Gold geometric shapes. Fundamentals of the composition: Golden section
Golden cross section
A person distinguishes the surrounding items 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, based on the construction of which is a combination of symmetry and the golden section, contributes to the best visual 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 respect 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 proportion (Lat. Proportio) call the equality of two relationships: a. : b. = c. : d..
Cut straight AU You can divide into two parts in the following ways:
- into two equal parts - AU : AC = AU : Sun;
- on two unequal parts in any respect (such parts of the proportion are not formed);
- so when AU : AC = AC : Sun.
Last and there is a gold division or division of a segment in extreme and middle.
The gold cross section is such a proportional division of the segment to unequal parts, in which the entire segment belongs to the most part, as most of the most relate to the smaller; or in other words, a smaller cut is so relate to more as bigger to everything
a. : b. = b. : c. or from : b. = b. : but.
Fig. one. 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.
Fig. 2. The division of the segment is straight along the golden section. BC. = 1/2 AB; CD = BC.
From the point IN Regenerated perpendicular equal to half AU. Point received FROM connected line with a point BUT. A segment is postponed on the resulting line Sunending in point D.. Section AD It is transferred to straight AU. The point obtained E. divide cut AU In the ratio of the golden proportion.
Segments of gold proportions are expressed by an endless irrational fraction AE \u003d 0.618 ... if AU take a unit Ve \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If cut AU To adopt for 100 parts, most of the segment is 62, and the smaller - 38 parts.
The properties of the gold cross section are described by the equation:
x. 2 - x. - 1 = 0.
Solution of this equation:
The properties of the golden section created around this number a romantic halo of mysteriousness and almost a mystical worship.
Second golden cross section
The Bulgarian Magazine "Fatherland" (№10, 1983) published the article Colorin Tsekova -Karandasha "On the second gold section", which follows from the main cross section and gives another ratio of 44: 56.
Such a proportion is found in architecture, and also takes place when constructing compositions of images of an extended horizontal format.
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Fig. 3. Building a second golden section |
The division is as follows. Section AU It is divided into the proportion of the golden section. From the point FROM Regenerated perpendicular Cd.. Radius AU There is a point D.which connects the line with a point BUT. Right angle ACD shares in half. From the point FROM Line is held before crossing the line AD. Point E. divide cut AD In terms of 56: 44. |
Fig. four. 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
To find segments of the gold proportion of the ascending and downward rows, you can use pentagram.
Fig. five. Building the right pentagon and pentagram
To build a pentagram, it is necessary to build the right pentagon. The way to build it developed a German painter and chart of Albrecht Dürer (1471 ... 1528). Let be O. - center of the circle, A. - point on circle and E. - Middle Cut OA. Perpendicular to radius OAdot ABOUT intersects with a circle at point D.. Using a circulation, postpone on the diameter of the segment CE = ED. The length of the side inscribed in the circumference of the correct pentagon is equal to DC. 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 ° at the top, and the base, laid down on the side divides it in the proportion of the golden section.
Fig. 6. Building gold
Triangle
We carry out straight AU. From the point BUT lay on it three times the segment ABOUT arbitrary magnitude through the resulting point R We carry out perpendicular to the line AU, perpendicular to the right and left from the point R lay off the segments ABOUT. Points received d. and d. 1 Connect straight with a point BUT. Section dD 1 Put on the line AD 1 , getting a point FROM. She divided the line AD 1 in the proportion of the golden section. Lines AD 1 I. dD 1 Enjoy to build a "golden" rectangle.
History of golden section
It is believed that the concept of the golden division introduced the scientific use of Pythagoras, an ancient Greek philosopher and mathematician (VI century. BC). There is an assumption that Pythagoras his knowledge of the golden division borrowed from Egyptians and Babylonians. And indeed, the proportions of the pyramid of Heops, temples, bas-reliefs, the objects of life and jewelry from the tombs of Tutankhamon show that the Egyptian masters used the ratios of gold division when creating them. The French 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 values \u200b\u200bof the golden division. The architect Hesier, 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 the 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.
Fig. 7. Dynamic rectangles
Plato (427 ... 347 BC) also knew about the golden division. His dialogue "Timy" is devoted to the mathematical and aesthetic views of the School of Pythagora and, in particular, the issues of golden division.
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 antique World. In Pompeary Circle (Museum in Naples), the proportions of gold division are also laid.
