a second golden section

The division is as follows. Cut AB is divided in the proportion of the golden section. From the point C is the perpendicular CD. The AV radius is point D, which connects the line with a point A. The forward an ACD angle is divided in half. From the point C is a line before crossing the AD line. Point E divides segment AD in relation to 56:44.

The division of the rectangle of the line of the second golden section

The figure shows the position of the line of the second golden section. It is in the middle between the line of the golden section and the middle line of the rectangle.

Golden Triangle (Pentagram)

To find the segments of the gold proportion of the ascending and downward rows, you can use the pentagram.

Building the right pentagon and pentagram

To build a pentagram, it is necessary to build the right pentagon. The way to build it has developed a German painter and chart of Albrecht Dürer. Let o be the center of the circle, A - the point on the circle and E is the middle of the segment OA. Perpendicular to the OA radius, subscribed at the point O, intersects with a circle at D. Using the circulation, postpone the segment of Ce \u003d ED on the diameter. The length of the side inscribed in the circumference of the correct pentagon is equal to DC. We put on the circumference of the segment DC and we get five points to draw the right pentagon. We connect the corners of the pentagon through one diagonals and get a pentagram. All diagonals of the pentagon shall form each other for segments related to the golden proportion.

Each end of a pentagonal star is a golden triangle. Its parties form an angle of 36 0 at the top, and the base, pending on the side divides it in the proportion of the golden section.

We carry out a straight AV. From the point and lay it on it three times the segment of an arbitrary value, through the resulting point P we carry out perpendicular to the AB line, on the perpendicular to the right and left on the point p laying the segments of O. The obtained points D and D 1 connect straight to the point A. Cut DD 1 I put it on the AD 1 line, getting the point S. It divided the AD 1 line in the proportion of the golden section. Lines AD 1 and DD 1 use to build a "golden" rectangle.

Building a golden triangle

History of golden section

And indeed, the proportions of the pyramid of Heops, temples, household items and jewelry from the tombs of Tutankhamon show that Egyptian masters used the ratios of gold division when creating them. The French architect Le Corbusier found that in the relief from the pharaoh of Network I in ABIDOS and in the relief depicting the Pharaoh Ramses, the proportions of the figures correspond to the values \u200b\u200bof gold division. The architect hashire depicted on the relief of a wooden board from the tomb of his name, keeps in the hands of measuring instruments in which the proportions of gold division are fixed.

The Greeks were skilled geometers. Even arithmetic trained their children with the help of geometric shapes. The square of Pythagora and the diagonal of this square were the basis for building dynamic rectangles.

Dynamic rectangles

Plato also knew about the golden division. Pythagorean Timy in the Plato dialogue says: "It is impossible that two things are perfectly connected to without the third, since a thing that fasten them should appear between them. This can be performed in the best way proportion, because if three numbers have the property that the mean is as follows to a smaller, as greater than the average, and, on the contrary, the smaller way refers to the average, as the average to more, then the last and the first will be average, and The average is the first and last. Thus, everything you need will be the same, and since it will be the same, then it will be a whole. " The earthly world Plato is building using the triangles of two varieties: anamental and unequal. Beautiful rectangular triangle He considers such in which hypotenuse is twice as much smaller of the cathetov (such a rectangle is half an equilateral, the main figure of Babylonians, in it is the ratio of 1: 3 1/2, which differs from the golden section by about 1/25, and called the Timerding "Rival Golden sections "). With the help of triangles, Plato builds four regular polyhedra, associating them with four terrestrial elements (earth, water, air and fire). And only the last of five existing correct polyhedra - a dodecahedron, all the twelve grades of which the right pentagons serve as a symbolic image of the heavenly world.

Ikosahedron and Dodecahedron

The honor of the discovery of the Dodecahedron (or, as the most universe itself, this quintessence of four elements, symbolized, respectively, tetrahedrome, octahedrome, Ikosahedrome and cube) belongs to Hippas, subsequently the dead during shipwreck. In this figure, many ratios of the golden section are really captured, so the main role was given to the main role in the heavenly world, which later insisted Brother Minject Luka Pacheti.

In the facade of the ancient Greek temple of Parfenon there are golden proportions. With its excavations, the circuses were found, which were used by architects and sculptors of the ancient world. In Pompeary Circle (Museum in Naples), the proportions of gold division are also laid.

Antique Circle Golden Section

In the antique literature, the gold division was first mentioned in the "beginning of" Euclide. In the 2nd book "Beginning" gives a geometric construction of gold division. After Euclide, the Golden Delival study was engaged in the gypsy (II century BC), PAPP (III century. AD) and others. In medieval Europe with gold division, they met the Arab translation "began" Euclida. Translator J. Campano from Navarre (III century) Made to transfer comments. The secrets of the golden division were jealously worried, stored in a strict mystery. They were known only dedicated.

In the Middle Ages, the Pentagram was demoned (as, however, much, which was revered by the Divine in the ancient paganism) and found shelter in the occult sciences. However, the revival again puts on the light and the pentagram, and the golden cross section. So, wide circuit in that period of statement of humanism found a scheme describing the structure of the human body.

To such a picture, in fact, Leonardo da Vinci repeatedly resorted to the pentagram. Its interpretation: the human body has divine perfection, because the proportions embedded in it are the same as in the main celestial figure. Leonardo da Vinci, an artist and a scientist, saw that Italian artists empiric experience is big, and there are few knowledge. He conceived and began writing a book on geometry, but at that time the book of the monk Luke Pacheti appeared, and Leonardo left his venture. According to contemporaries and historians of science, Luka Pacheti was a real luminary, the greatest mathematician of Italy in the period between Fibonacci and Galileem. Luka Pacheli was a student of the artist Piero della Franni, who wrote two books, one of which was called "about the future in painting." He is considered the creator of the descriptive geometry.

Luka Pacheti perfectly understood the importance of science for art.

In 1496, at the invitation of the Duke of Moro, he comes to Milan, where he lectures in mathematics. In Milan, at the courtyard of Mora, Leonardo da Vinci worked at that time. In 1509, in Venice, the book of Luke Pachet "On Divine Proportion" was published (De Divina Proportione, 1497, ed. In Venice in 1509) with brilliantly fulfilled illustrations, in view of which they believed that Leonardo da Vinci did them. The book was an enthusiastic anthem of the golden proportion. Such proportion is only one, and the uniqueness is the highest property of God. In it embodied the holy trinity. This proportion cannot be expressed by an affordable number, remains hidden and mystery and the mathematicians themselves is called irrational (so God cannot be determined nor clarified by words). God never changes and represents everything in everything and everything in every one of its part and the golden section for any continuous and definite value (regardless of whether it is large or small) the same thing can not be changed nor In one perceived by reason. God called for the Being of Heavenly Virtue, otherwise called the fifth substance, with its help and four other simple bodies (four elements - land, water, air, fire), and on their basis caused every other thing in nature; So our sacred proportion, according to Plato in Timea, gives a formal being to the very sky, because it is attributed to the type of body called a dodecahedron, which cannot be built without a golden section. These are the arguments of Pachet.