Fig. eight. Antique Circle Golden Section
In reached us antique literature The gold division is first mentioned in the "beginning of" Euclidea. In the 2nd book "Beginning", the geometric construction of the gold division of the Euclid Don is a study of the golden division was made by the gypsum (II century. BC), PAPP (III century. AD) and others. In medieval Europe with gold The division got acquainted on 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 era of the Renaissance, interest in the gold division among scientists and artists in connection with its use both in geometry and in art, especially in the architecture of Leonardo da Vinci, an artist and a scientist, saw that Italian artists have a big empirical experience, 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.
Luke Pacheti perfectly understood the importance of science for art. In 1496, at the invitation of the Duke 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 "Divine Proportion" was published with brilliantly fulfilled illustrations, due to what Leonardo da Vinci did them. The book was an enthusiastic anthem of the golden proportion. Among the most advantages of the golden proportion to the monk of Luka Pacholi did not fail to call her "Divine Essence" as the expression of the divine trinity God Son, God Father and God's Holy Spirit (I understood that the small segment is the personification of God's Son, a bigger segment - God Father, and the whole cut - God of the Holy Spirit).
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. So he gave this division name 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 one who knows anything, taught this other 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 growth of a person is divided into the 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, build the scale of segments of the gold proportion of the ascending and downward rows
Fig. nine. Building a gold proportion segment scale
In the next centuries, the golden proportion rule turned into an academic canon and, when in the art, the struggle against academic routine began in art, in the heat of struggle "and the child was splashing with water." 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".
Fig. 10. Golden proportions in human body parts
Fig. eleven. Golden proportions in a person figure
Cainsing did 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 the 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.
Fibonacci row
With the history of the Golden section indirectly, the name of the Italian mathematics of the monk Leonardo from Pisa, more famous under the name Fibonacci (son Bonachci). He traveled a lot in the east, introduced Europe with Indian (Arabic) figures. In 1202, his mathematical work "Book On Abak" (counting board) was published, in which all the tasks known at that time were collected. One of the tasks read "How many rabbit pairs in one year from one couple will be born." Reflecting on this topic, Fibonacci lined up such a series of numbers:
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Months |
etc. |
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Couple rabbits |
etc. |
A number of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Known as a number of fibonacci. The peculiarity 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 indicated by the symbol. F.. Only this ratio is 0.618: 0.382 - it 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 relate to more as greater than everything.
Fibonacci also dealt with the decision of the practical needs of the trade: With which the smallest number of Giri, you can weigh the goods? Fibonacci proves that the optimal is such a system of Irrows: 1, 2, 4, 8, 16 ...
Generalized golden cross section
A number of Fibonacci could only remain a mathematical incident, if it were not for the fact that all researchers in the golden division in the plant and in the animal world, not to mention the art, invariably came to this series as an arithmetic expression of the law of the golden division.
Scientists continued to actively develop the theory of Fibonacci numbers and a golden section. Yu. Matyatsevich using Fibonacci numbers solves the 10th problem of Hilbert. Elegant methods of solving a number of cybernetic tasks (search theory, games, programming) using Fibonacci and Golden Section are arising. Even Mathematical Fibonachchi-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.
Fibonacci row (1, 1, 2, 3, 5, 8) and the Outlook the same "binary" range of Girric 1, 2, 4, 8, 16 ... 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 two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... whether it is impossible to find the general mathematical formula from which it turns out and " binary "row, and a number of fibonacci? Or maybe this formula will give us new numeric sets with some new unique properties?
Indeed, set the numeric parameter S.which can take any values: 0, 1, 2, 3, 4, 5 ... Consider a numeric row S. + 1 of 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 separating from the previous on S. Steps. If a n.-y member of this series we denote by φ S. (n.), then we get the general formula φ S. (n.) = φ S. (n. - 1) + φ S. (n. - S. - 1).
Obviously, when S. \u003d 0 From this formula, we get a "binary" series, when S. \u003d 1 - Fibonacci row, with S. \u003d 2, 3, 4. New rows of numbers that were called S.- Located Fibonacci.
In general gold S.-Pages there is a positive root of the gold equation S.-Exteen X. S + 1 - x. S. - 1 = 0.
It is easy to show that when S. \u003d 0 It turns out the division of the segment in half, and when S. \u003d 1 -Baby classic gold cross section.
Relationships of neighboring S.- Located fibonacci with absolute mathematical accuracy coincide in the limit with gold S.-Porporations! Mathematics in such cases say gold S.-Schens are numeric invariants S.- Located Fibonacci.
Facts confirming the existence of gold S.-sens in nature, leads the Belarusian scientist E.M. Sorok. In the book "Structural Harmony of Systems" (Minsk, "Science and Technology", 1984). It turns out that, for example, that well-studied double alloys have special, pronounced functional properties (thermally resistant, solid, wear-resistant, resistant to oxidation, etc.) only if the specific gravity of the source components are connected to each other of one of Golden S.-Pages. This allowed the author to push the hypotheses e. that gold S.-Engthening there are numeric 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 learning the processes in self-organizing systems.
Using gold codes S.-Pages can be expressed by any real number in the form of the amount of the Golden degrees S.-Pages with whole coefficients.
The fundamental difference of this method of encoding the numbers is that the foundations of new codes that are gold S.-Proids, S. \u003e 0 turn out to be 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 opening with Pythagorians, irrational numbers appeared, irrational numbers appeared. For example, in decimal, five-hour, binary and other classical positional systems, natural numbers were chosen as a kind of primary source - 10, 5, 2, from which all other natural, as well as rational and irrational numbers were constructed.
A kind of alternative to existing methods of note is a new, irrational system, as a primary source, the start of the number of which is chosen an irrational number (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 amount of degrees of any of the gold S.-Pages. 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 kind of form has been formed, grew, sought to take a place in space and keep themselves. This desire finds implementation mainly in two versions - growth up or pouring on the surface of the earth and spiral spiral.
The shell is spinled on the helix. If it is deployed, it turns out the length, 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.
Fig. 12. Spiral Archimedes
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.
Gethete also 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 rods spiral spiral. 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."
Among the roadside herbs there is no remarkable plant - chicory. I look at it carefully. From the main stem, the process was formed. Immediately the first sheet is located.
Fig. 13. Chicory
The process makes a strong release into space, stops, produces a sheet, but already shorter than the first, again makes a release into space, but already less power, 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 growth, the conquest of space, the plant retained certain proportions. The impulses of its growth gradually decreased in the proportion of the Golden Section.
Fig. fourteen. Lizard niphelistic
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.
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.
Fig. fifteen. Egg bird
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. It he introduced the term morphology into scientific use.
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 energy transitions elementary particles, in the structure of some chemical compounds, in planetary and space systems, in 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.
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 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 an 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.
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Composition - This is the distribution of objects and figures in space, the establishment of the ratio of volumes, light and shadows, colors, etc. There are different ways and rules for creating a harmonious composition. When we look around the naked eye, our brain quickly snags interesting stories, objects. The camera fixes everything. Therefore, this is your task to choose the main object, make the center of attention in the frame, and the others surrounding it in the background or make part of the story you want to "tell" with your picture.
Photos with a properly chosen composition make the look at them look and consider the details. They tell the story, create a mood and make you think.
Golden section in photography - The main and powerful tool for obtaining dynamic, interesting pictures. The Rule of the Golden section is found in nature, and everywhere. He knew about it in ancient Egypt. The proportions of the pyramid of Heops, temples, bas-reliefs, household items and jewelry from Tutankhamon's tomb testify that Egyptian masters used the ratios of the golden section when creating them. In the facade of the ancient Greek temple of Parfenon, gold proportions are also present. This phenomenon studied and began to apply a famous scientist, artist and sculptor in practice. Leonardo da Vinci.
For those who want to learn more - video:
We will affect only practical part Applications of the golden section in the photo. The frame is conditionally divided into three parts horizontally and vertical:
When crossing the horizontal and vertical line, a special point is formed - "Power point" or "Node of Attention". There are four of them - it is at these points that the main objects of the frame are better, it is for them that there is a look regardless of independence from the frame format or painting.
Practical tips:
- If the horizon is located on one of the horizontal lines, the frame will look harmonious. But on which of the lines, on upper or nower?
- If you want to concentrate the attention of the viewer on earth or water - then better on upper.
- If you make an emphasis on an interesting, expressive sky - then nower.
- If you take off the portrait, then it is better to post your eyes on upper Horizontal line.
- If you shoot a person in full growth, then it is better to post it on right or leva vertical lines.
- It is very important to keep track of what direction a person is directed, or where his eyes are directed. For example, if a person looks left, accordingly, it must be placed on right horizontal Lines that the space remains before him.
In modern cameras, there is already a prompt function that displays the lines of the arbitrarier rules on the monitor or in the viewfinder.
Place important parts of the composition along the lines, and most importantly - on their intersection.
The golden section can be traced not only in a rectangular grid, but also in diagonals or spirals. The principle of the location of the items is the same, by the main lines and at the points of their intersections.
Diagonal gold cross section
Applying the Rule of the Golden Section is diagonally and we obtain a rectangle consisting of three sectors. This rectangle can be rotated as you like if you build your frame so that three different objects are approximately located in these sectors, and the main objects in larger sections will look very harmonious.
This rule is used if you have several areas that differ in the frame.
Spiral golden cross section
Spirals are very common in nature. 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. Goethe called the spiral - the "curve of life."