Leonardo da Vinci also paid a lot of attention to studying the golden division. He produced a sequence of a stereometric body formed by the right pentagons, and each time he received rectangles with the relations of the parties in the gold division. Therefore, he gave this division the name of the golden cross section. So it still keeps the most popular.

At the same time in the north of Europe, in Germany, Albrecht Durer worked on the same problems. It makes an outline of introduction to the first variant of the treatise on the proportions. Durer writes: "It is necessary that anyone who knows anything, taught them the others who need it. That I am removed to do. "

Judging by one of the letters of Durera, he met with Luke Pachet during his stay in Italy. Albrecht Dürer develops in detail the theory of the proportions of the human body. An important place in its system relations Durer took the golden cross section. The human growth is divided into golden proportions of the belt line, as well as a line spent through the tips of the middle fingers of lowered hands, the lower part of the face - mouth, etc. The proportional durer circuit is known.

Great astronomer XVI century. Johan Kepler called the Golden section by one of the treasures of geometry. It first draws attention to the value of the golden proportion for botany (plant growth and their structure).

Kepler called the gold proportion continuing himself "it works so," he wrote, - that two younger members of this endless proportion in the amount give the third dick, and any two last member, if they are folded, give the following member, and the same proportion remains to Infinity. "

The construction of a series of segments of the golden proportion can be made both towards the increase (increasing row) and in the direction of the reduction (downward row).

If on a straight arbitrary length, postpone the segment m. , Next lay off the segment M. . Based on these two segments, we build the scale of segments of the gold proportion of the ascending and downward rows.

Building a gold proportion segment scale

In the next centuries, the Golden proportion rule turned into an academic canon and, when the fight against academic routine began in art, in the art of the struggle, "together with water, a child was splashed." The golden cross section was "openly" again in the middle of the XIX century.

In 1855, the German investigator of the Golden section, Professor Zeasing, published his work "aesthetic research." It was exactly what was happening with Cainling, which was to inevitably happen to occur with the researcher, who considers the phenomenon as such, without communication with other phenomena. He absolutized the proportion of the Golden section, declaring it universal for all the phenomena of nature and art. Casesing had numerous followers, but there were opponents who declared his doctrine on the proportions of "mathematical aesthetics".

Ceasing has done a colossal job. He measured about two thousand human bodies and came to the conclusion that the golden cross section expresses the average statistical law. Body division Pup point is the most important indicator of the golden section. The proportions of the male body fluctuate within the middle ratio of 13: 8 \u003d 1.625 and are somewhat closer to the golden section than the proportions of the female body, in respect of which the average value of the proportion is expressed in the ratio of 8: 5 \u003d 1.6. In a newborn, the proportion is ratio of 1: 1, by 13 years it is equal to 1.6, and by 21 years equals male. The proportions of the golden section appear in relation to other parts of the body - the length of the shoulder, forearm and brushes, brushes and fingers, etc.

The justice of its theory Ceasing was checked in Greek statues. In the most detailed, he developed the proportions of Apollo Belvedere. We have been studied by Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, poetic sizes. Cainling gave the definition of the golden section, showed how it is expressed in the segments of direct and in numbers. When the numbers expressing the lengths of the segments were obtained, Cainsing saw that they constitute a number of fibonacci, which could be continued to infinity in one and the other side. The next book was the name "Golden division as a major morphological law in nature and art." In 1876, a small book was published in Russia, almost a brochure, with the statement of this work of Ceasing. The author covered under the initials Yu.F.V. In this edition, no product of painting is not mentioned.

IN late XIX. - early XX centuries. A lot of purely formalistic theories appeared on the use of a gold section in works of art and architecture. With the development of design and technical aesthetics, the law of the Golden section has spread to the design of machines, furniture, etc.

Golden section and symmetry

The golden cross section cannot be viewed in itself, separately, without a symmetry. Great Russian crystallograph of G.V. Wulf (1863-1925) considered the golden cross section by one of the manifestations of symmetry.

The gold division is not a manifestation of asymmetry, something opposite symmetry. According to modern ideas, the gold division is asymmetric symmetry. The science of symmetry includes such concepts as static and dynamic symmetry. Static symmetry characterizes peace, equilibrium, and dynamic - movement, growth. So, in nature, static symmetry is represented by the structure of crystals, and in art characterizes peace, equilibrium and immobility. Dynamic symmetry expresses activity, characterizes movement, development, rhythm, it is evidence of life. Static symmetry is characteristic of equal segments, equal values. The dynamic symmetry is typical of the increase in segments or their decrease, and it is expressed in the values \u200b\u200bof the golden section of an increasing or decreasing range.

Fibonacci row

With the history of the golden section indirectly connected the name of the Italian mathematics of the monk Leonardo from Pisa, better known under the name Fibonacci. He traveled a lot in the east, introduced Europe with Arabic figures. In 1202, his mathematical work "Book On Abak" (billboard) was published, in which all the tasks known at that time were collected.

A number of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Known as a number of fibonacci. The feature of the sequence of numbers is that each member, starting from the third, is equal to the sum of the two previous 2 + 3 \u003d 5; 3 + 5 \u003d 8; 5 + 8 \u003d 13, 8 + 13 \u003d 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers of the series is approaching the ratio of gold division. So, 21: 34 \u003d 0.617, and 34: 55 \u003d 0.618. This ratio is denoted by the symbol of F. Only this ratio - 0.618: 0.382 - gives continuous division of a straight line in a gold proportion, an increase in it or a decrease to infinity when a smaller cut does so much as greater than anything.

As shown in the lower figure, the length of each joint of the finger is correlated with the length of the next joint by the proportion of F. The same ratio is manifested in all fingers and legs. This relationship is somehow unusual, because one finger is longer than the other without any visible pattern, but it's not by any chance, as it is not by chance everything is in the human body. The distances on the fingers marked from A to B to C to D to E, everyone correlate with each other by the proportion of F, as well as the phalange of the fingers from F to G to H.

Take a look at this skeleton of frogs and see how each bone corresponds to the model of the proportion F exactly as in the human body.