Using this helix when building a composition in a frame (it can be turned upside down or in the other direction), we will get a frame with a clear item in the center of the Spiral.
Photo: John Lemieux)
More photograph and experiment. Good luck!
The Bulgarian Magazine "Fatherland" (No. 10, 1983) published the article Coloring Tsekova-pencil "On the second gold section", which follows from the main cross section and gives another ratio of 44: 56.
Such a proportion is found in architecture, and also takes place when constructing compositions of images of an extended horizontal format.
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
To find segments of the gold proportion of the ascending and downward rows, you can use pentagram.
To build a pentagram, it is necessary to build the right pentagon. The way to build it developed a German painter and chart of Albrecht Dürer (1471 ... 1528). Let be O. - center of the circle, A. - point on circle and E. - Middle Cut OA. Perpendicular to radius OAdot ABOUTintersects with a circle at point D.. Using a circulation, postpone on the diameter of the segment CE = ED. The length of the side inscribed in the circumference of the correct pentagon is equal to DC. 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 ° at the top, and the base, laid down on the side divides it in the proportion of the golden section.
We carry out straight AU. From the point BUT lay it on it three times the segment of an arbitrary value, through the point R We carry out perpendicular to the line AU, perpendicular to the right and left from the point R lay off the segments ABOUT. Points received d. and d1 Connect straight with a point BUT. Section dD1. Put on the line AD1, getting a point FROM. She divided the line AD1 in the proportion of the golden section. Lines AD1 and dD1. Enjoy to build a "golden" rectangle.
Golden cross section
1. Introduction 2. . Golden section - harmonic proportion
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. Second golden cross section
four . Zo lot Triangle (Pentagram)
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History of golden section
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. Golden section and symmetry 7. Fibonacci row 8. . Generalized golden cross section
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. Principles of formation in nature
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. Human body and golden cross section
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. Golden section in sculpture
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. Golden section in architecture
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. Golden section in music
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. Golden section in poetry
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. Golden section in fonts and household subjects
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. Optimal physical parameters of the external environment
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. Golden section in painting
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Golden section and image perception
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Golden section in photos
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. Golden section and space 2 1. Conclusion 2 2. . Bibliography
Introduction
People from a long time worried about the question whether such elusive things are subordinate as beauty and harmony, 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 to know what is a golden section and establish where humanity has found the use of goldsection. You probably paid attention to that we are not the same in the subjects and phenomena of the surrounding reality. Disorder, shapelessness, disproportion is perceived by us as ugly and produce an outdating impression. And objects and phenomena, which are peculiar to the measure, feasibility and harmony are perceived as beautiful and cause a feeling of admiration, joy, raise the mood.
A person in its activities is constantly facing objects using a golden cross section.There are things that cannot be explained. Here you come to an empty bench and sit on it. Where do you sit 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 produced a "golden cross section." Ancient Egypt and Babylon, in India and China, knew about the golden section, in India and China. 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 their immortal sculptures. Plato told that the universe was arranged according to the "golden section." And Aristotle found the compliance of the "golden section" to the ethical law. The highest harmony of the "golden section" will preach Leonardo da Vinci and Michelangelo, because the beauty and "golden cross section" are the same thing. And Christian mystics will draw on the walls of their monasteries of the Pentagram "Golden Section", fleeing from the devil. At the same time, scientists - from Pachol. and to Einstein - will search, but will not find its exact meaning. The endless series after the comma is 1,6180339887 ... Strange, mysterious, inexplicable thing: This divine proportion is mystically accompanied by everything alive. Non-fat nature does not know what a "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 living and everything is beautiful - everything obeys the divine law, the name of which is the "golden cross section." So what is the "golden cross section"? .. What is this ideal, divine combination? Maybe this is the law of beauty? Or is it still a mystical secret? 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 go 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 this is the people of the exact sciences - first of all, 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, based on the construction of which is a combination of symmetry and the golden section, contributes to the best visual 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 respect 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: A: B \u003d C: D.
Cut the straight AB can be divided into two parts in the following ways:
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into two equal parts - AV: AS \u003d AV: Sun;
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on two unequal parts in any respect (such parts of the proportion are not formed);
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thus, when AV: AC \u003d AC: Sun.
Last and there is a golden division.
The gold cross section is such a proportional division of the segment to unequal parts, in which the entire segment belongs to the most part, as most of the most relate to the smaller; or in other words, a smaller cut is so relate to more as bigger to everything
a: B \u003d B: C or C: B \u003d B: a.
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.
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.
The segments of the gold proportion are expressed by an infinite fraction of ae \u003d 0.618 ... if Av to take per unit, be \u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the cut is taken for 100 parts, most of the segment is 62, and the smaller - 38 parts.
The properties of the gold section are described by the equalization: x2 - x - 1 \u003d 0. Solution of this equation:
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Second golden cross section The Bulgarian magazine "Fatherland" published an article by the color of the Tsekova-pencil "On the second gold section", which follows from the main cross section and gives another ratio of 44: 56. Such a proportion is found in architecture. 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 terms of 56: 44. 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 To find the segments of the gold proportion of the ascending and downward rows, you can use the 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® at the top, and the base deflected on the side divides it in the proportion of the golden section. We carry out a straight AV. From the point and lay 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 out the sections of O. The obtained points D and D1 connect straight with the point A. Cut DD1 laying on Line AD1, receiving a point C. She divided the AD1 line in the proportion of the golden section. Lines AD1 and DD1 use to build a "golden" rectangle. History of golden section
It is believed that the concept of gold division introduced Pythagoras, ancient Greek philosopher and mathematician into scientific use. There is an assumption that Pythagoras his knowledge of the golden division borrowed from Egyptians and Babylonians. 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 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 values \u200b\u200bof the golden division. The architect Hesier, 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 the 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. Plato also knew about the golden division. The Pythagorean Time in the same name of the Plato dialog says: "It is impossible that two things are perfectly connected to without the third, since it should appear between them, which would bond them. This can be performed in the best way proportion, because if three numbers have the following property, that the average This refers to a smaller, as greater to the middle, and, on the contrary, less refers to the average, as the average to more, then the last and the first will be average, but the average - the first and last. Thus, everything you need will be the same, but Since it will be the same, it will make a whole. " Earth Mir Plato is building using the triangles of two varieties: anamental and unequal. He considers the most beautiful rectangular triangle, in which hypotenuse is twice as much of the smaller cathetov (such a rectangle is half an equilateral, the main figure of Babylonians, in it is the ratio of 1: 3 1/2 , differing from the golden section by about 1/25, and called the Timerding "rival of the golden section"). 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 opening of the Dodecahedron (or, as the most universe itself, this quintessence of the four elements, symbolized, respectively, tetrahedrome, octahedrome, Ikosahedrome and cube) belongs to Hippas, subsequently the dead during shipwreck. In this figure, many relationships 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. In the antique literature, the gold division was first mentioned in the "beginning of" Euclide. The 2nd book "Beginning" is given a geometric construction of gold division.. After Euclide, the gold division was engaged in the Hyplic (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 circulation in that period of statement of humanism has found a scheme describing the structure of the human body: To such a picture, essentially reproducing the pentagram, Leonardo da Vinci has repeatedly resorted. Its interpretation: the human body has Divine perfection, for the proportions laid 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 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, Luke Pachet's book "On the 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 mathematicians themselves is called irrational (so God can not 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 certain value (regardless of whether it is large or small) one and the same, can not be changed, nor in different 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 based on them 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 dodecahedrome, which is impossible to build without a golden cross 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 how to teach this others who need it. This is what I am removed." 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 growth of a person is divided into the 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 member 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 is preserved 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 to lay the segment M. On the basis of these two sections, we build the scale of segments of the gold proportion of the ascending and downward rows. 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, "the child also splashed with water." The Golden section was "open" again in the middle of the XIX century. In 1855, the German investigator of the Golden section, Professor Casesing, 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 about 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 the 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 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 an 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. Fibon row AC. C. AND
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" (counting board) 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 peculiarity 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 indicated 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 is similar to more as greater than everything. 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 connection 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 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 section. Y. Matyatsevich using Fibonacci numbers solves 10- Yu Hilbert's problem. There are methods for solving a number of cybernetic tasks (theory of search, games, programming) using Fibonacci numbers and a golden section. Even Mathematical Fibonachchi-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 an open "binary" range of weights 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 two previous numbers 2 \u003d 1 + 1, 3 \u003d 2 + 1, 5 \u003d 3 + 2 .... It is impossible to find the general mathematical formula from which it turns out and " binary "row, and a number of fibonacci? Or maybe this formula will give us new numeric 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 numeric row, S + 1 of the first members of which are units, and each of the following is equal to the sum of the two members of the previous one and Defending the previous steps. 