Generalized golden cross section

Scientists continued to actively develop the theory of Fibonacci numbers and a golden cross section. Yu. Matyatsevich using Fibonacci numbers solves the 10th problem of Hilbert. There are methods for solving a number of cybernetic tasks (theory of search, games, programming) using Fibonacci numbers and a golden section. Even Mathematical Fibonacci-Association is created in the USA, which since 1963 produces a special magazine.

One of the achievements in this area is the discovery of generalized Fibonacci numbers and generalized golden sections.

A row of fibonacci (1, 1, 2, 3, 5, 8) and, with the same "binary" range of the weight of 1, 2, 4, 8, at first glance, completely different. But their construct algorithms are very similar to each other: in the first case, each number is the sum of the previous number with itself 2 \u003d 1 + 1; 4 \u003d 2 + 2 ..., in the second - this is the sum of the two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 ... whether it is impossible to find a common mathematical formula from which "binary »A number, and a number of fibonacci? Or maybe this formula will give us new numerical sets with some new unique properties?

Indeed, we define a numeric parameter S, which can take any values: 0, 1, 2, 3, 4, 5 ... Consider a numerical series, S + 1, the first members of which are units, and each of the following is equal to the sum of the two members of the previous And the steps away from the previous one. If the N-th member of this series we denote through? S (n), will we get a general formula? S (n) \u003d? S (n-1) +? S (n-S-1).

It is obvious that at s \u003d 0 of this formula, we obtain a "binary" series, with s \u003d 1 - a number of fibonacci, at s \u003d 2, 3, 4. New numbers of numbers that were called S-numbers Fibonacci.

In general, the Gold S-proportion is the positive root of the equation of the Gold S-section x S + 1 -X S -1 \u003d 0.

It is not difficult to show that at s \u003d 0 it turns out the division of the segment in half, and at s \u003d 1 - a familiar classic golden cross section.

The relations of neighboring Fibonacci S-numbers with absolute mathematical accuracy coincide in the limit with gold S-proportions! Mathematics in such cases say that gold S-sections are numeric invariants of Fibonacci S-numbers.

Facts confirming the existence of gold S-sections in nature, leads the Belarusian scientist E.M. Soroka in the book "Structural Harmony Systems" (Minsk, "Science and Technology", 1984). It turns out that, for example, well-studied double alloys have special, pronounced functional properties (thermally resistant, solid, wear-resistant, resistant to oxidation, etc.) only if the specific weights of the source components are connected to each other of one from gold s-proportions. This allowed the author to put forward the hypothesis that the gold s-cross sections are numerical invariants of self-organizing systems. Being confirmed, experimentally, this hypothesis may have a fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

Using the codes of the gold S-proportion, you can express any valid number as the sum of the degrees of gold S-proportions with integer coefficients.

The fundamental difference of this method of encoding the numbers is that the grounds for new codes, which are gold S-proportions, with s\u003e 0 are irrational numbers. Thus, new surgery systems with irrational bases as it were "from head to feet" historically established hierarchy of relations between the numbers rational and irrational. The fact is that at first there were "open" numbers of natural; Then their relationships are rational numbers. And only later, after discovering by Pythagoreans, irrational numbers appeared on the light. For example, in decimal, five-hour, binary and other classical positional systems, natural numbers were chosen as a kind of primary acquisitions: 10, 5, 2, of which all other natural, as well as rational and irrational numbers were designed for definite rules.

A kind of alternative to existing methods of number is the new, irrational, system, as the primary acquisition of which the irrational number is selected (which is reminding, the root of the golden section equation); Other valid numbers are already expressed through it.

In such a system of number, any natural number is always imagined in the form of the ultimate - and not infinite, as they thought earlier! - The sums of degrees of any of the gold S-proportions. This is one of the reasons why the "irrational" arithmetic, possessing an amazing mathematical simplicity and grace, as if absorbed the best qualities of the classic binary and "fibonaccia" arithmetic.

Forming principles in nature

Everything that has acquired some form, has been formed, grew, sought to take a place in space and keep themselves. This desire finds the implementation mainly in two versions: growth up or pouring on the ground surface and spiral spiral.

The shell is spinled on the helix. If it is deployed, then the length is obtained, a little inferior to the length of the snake. A small ten-centimeter shell has a spiral of 35 cm long. The spirals are very common in nature. The idea of \u200b\u200bthe golden section will be incomplete if not to say about the spiral.

The shape of spirally curled shell attracted the attention of Archimedes. He studied it and brought the spiral equation. The spiral drawn through this equation is called his name. The increase in its step is always evenly. Currently, the Archimeph Spiral is widely used in the technique.

Another Goethe emphasized the trend of nature to spiral. The screw and spiral arrangement of the leaves on the branches of the trees was noticed for a long time.

The spiral saw in the location of sunflower seeds, in pine cones, pineapples, cactus, etc. The joint work of nerds and mathematicians shed light on these amazing phenomena of nature. It turned out that in the location of the leaves on the branch (phyotaxis), the seeds of sunflower, the pine cones show itself a number of fibonacci, and therefore, the law of the golden section is manifested. Spider breaks a spiral cobweb. Hurricane spiral spiral. A frightened flock of reindeer is running around the spiral. DNA molecule is twisted with a double helix. Goethe called the spiral of the "curve of life."

Row of Mandelbroota

Golden spiral is closely related to cycles. Modern science of chaos is studying simple cyclic operations with feedback and the fractal forms generated by them, unknown earlier. The drawing shows a known series of mandelbroke - a page from the vocabulary z.the limbs of individual patterns called Julian rows. Some scientists associate a series of mandelbroke with the genetic code of cell cores. A consistent increase in cross sections reveals amazing fractals by its artistic complexity. And here there are also logarithmic spirals! This is all the more important as a series of mandelbrot, and Julian ranks are not an invention of the human mind. They arise from Plato's primitiveness. As the doctor R. Penrose said, "They are like Mount Everest"

Among the roadside herbs is growing no noticeable plant - chicory. I look at it carefully. From the main stem, the process was formed. Immediately the first sheet is located.

The process makes a strong release into space, stops, produces a sheet, but already in short, it makes the emission of the one again, but already less strength, releases a leaflet of even smaller size and emissions again.

If the first emission is taken for 100 units, the second is 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subordinate to the golden proportion. In the growth, the conquest of space plant retained certain proportions. The impulses of its growth gradually decreased in the proportion of the Golden Section.