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). Obviously, at s \u003d 0 of this formula, we obtain a "binary" row, with s \u003d 1 - a number of fibonacci, at s \u003d 2, 3, 4. New numbers of numbers that were called Fibonacci S-numbers. In general gold S-proportion There is a 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 -Baby classical 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. Soroko in the book "Structural Harmony Systems" (Minsk, "Science and Technology", 1984). It turns out that, for example, that well-studied double alloys have special, pronounced functional properties (thermally resistant, solid, wear-resistant, resistant to oxidation, etc.) only if the specific gravity of the source components are connected to each other of one of Gold S-proportions. This allowed the author to put forward the hypothesis that the gold S-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 learning the 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 with irrational bases as it were, "from heads to the legs" historically established hierarchy of relations between the numbers rational and irrational. The fact is that first were "open" numbers of natural; Then their relationships are rational numbers. And only later - after opening with Pythagorians, irrational numbers appeared, irrational numbers appeared. For example, in decimal, five-hour, binary and other classical positional systems, natural numbers were chosen as a kind of primary source - 10, 5, 2, from which all other natural, as well as rational and irrational numbers were constructed. A kind of alternative to existing methods of note is a new, irrational system, as a primary source, the start of the number of which is chosen an irrational number (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 "irrational" arithmetic, possessing an amazing mathematical simplicity and grace, as if the best qualities of the classic binary and fibonaccium arithmetic imagined. Forming principles in nature Everything that has acquired some kind of form has been formed, grew, sought to take a place in space and keep themselves. This desire finds implementation mainly in two versions - growth up or pouring on the surface of the earth and spiral spiral. The shell is spinled on the helix. If it is deployed, it turns out the length, 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. Gethete also 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 rods spiral spiral. 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." Zo Golden Spiral is closely related to cycles. Modern science The chaos studies simple cyclic operations with feedback and the fractal forms generated by them, previously unknown. Figure 6 shows a known series of mandelbrot - a page from the infinity 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." The spiral is closely related to the cycles. Contemporary science of chaos is studying simple cyclic operations with feedback and they generated by fractal
Among the roadside herbs there is no remarkable plant - chicory. I look at it carefully. From the main stem, the process was formed. Immediately the first sheet is located.
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| Fig. . Chicory |
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.
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| Fig. . Lizard niphelistic |
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.
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| Fig. . Egg bird |
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 figures In the snowflakes, it is also always unconscious on the perfect clear formula of the golden section.
In the microworld, 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 certain sequence. In each corner of the Ikosahedron, 12 units of protein cells are located in the form of a pentagonal prism and from these angles are shi-like structures.
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| Adino virus |
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 intention and a reasonable project has been implemented. Performances with the most advanced architectural projects created by people. For example, projects created by the brilliant architect bookminster Fuller. The three-dimensional models of the Dodecahedron and Ikosahedron are also present in the structure of the skeletons of single-celled sea microorganisms of radiolearies (radiation), the skeleton is made of silica. Radiolarians form their body very exquisite, unusual beauty. Their shape constitutes the right dodecahedron. And from each of his 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. It he introduced the term morphology into scientific use. 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 "gold" symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in 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.
If you take the center of the Pupa's human body, and the distance between the feet of a person and the PUP point per unit of measurement, then the human height 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 crown is 1: 1.618
Facial Height / Face Width
The central point of the lip connection to the nose base / nasal length.
Facial height / Distance from chin tip to central lip connection point
Mouth width / nasal width
Nose width / Distance between nostrils
Distance between Pupils / Distance between Brows It is enough just to bring your palm now to yourself and carefully look at the index finger, and you immediately find in it the formula of the Golden Section.
Each finger of our hand consists of three phalange.Summum of the first first phalanx of the finger in the ratio from all thumbs and gives the number of golden section (except thumb).
In addition, the ratio between the middle finger and the little finger is also equalthe number of golden cross sectionA 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. While all these numbers 2, 3, 5 and 8 are the numbers of Fibonacci sequence.
It should also be noted that most people have the distance between the ends of the hands of the hands equally. Truths of the golden section within us and in ourspace
The peculiarity of the bronchi, components of human lungs, 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 man there is an organCochlea. ("Snail"), which performs the function of transmitting sound vibration. This boutine structure is filled with liquid and is also created in the form of a snail containing a stable logarithmic form of a spiral \u003d 73? 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 you take the average 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 respect to time cycles and changes in blood pressure are optimized in the same principle - the law of the golden proportion.