Chicory

Many butterflies have the ratio of the sizes of the chest and abdominal parts of the body corresponds to the golden proportion. Folding the wings, the night butterfly forms the correct equilateral triangle. But it is worth divorcing the wings, and you will see the same principle of body membership at 2, 3, 5, 8. Dragonfly was also created according to the laws of the golden proportion: the ratio of the length of the tail and the housing is equal to the ratio of the total length to the tail length.

In a lizard at first glance, pleasant for our eye proportion - the length of her tail is as follows to the length of the rest of the body, like 62 to 38.

Lizard niphelistic

Both in the plant, and in the animal world persistently breaks through the formative tendency of nature - symmetry relative to the direction of growth and movement. Here, the golden cross section is manifested in the proportions of parts perpendicular to the direction of growth.

Nature made division into symmetric parts and gold proportions. In parts manifests the repetition of the structure of the whole.

Of great interest is the study of the forms of bird eggs. Their all sorts of forms ranges between two extreme types: one of them can be entered into a rectangle of the golden section, the other in a rectangle with a module 1.272 (golden proportion root)

Such forms of bird eggs are not random, since it is currently established that the form of eggs described by the ratio of the golden section corresponds to higher strength characteristics of the egg membrane.

The tissue of elephants and extinct mammoths, claws of lions, and the clusters of parrots are logarithmic forms and resemble the shape of the axis, prone to contact the spiral.

Forms based on "Pentagonal" symmetry (starfishes, marine hedgehogs, flowers) are widespread.

The golden cross section is present in the structure of all crystals, but most of the microscopic crystals are small, so we cannot see their naked eye. However, snowflakes, also constituting aquatic crystals, are quite accessible to our gaze. All the exquisite beauty of the shapes that form snowflakes, all axes, circles and geometric shapes in snowflakes are also always, without exception, are built according to a perfect clear formula of the golden section.

In the micrometer, three-dimensional logarithmic forms, built on gold proportions, are common everywhere. For example, many viruses have a three-dimensional geometric shape of the Ikosahedron. Perhaps the most famous of such viruses is the adeno virus. The protein sheath of the adeno virus is formed from 252 units of protein cells located in a specific sequence. In each corner of the Ikosahedron, 12 units of protein cells are located in the form of a pentagonal prism, and engaging structured structures extend from these angles.

Adino virus

For the first time, the golden cross section in the structure of viruses was found in the 1950s. Scientists from London Birkbek College A. Klog and D. Kaspar. The first logarithmic form revealed the Polyo virus. The form of this virus turned out to be similar to the form of the Rhino virus.

The question arises: how do viruses form such complex three-dimensional forms, the device of which contains a gold cross section, which even our human mind construct quite difficult? The discoverer of these forms of viruses, virologist A. K G gives such a comment: "Dr. Kaspar and I have shown that for the spherical shell of the virus, the most optimal form is the symmetry of the type of the shape of the Ikosahedron. Such an order minimizes the number of binding elements ... Most of the geodesic hemispherical cubes of the wagers of fuller are built on a similar geometric principle. Installation of such cubes requires an extremely accurate and detailed explanation scheme, while unconscious viruses themselves build a complex shell of elastic, flexible protein cellular units. "

Clue's commentary once again reminds of extremely obvious truth: in the structure of even a microscopic organism, which scientists classify as "the most primitive form of life", in this case in the virus, there is a clear idea and a reasonable project is carried out. This project is non-performing in its excellence and accuracy with the most advanced architectural projects created by people. For example, projects created by the brilliant architect bookminster Fuller.

Three-dimensional models of the Dodecahedron and Ikosahedron are also present in the structure of skeletons of single-celled sea microorganisms of radiolearies (radiation), the skeleton of which is created from silica.

Radiolaries form their body very exquisite, unusual beauty. The form of them constitutes the correct dodecahedron, and from each of its angle, pseudo-satisfaction-finiteness and other unusual form-growths germinate.

Great Goethe, poet, naturalist and artist (he painted and wrote watercolor), dreamed of creating a unified teaching on the form, education and transformation of organic bodies. This is he introduced into a scientific source. The term morphology.

Pierre Curie at the beginning of our century formulated a number of deep ideas of symmetry. He argued that it was impossible to consider the symmetry of any body without taking into account the symmetry of the environment.

The patterns of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systemsIn the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human and body bodies as a whole, and also manifest themselves in biorhythms and the functioning of the brain and visual perception.

Human body and golden cross section

All the bones of man are withstanding in the proportion of the golden section. The proportions of various parts of our body make up a number, very close to the golden section. If these proportions coincide with the formula of the golden section, the appearance or body of a person is considered perfectly folded.

Golden proportions in human body parts

If you take the center of the human body, the Pup point, and the distance between the feet of a person and the PUP point per unit of measurement, then the human growth is equivalent to the number 1.618.

• the distance from the shoulder level to the head of the head and the size of the head is 1: 1.618;
• the distance from the Pup point to the head of the head and from the shoulder level to the scalp is 1: 1.618;
• the distance of the PUP point to the knees and from 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 accurate presence of a gold proportion in the person of a person is the ideal of beauty for a human gaze;
• the distance from the tip of the chin to the upper line of eyebrows and from the upper line of eyebrows to the top is 1: 1.618;
• facial height / face width;
• central lip connection point to the base of the nose / nasal length;
• face height / distance from the tip of the chin to the central point of the lip connection;
• mouth width / width of the nose;
• nose width / distance between nostrils;
• distance between pupils / distance between the eyebrows.

It is enough just to bring your palm now to yourself and carefully look at the index finger, and you immediately find it in it the formula of the Golden section.

Each finger of our hand consists of three phalanges. The sum of the length of the two first phalanches of the finger in the ratio from the entire thumb and gives the number of the golden section (except for the thumb).

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

A person has 2 hands, fingers on each hand consist of 3 phalanges (except for thumb). On each hand there are 5 fingers, that is, only 10, but with the exception of two two-phase thumbs only 8 fingers are created according to the principle of the golden section. Then, all these numbers 2, 3, 5 and 8 are the numbers of Fibonacci sequences.

It should also be noted that most people have the distance between the ends of the hands of the hands equally.

The truths of the golden section within us and in our space. The peculiarity of the bronchi, components of the lung man, is enclosed in their asymmetry. Bronchi consist of two main respiratory tract, one of which (left) is longer, and the other (right) is shorter. It was found that this asymmetry continues in the branches of the bronchi, in all smaller respiratory tract. Moreover, the ratio of the length of short and long bronchi is also a golden cross section equal to 1: 1.618.

In the inner ear of a person there is a cochlea ("snail"), which performs the function of transmitting sound vibration. This boniform structure is filled with liquid and also created in the form of a snail containing a stable logarithmic form of a spiral \u003d 73 0 43 ".

Blood pressure changes during the work of the heart. It reaches the greatest magnitude in the left ventricle of the heart at the time of its compression (systole). In the arteries during the systole of the ventricles of the heart, the blood pressure reaches the maximum value of 115-125 mm mercury column in a young, healthy person. At the time of relaxation of the heart muscle (diastole), the pressure decreases to 70-80 mm Hg. The ratio of the maximum (systolic) to minimal (diastolic) pressure is equal to an average of 1.6, that is, close to the golden proportion.

If we take a single blood pressure in the aorta per unit, then the systolic pressure of blood in the aorta is 0.382, and the diastolic 0.618, that is, their ratio corresponds to the gold proportion. This means that the work of the heart with regard to time cycles and changes in blood pressure are optimized by the same principle of the law of the golden proportion.

The DNA molecule consists of two vertically intertwined spirals. The length of each of these spirals is 34 angstroms, width 21 angstrom. (1 angstrom - one velomillion share of centimeter).

DNA molecule spiral structure

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

Golden section in sculpture

Sculpture structures, monuments are erected to perpetuate significant events, keep the names of the famous people in the memory of descendants, their exploits and acts. It is known that in ancient times the basis of the sculpture was the theory of proportions. The relations of the parts of the human body were binding to the formula of the golden section. The proportions of the Golden section create the impression of harmony, beauty, so the sculptors used them in their works. Sculptors argue that the waist shares the perfect human body against the "golden section". For example, the famous statue of Apollo Belvedere consists of parts divided by gold relationships. The Great Ancient Greek Sculptor FIDI often used the Golden Section in his works. The most famous of them were the statue of Zeus Olympic (which was considered one of the wonders of the world) and Athens Parthenon.

The golden proportion of the statue of Apollo Belvederensky is known: the growth of the depicted person is divided by an umbilical line in a golden cross section.

Golden section in architecture

In the books about the "golden section" you can find a remark that in architecture, as in painting, it all depends on the position of the observer, and if some proportions in the building on one side seem forming the "golden cross section", then from other points of view they Will look different. The "golden section" gives the most calm ratio of the sizes of certain lengths.

One of the most beautiful works of ancient Greek architecture is Parthenon (V C. BC).

In the figures, a number of regularities associated with the golden section are visible. The proportions of the building can be expressed through various degrees of the number F \u003d 0.618 ...

Parthenon has 8 columns on short sides and 17 long. The protrusions are made entirely of the squares of the pentyl marble. The nobility of the material from which the temple was built, allowed to limit the application of the usual coloring in the Greek architecture, it only emphasizes the details and forms color background (blue and red) for sculpture. The ratio of the height of the building to its length is 0,618. If you make a division of Parfenon in the "Golden section", we will receive certain protrusions of the facade.

On the Plan of Parfenon, you can also notice "Golden Rectangles".

Golden ratio We can see and in the building of the Cathedral of the Paris Mother of God (Notre Dame de Paris), and in the pyramid of Heops.

Not only Egyptian pinamides are postponed in accordance with the consolidated golden cross section; The same phenomenon is indiscriminately at Mexican pipamid.

For a long time, it was believed that the architect ancient Russia was built on everything "on the eyes", without special mathematical calculations. However, the latest studies have shown that Russian architects knew the mathematical proportions well, the analysis of the geometry of the ancient temples.

The famous Russian architect M. Kazakov in his work widely used the "golden section". His talent was multifaceted, but in more than He revealed in numerous projects of residential buildings and estates. For example, the "golden section" can be found in the architecture of the Senate Building in the Kremlin. According to the project M. Kazakov, a 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 project M.F. Cossack

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

House Pashkova

V. Bazhenova's excellent creation firmly entered the ensemble of the center of modern Moscow, enriched him. The outdoor view of the house has been preserved almost unchanged to the present day, despite the fact that it burned heavily in 1812. When restoring the building acquired more massive forms. The internal layout of the building is not preserved, which only draws the drawing of the lower floor.

Many statements of the architecture deserve attention and these days. About her beloved art V. Bazhenov said: "The most important architecture has three subjects: the beauty, the peace of mind and strength of the building ... To achieve this, it serves as a leadership knowledge of proportion, perspective, mechanics or physics in general, and all of them the general leader is reasonable."

Golden section in music

Any musical work has a temporary stretch and shares some "aesthetic milestones" to individual parts that pay attention and facilitate perception in general. These milestones can be dynamic and intonation culmination points of the musical work. Separate time intervals of the musical work connected by the "culmination event" are usually in the ratio of the golden section.

Back in 1925, art historian L.L. Sabaneev, analyzing 1770 musical works of 42 authors, showed that the overwhelming majority of outstanding essays can be easily divided into parts or on the topic, or by intonational system, or by the lads, which are between themselves in relation to the golden section. Moreover, than a talented composer, the greatest sections found in the larger number of works. According to Sabaneva, the golden cross section leads to the impression of a special harness of the musical composition. This result of Sabaneev checked on all 27 Etudes of Chopin. He discovered 178 gold sections in them. It turned out that not only large parts of the etudes are divided into duration in relation to the golden section, but also part of the etudes inside are often divided in the same relation.

Composer and scientist MA Marutaev calculated the number of clocks in the famous Sonata "Appassionate" and found a number of interesting numerical ratios. In particular, in the development of a central structural unit of Sonatata, where themes are intensively developing and replaced by each other of the tonality - two main sections. In the first - 43.25 tact, in the second - 26.75. Attitude 43.25: 26.75 \u003d 0.618: 0.382 \u003d 1.618 gives a golden cross section.

The greatest number of works in which there is a gold cross section, at the island (95%), Beethoven (97%), Haidna (97%), Mozart (91%), Chopin (92%), Schubert (91%).

If music is a harmonious streamlining of sounds, then poetry is a harmonious streamlining of speech. Clear rhythm, natural alternation of shock and unstressed syllables, ordered dimension of poems, their emotional saturation make poetry the native sister of musical works. The golden cross section in poetry is primarily manifested as the presence of a certain moment of poem (culmination, the semantic fracture, the main thought of the work) in the row account for the division of the total number of lines of poem in the golden proportion. So, if the poem contains 100 lines, then the first point of the gold section falls on the 62nd line (62%), the second - on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, and including "Eugene Onegin," - the subtlest compliance with the golden proportion! Works of Shota Rustaveli and M.Yu. Lermontov is also built on the principle of the golden section.