The DNA molecule consists of two vertically twisted 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 ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the Golden section 1: 1,618
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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 contacted with the formula of the Golden section. The Golden Spection methods create the impression of the harmony of beauty, so the sculptors used them in their works. Culpors claim that the waist shares the perfect human body with respect to the "golden section". So, for example, the famous statue of Apollon Belvedere consists of parts divided by gold relations. Estimate ancient Greek sculptor FIDI often used the "golden cross 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 Parfenos.  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 "gold section" you can find a comment that in architecture, as in painting, it all depends on the position of the observer, and that if some proportions in the building on one side seem to form "golden cross section", then from other points Vision they will look different. The "gold cross section" gives the most calm ratio of the size 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 produce a division of Parfenon on the "golden section", we will get those or other protrusions of the facade. On the plan of Parfenon, you can also notice "golden rectangles":  Golden ratio we can see in the building of the Cathedral of the Parisian 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, they believed that architect Ancient Russia They built everything "on the eyes", without special mathematical calculations. but newest research They showed that Russian architects knew well mathematical proportions, as evidenced by the analysis of the geometry of the ancient temples. The famous Russian architect M. Cossacks in his work widely used the "golden cross section". His talent was multifaceted, but to a greater extent he revealed in numerous projects of residential buildings and manshes. For example, the "gold cross 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 N.I. Pirogova (Leninsky Prospekt, d.  Petrovsky Palace in Moscow. Built according to the project M.F. Kazakov.  Another architectural masterpiece of Moscow - House Pashkov - is one of the most perfect works of architecture V. Bazhenova.  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 hard in 1812 When restoring the building acquired more massive forms. The internal layout of the building is not preserved, which only gives the drawing of the lower floor. Many of the statements of the architecture deserve attention and now. About his beloved art V. Bazhenov said: "Architecture - the most important has three subjects: the beauty, the peace of mind and strength of the building ... to achieve this is the leadership knowledge of proportion, prospect, mechanics or physics in general, and all of them are a common leader." |
Any musical work has a temporary stretch and shares some "aesthetic milestones" to individual parts that pay attention to and facilitate perception in general. These milestones can be dynamic and intonation culmination points of the musical work. The individual temporal 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. Sabaneyev, 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 that are among themselves 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 M.A. Marutayev calculated the number of clocks in the famous Sonate "Appassionate" and found a number of interesting numerical relations. 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 largest number of works in which there is a gold 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, logical alternation of shock and unstressed syllables, Ordered dimension of poems, their emotional saturation make poetry a native sister of musical works. The golden cross section in poetry is primarily manifested as the presence of a certain point in the poem (climax, the semantic fracture, the main thought of the work) in the row, which falls on the dividing point of the total number of lines of poem in the gold 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 of 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
golden section he determined the place forf. -In-like cuts on the buildings of their famous violins. Golden section in poetry Poetry Pushkin Studies of poetic works from these positions are just beginning. And you need to start with poetry A.S. Pushkin. After all, his works - a sample of the most prominent creations of Russian culture, sample highest level harmony. With poetry A.S. Pushkin, we will begin searching for a 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 makes poetry a native sister of musical works. Each verse has its own musical form - with 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 from this point of view 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": The picture time was watching a shoemaker
And in the shoes, the error indicated;
Taking immediately the brush, the artist corrected,
Here, touch, the shoemaker continued:
"It seems to me that face is a little crooked ...
And this chest is not too naga?
Here the apelles interrupted impatiently:
"Sudi, friend, not higher than boot!"
I have a buddy at the commemorate:
I do not know what it is
Was a connoisseur, at least Strict Non in words
But his devil carries to judge the light:
Try it to judge boots!
The famous poem of Lermontov "Borodino" is divided into two parts: the entry facing the storytellor and occupies only one storm ("Tell me, uncle, because there is no wonder ..."), and the main part representing an independent integer that disintegrates two equivalent parts. In the first of them, the expectation of the battle is described with the growing tension, in the second - the battle 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, there was a day!" It is this phrase that is a "climax 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. Poetry Shota Rustaveli 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 refers 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 species: a - semi-edible with equal segments and an even number of 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 born the new kind Arts - Cinema, who has entered the playwork of action, painting, music. IN outstanding works The cinema is legitimate to seek the manifestations of the golden section. The first made the creator of the masterpiece of world cinema "Bramenos 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 The special type of visual art of ancient Greece should be distinguished by 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 standing person of the running, 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 Sound volume.
It is known that the maximum volume of sound, which causes pain, is 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 it is divided with 50 decibelles with a square of gold proportion of 2.618, then we obtain 20 decibell, which corresponds to the whisper of the person.
Thus, all the characteristic volume parameters are interconnected through a gold proportion.
Air humidity. At a temperature of 18-20®, 40-60% humidity interval 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).
Air pressure. With a 0.5 MPa air pressure, a person has unpleasant sensations, its physical and psychological activity deteriorates. At a pressure of 0.3 - 0.35 MPa is allowed only short-term work, and at a pressure of 0.2 MPa, no more than 8 minutes are 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.