Stradivari wrote that with the help of a golden cross section, it determined the places for F-shaped cuts on the buildings of its famous violins.

Golden section in poetry

Studies of poetic works from these positions are just beginning. And you need to start with poetry A.S. Pushkin. After all, his works are a sample of the most prominent creations of Russian culture, a sample of the highest level of harmony. From poetry A.S. Pushkin We will begin searching for the golden proportion - Merila Harmony and Beauty.

Much in the structure of poetic works Rodinitis This kind of art with music. Clear rhythm, natural alternation of shock and unstressed syllables, ordered dimension of poems, their emotional saturation make poetry the native sister of musical works. Each verse has its own musical shape, its rhythm and melody. It can be expected that in the structure of poems, some features of musical works, the patterns of musical harmony, and, consequently, the golden proportion.

Let's start with the magnitude of the poem, that is, the number of rows in it. It would seem that this parameter of the poem may change arbitrarily. However, it turned out that it was not. For example, conducted by N. Vasyutinsky Analysis of the poems A.S. Pushkin showed that the dimensions of the verses are distributed quite unevenly; It turned out that Pushkin clearly prefers dimensions in 5, 8, 13, 21 and 34 lines (numbers of Fibonacci).

Many researchers were noticed that poems are similar to music; They also have culmination items that share a poem in the proportion of the golden section. Consider, for example, a poem A.S. Pushkin "Supozhnik":

Conduct the analysis of this parable. The poem consists of 13 lines. It allocates two semantic parts: the first in 8 lines and the second (moral parable) in 5 lines (13, 8, 5 - Fibonacci numbers).

One of the last poems of Pushkin "Not expensive I appreciate the loud rights ..." consists of 21 lines and two semantic parts are allocated in it: at 13 and 8 lines:

Not expensive I appreciate Loud rights, From koi, not one spins the head. I do not pour about the gods refused I need to challenge taxes Or interfere with the kings to do with each other; And little grief to me, whether printing freely Ferriching Olukhov, Ile sensitive censorship In journal plans, Balaguore is constrained. All this, you see, words, words, words. Other, best, I am expensive rights: Other, the best, demand me freedom: Depend on the king, depend on the people - Is it all equal to us? God with them. Report not to give myself only yourself Serve for power, for livery Do not bend no conscience, nor thoughts, nor neck; By whim, your wander here and there, Divine Divine Beauty And before creating arts and inspiration Trembling joyfully in the delights of the dying That's happiness! Here is the right ...

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

Represents an undoubted interest analysis of the novel "Eugene Onegin", made by N. Vasyutinsky. This novel consists of 8 chapters, in each of them, an average of about 50 poems. The most perfect, most honed and emotionally rich is the eighth chapter. In it, 51 verse. Together with the letter of Eugene to Tatiana (60 lines), this is exactly the number of Fibonacci 55!

N. Vasyutinsky states: "The culmination of the chapter is the explanation of Eugene in love for Tatiana - the line" pale and go out ... here is bliss! " This line divides the entire eighth chapter into two parts: in the first 477 rows, and in the second - 295 lines. Their attitude is 1,617! The finest compliance with the magnitude of the gold proportion! This is a great harmony miracle committed by the genius of Pushkin! ".

E. Rosenov analyzed many poetic works by M.Yu. Lermontova, Schiller, A.K. Tolstoy and also discovered the "golden cross section" in them.

The famous poem of Lermontov "Borodino" is divided into two parts: the introduction facing a storytellor, which occupies only one storm ("Tell me, uncle, because there is no wonder ..."), and the main part representing an independent integer that disintegrates into two equivalent parts. In the first of them, it is described with increasing voltage, the waiting for the battle, in the second - the fight itself with a gradual decline in tension by the end of the poem. The border between these parts is a climax point of the work and is accounted for a point of dividing by its golden cross section.

The main part of the poem consists of 13 seven, that is, out of 91 lines. Sharing it with a golden cross section (91: 1.618 \u003d 56,238), we are convinced that the division point is at the beginning of the 57th verse, where there is a short phrase: "Well, I was a day!" It is this phrase that is a "culminating item of excited expectation", finalizing the first part of the poem (waiting for the battle) and its opening part (a description of the combat).

Thus, the golden cross section plays in poetry a very meaningful role, highlighting the culmination point of the poem.

Many researchers of the poem Shota Rustaveli "Vityaz in Tiger Shkure" celebrate exceptional harmony and melodiousness of his verse. These properties of the poem Georgian scientist, academician G.V. Tsereteli relates at the expense of conscious use by the poet of the golden section both in the formation of the form of the poem and in the construction of her poems.

Rustaveli's poem consists of 1587 stanza, 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 semi-edge. All semi-possessions are divided into two segments of two types: a - semi-edible with equal segments and an even syllables (4 + 4); B - semi-edible with asymmetric division into two unequal parts (5 + 3 or 3 + 5). Thus, in the semi-horsepie, the ratios 3: 5: 8 are obtained, which is approaching the golden proportion.

It was established that in the poem Rustaveli from 1587 stanza more than half (863) were built on the principle of the golden section.

Nowadays, a new type of art was born - cinema, which entered the playwright of actions, painting, music. In the outstanding works of cinema, it is legitimate to seek the manifestations of the golden section. The first was the creator of the masterpiece of the world cinema "Bronnosets Potemkin" film director Sergey Eisenstein. In the construction of this painting, he managed to embody the basic principle of harmony - a golden cross section. As Eisenstein himself notes, the red flag on the mast of the rebel battleship (the point of the film's appoge) is placed at the point of the golden proportion, counted from the end of the film.

Golden section in fonts and household subjects

Special view visual arts Ancient Greece should highlight the manufacture and painting of all sorts of vessels. In elegant form, the proportions of the golden section are easily guessed.

In painting and sculpture of temples, the ancient Egyptians were most often portrayed by gods and pharaohs. The canons of the image of the standing person walking, sitting, etc. were installed. Artists were obliged to memorize individual forms and schemes of the image on tables and samples. Artists of ancient Greece made special travel to Egypt to learn the ability to use the canon.

Optimal physical parameters of the external environment

It is known that the maximum sound volumewhich causes pain, equal to 130 decibellam. If we split this interval with a gold proportion of 1.618, then we obtain 80 decibell, which are characteristic of the volume of the human scream. If now 80 decibell split the gold proportion, then we get 50 decibell, which corresponds to the volume of human speech. Finally, if you divide 50 decibelles with a square of the golden proportion of 2.618, then we obtain 20 decibell, which corresponds to the human whistery. Thus, all the characteristic volume parameters are interconnected through a gold proportion.

At a temperature of 18-20 0 C interval humidity 40-60% is considered optimal. The boundaries of the optimal range of humidity can be obtained if the absolute humidity is 100% twice to divide the golden cross section: 100 / 2,618 \u003d 38.2% (lower bound); 100 / 1,618 \u003d 61.8% (Upper border).

For air pressure 0.5 MPa in humans there are unpleasant sensations, its physical and psychological activity deteriorates. At a pressure of 0.3-0.35 MPa, only short-term work is allowed, and at a pressure of 0.2 MPa, no more than 8 minutes is allowed. All these characteristic parameters are related to the golden proportion: 0.5 / 1,618 \u003d 0.31 MPa; 0.5 / 2.618 \u003d 0.19 MPa.

Boundary parameters outdoor air temperature, within which a normal existence may be possible (and, the main thing, it became possible the origin) of a person is the temperature range from 0 to + (57-58) 0 C. Obviously, on the first border of the explanation can not be given.

We divide the specified range of positive temperatures with a golden cross section. At the same time, we obtain two boundaries (about the boundaries are characteristic of the human body with temperatures): The first corresponds to the temperature, the second border corresponds to the maximum possible temperature of the outer air for the human body.

Golden section in painting

Even in the era of the Renaissance, the artists discovered that any picture has certain points involuntarily attaching our attention, the so-called visual centers. It is absolutely no matter what format has a picture 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 of the artists of that time was called the "Golden section" of the paintings.

Turning to the examples of the Golden section in painting, it is impossible not to stop your attention on the work of Leonardo da Vinci. His personality is one of the riddles of history. Leonardo da Vinci himself said: "Let no one, without being a mathematician, will not bother to read my works."

He gained the fame of an unsurpassed artist, a great scientist, genius, anticipating many of the invention, which were not implemented up to the XX century.

There is no doubt that Leonardo da Vinci was a great artist, it was already recognized by his contemporaries, but his personality and activity will remain covered by the secret, since he left the descendants not a coherent presentation of his ideas, but only numerous handwritten sketches, notes in which they say " All in the world. "

He wrote to the right left with an unintelligible handwriting and left hand. This is the most famous, of the existing, sample mirror writing.

Portrait of Monta Lisa (Dzokonda) long years Attracts the attention of researchers who discovered that the composition of the pattern is based on gold triangles, which are parts of the correct star pentagon. There is a lot of versions about the history of this portrait. Here is one of them.

Once Leonardo da Vinci received an order from the banker Francesco Djokondo's business. Write portrait of a young woman, banker wife, Monsa Lisa. The woman was not beautiful, but it was attracted simplicity and naturalness of the appearance. Leonardo agreed to write a portrait. His model was sad and sad, but Leonardo told her a fairy tale, hearing, she became a living and interesting.

STORY. He lived, there was one poor man, he had four sons: three smart, and one of them and so, and Syak. And the death came after his father. Before parting with life, he called the children to him and said: "My sons, I will die soon. As soon as you shore me, dampen the hut and go to the edge of the world to extract happiness. Let each of you learn something to feed myself ". The father died, and the sons were separated by the light, having agreed three years later to return to the pool of the native grove. The first brother came, who learned to carpent, chuckled a tree and squeezed him, made a woman from him, moved a little and waiting. The second brother returned, saw a wooden woman and, since he was a tailor, she dressed her one minute: as a skillful master, 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 came the fourth brother. He could not worry and sew and sew, he only knew how to listen to what the land, trees, herbs, animals and birds say, knew the course of heavenly bodies and could still sing wonderful songs. He fell off the song from which brothers joined the bushes. He lived the song to this woman, she smiled and sighed. The brothers rushed to her and everyone 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 - be brothers. And you, that inhaled the soul in me and taught to enjoy life, you need one of my life. "

Ending a fairy tale, Leonardo looked at Montu Lisa, her face was lit by light, her eyes shone. Then, exactly awakened from sleep, she sighed, spent her hand in her face and she went to her place without words, folded his hands and took the usual posture. But the case was done - the artist awakened an indifferent statue; The smile of bliss, slowly disappearing from her face, remained in the corners of the mouth and tremble, giving the face an amazing, mysterious and slightly crap expression, like a person who learned the secret and, carefully keeping her, cannot hold back the celebration. Leonardo silently worked, fearing to miss this moment, this ray of the sun, illuminated his boring model ...

It is difficult to note that they have noticed in this masterpiece of art, but everyone spoke about the deep knowledge of Leonardo structure of the human body, thanks to which he managed to catch this, as if mysterious, smile. They spoke about the expressiveness of individual parts of the painting and about the landscape, unprecedented satellite portrait. Interpreted about the naturalness of the expression, about simplicity of poses, about the beauty of the hands. The artist did even unprecedented: the picture shows the air, it envelops the figure with a transparent haze. Despite the success, Leonardo was gloomy, the situation in Florence seemed to the artist, he gathered on the road. He did not help him reminders of surging orders.

Golden section in the painting of I.I. Shishkin "Pine Grove". On this famous picture of I.I. Shishkin is obviously visible to the motifs of the golden cross section. Brightly illuminated by the sun pine (standing in the foreground) divides the length of the painting along the golden cross section. To the right of the pine - the shed lighted by the sun. It divides the right side of the painting horizontally according to the golden section. To the left of the main pine is a lot of pines - if desired, you can successfully continue the division of the painting on the golden section and on.

Pine grove

The presence in the picture of bright verticals and horizontals dividing it in relation to the golden section gives it the nature of the balance and tranquility in accordance with the artist's idea. When the artist's idea is different, if, say, it creates a picture with a rapidly developing action, such a geometric scheme (with the predominance of verticals and horizontals) becomes unacceptable.

IN AND. Surikov. "Boar Morozova"

The role of it is assigned the middle part of the picture. It is riddled by the point of top takeoff and the point of the lowest complication of the plot of the picture: takes off the hands of the frost with a bonde-hollow congestion, like a high point; A hand stretched to the same boyfriend, but this time the hand of the old woman is a wonderland, a hand, from under which, together with the last hope for salvation, slips the end of the sledge.

And what about the "highest point"? At first glance, we have a seeming contradiction: after all, the cross section of the 1 in 1, which is 0.618 ... From the right edge of the picture, it does not pass through the hand, not even through the head or eye of the boylaws, but it turns out somewhere before the mouth of the boyar.

The golden cross section cuts here really on the most important thing. In it, and it is in him - the greatest power of Morozova.

There is no painting more poetic than Painting Botticelli Sandro, and there is no picture of the great Sandro painting more famous than his "Venus". For Botticelli, his Venus is the embodiment of the idea of \u200b\u200bthe universal harmony of the "Golden section", which dominates in nature. Proportional analysis of Venus convinces us in this.

Venus

Rafael "Athens School". Rafael was not a mathematician scientist, but, like many artists of that era, possessed considerable knowledge of geometry. In the famous fresco "Athens School", where the Society of Great Philosophers of Antiquity is to be in the Temple of Science, our attention is attracted by a group of Euclide - the largest ancient Greek mathematics, which breaks up a complex drawing.

The cunning combination of two triangles is also built in accordance with the proportion of the golden section: it can be inscribed in a rectangle with the aspect ratio of 5/8. This drawing is amazingly easily inserted into the upper part of the architecture. The upper corner of the triangle rests on the castle stone of the arch in the near the viewer to the viewer, the lower - to the point of departure of the prospects, and the side portion denotes the proportions of the spatial gap between the two parts of the archer.

Golden spiral in the painting of Raphael "Beating babies." Unlike the golden section, the feeling of the dynamics, the unrest manifests itself, perhaps the strongest in another simple geometric figure - spirals. The multifigure composition made in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, just distinguished by dynamism and drama of the plot. Raphael never brought his idea to completion, but his sketch was engraved by an unknown Italian schedule Markantinio Raymondi, which on the basis of this sketch and created the Engraving "Beating Infants".

Massacre of the innocents

If on the preparatory sketch of Raphael to mentally spend the lines coming from the semantic center of the composition - the points where the warrior's fingers closed around the ankle of the child, along the figures of the child, a woman pressed to themselves, a warrior with a laid sword and then along the figures of the same group on the right side Sketch (in the figure, these lines were performed in red), and then connect these pieces with a curve with a dasy, then a gold spiral is obtained with very high accuracy. This can be checked by measuring the ratio of the lengths of sections of the spiral veil on direct, passing through the start of the curve.

Golden section and image perception

On the ability of a person's visual analyzer to allocate objects constructed according to the golden section algorithm, like beautiful, attractive and harmonious, known for a long time. The golden cross section gives the feeling of the most perfect single whole. The format of many books corresponds to the golden cross section. It is selected for windows, pictorial canvases and envelopes, brands, business cards. A person can not know anything about the number F, but in the structure of objects, as well as in the sequence of events, it subconsciously finds elements of the golden proportion.

Studies were conducted in which the subjects were proposed to choose and copy rectangles of various proportions. Three rectangles were asked: the square (40:40 mm), the Golden section rectangle with the attitude of the parties 1: 1.62 (31:50 mm) and a rectangle with extended proportions 1: 2.31 (26:60 mm).

When selecting rectangles in the usual state in 1/2 cases, preference is given to the square. Right hemisphere prefers a golden cross section and rejects a stretched rectangle. On the contrary, the left hemisphere is to the extended proportions and rejects the golden cross section.

When copying these rectangles, the following was observed: when an actively right hemisphere - proportions in copies were maintained most accurately; With the activity of the left hemisphere - the proportions of all rectangles were distorted, the rectangles were stretched (the square was drawn as a rectangle with the ratio of the parties 1: 1.2; the proportions of the elongated rectangle increased dramatically and reached 1: 2.8). The proportions of the "golden" rectangle are most distorted; Its proportions in copies became the proportions of the rectangle 1: 2.08.

When drawing their own drawings, proportions close to the golden section are dominated, and elongated. On average, the proportions are 1: 2, while the right hemisphere prefers the proportions of the golden section, the left hemisphere is departed from the proportions of the golden section and pulls out the pattern.

Now draw a few rectangles, measure their parties and find the aspect ratio. What hemisphere do you have prevailing?

Golden section in photography

An example of using a gold cross section in the photo is the location of key components of the frame at points, which are located in 3/8 and 5/8 from the edge of the frame. It is possible to illustrate this as follows: the photo of the cat, which is located in an arbitrary location of the frame.

Now it is possible to conditionally divide the frame on the segments, in the proportion of 1.62 total length from each side of the frame. In places intersection of segments and there will be basic "visual centers", in which it is worth placing the necessary key elements of the image. We transfer our cat to the point of "visual centers".

Golden section and space

From the history of astronomy, it is known that I. Titius, a German astronomer of the XVIII century, with the help of this series found patterns and order within the distance between the planets of the solar system.

However, one case, which, it would seem to contradict the law: There was no planet between Mars and Jupiter. Conducted observation of this section of the sky led to the opening of the belt of asteroids. It happened after the death of Tizius at the beginning of the XIX century. Pyad Fibonacci is widely used: it is helpful to architectonics and living beings, and man-made structures, and the structure of galaxies. These facts are evidence of the independence of the numerical series from the conditions of its manifestation, which is one of the signs of its versatility.

Two gold spirals of the galaxies are compatible with David's star.

Pay attention to the stars overlooking the Galaxy in the White Spiral. Exactly by 180 0 from one of the spirals, another deploying spiral comes ... For a long time astronomers simply believed that everything that is there is what we see; If something apparently, it exists. They were either completely not noticed by the invisible part of reality, or they did not consider it important. But the invisible side of our reality in reality is much more visible and, probably, more importantly ... in other words, the visible part of reality is much less than one percent of the whole - almost nothing. In fact, our real house is an invisible universe ...

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

Conclusion

Nature, understood as the whole world in the manifold of its forms, is, as it were, of two parts: alive and inanimate nature. For the creations of inanimate nature, high stability is characterized, weak variability, judging by human life. A man is born, lives, agitates, dying, and granite mountains remain the same and planets rotate around the Sun as well as in the time of Pythagora.

The world of wildlife appears before us completely different - movable, volatile and amazingly diverse. Life demonstrates us a fantastic carnival of diversity and uniqueness of creative combinations! The world of inanimate nature is primarily the world of symmetry, who has sustainability and beauty. The world of nature is primarily the world of harmony in which the "golden section law" operates.

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

In this paper, the influence of the properties of the "Golden section" on the living and not live nature, on the historical course of the development of the history of mankind and the planet as a whole. Analyzing the gradation of the process of knowledge of the world, the discovery of all its 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 systems of nature, the laws of growth are not very diverse and traced in a variety of formations. This is manifested by the unity of nature. The idea of \u200b\u200bsuch unity, based on the manifestation of the same patterns in the heterogeneous phenomena of nature, has retained its relevance from Pythagora to the present day.