Outdoor temperature. The boundary parameters of the outdoor temperature, within which the normal existence is possible (and, the main thing, it became possible) of a person is the temperature range from 0 to + (57-58) ®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. In this case, we get two borders:
Both borders 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 outdoor 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 causing 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 picture. 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 daring to read my work."
He gained the fame of an unsurpassed artist, a great scientist, a 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 with a secret, since he left the descendants not a coherent presentation of his ideas, but only numerous handwritten sketches, notes in which they say " Everyone 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 of a mirror letter.
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 de Le Jokondo to write a portrait of a young woman, a 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 himself and said: "My sons, I will die soon. As soon as you shore me, get a 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, quit the tree and shed 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 did not know how to go and sew and sew, he could only listen to what the earth, trees, herbs, animals and birds, knew the course of heavenly bodies and still knew how to 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 a father." You dressed me, and you decorated - be my brothers.
And you, that breathed in me soul and taught to enjoy life, you are alone for all my life".
Ending a fairy tale, Leonardo looked at Montnu Lisa, her face lit up with light, 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 storing 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.
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Golden section in the picture I. I. Shishkin "Pine Grove" In this famous picture of I. I. Shishkin, the motives of the golden section are obviously visible. 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. |
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The presence in the picture of bright verticals and horizontals dividing it with respect to the golden section gives it the nature of balance and tranquility, in accordance with the artist's plan. 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.
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 V. I. Surikov. "Boaying Morozova". The roles of it is assigned the middle part of the painting. It is riddled by the point of top takeoff and the point of the lowest decline in the plot of the paintings. 1) It takes off the hands of frost with a bonde-hollow congestion as the highest point. 2) It is a helplessly 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 sink. And what about the "highest point"? At first glance, we have aware of the contradiction: after all, the A1B1 cross section, which is 0.618 ... From the right edge of the picture, it does not pass through the hand, not even through the head or the eye of the boylaws, but it turns out somewhere before the mouth of the boyar! |
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| The portrait of Mona Lisa attracts the fact that the composition of the drawing is built on "golden triangles" (more precisely on triangles, which are slices of the right star pentagon). | |
There is no painting more precisely than the painting of 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.  Raphael "Athens School" Rafael was not a scientist-mathematician, 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 the 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" |
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In contrast to the golden section, the feeling of the dynamics, the excitement 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, however, his sketch was engraved by an unknown Italian schedule Markantinio Raymondi, who, on the basis of this sketch, created the engraving "beating babies".
If on the preparatory sketch of Raphael to mentally spend the lines coming from the semantic center of the composition - the points where the fingers of the warrior close around the ankle of the child, - along the figures of a child, a woman pressering him to themselves, a warrior with a sword listed and then along the figures of the same group in the right Pieces of the sketch (in the figure these lines are performed in red), and then connect these pieces with a curve with a dotted line, then a gold spiral is obtained with very great 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.
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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 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 side ratio of 1: 1.62 (31:50 mm) and a rectangle with elongated 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, left hemisphere Learn to extended proportions and rejects a golden cross section. When copying these rectangles, the following was observed. When the right hemisphere is actively, the 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? |
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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 edges of the frame. You can illustrate it with the following example.
Here is a photo of a 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.Ticius, a German astronomer of the XVIII century, with the help of this series found regularity and order within the distance between the planets of the solar system.
However, one case, which seemed to contradict the law: Between Mars and Jupiter there was no planet. Considered observation of this section of the sky led to the opening of the asteroid belt. 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 at 180® 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 important. ... in other words, the visible part of reality is significantly less than one percent of the whole - almost nothing. In fact, our real house is an invisible universe ... In the universe everything famous humanity Galaxies 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, consists of both two parts: live and non-fat nature. For the creations of inanimate nature, high stability is characterized by weak variability, if judged on scale human life. A person is born, living, 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" is valid. In the modern world, science is of particular importance in connection with the enhancement of human impact. Important tasks on modern stage are the search for new ways the coexistence of man and nature, 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 a living and non-living nature was considered, on the historical course of the development of the history of mankind and the planet as a whole. Analyzing all the above, one can once again be mowed by the grandeur 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 structural andfunctionan wow perfection of the whole and its parts in art, science, technology and nature. It can be expected that the laws of development various systems Nature, growth laws are not very diverse and traced in a wide variety of students. 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 our dayth. 51.
Aerography is based on the same "pillars" as other arts.
All our world can be described by numbers. Many numbers play such a significant role in this description that they have their own names: PI, expansion (E), etc. Among these "nominal" numbers are very wonderful. Mathematics, artists, architects at different times called him " golden number"," Divine number "," Divine section ". The term "golden section" came up with Claudius Ptolemy, and he became popular thanks to Leonardo da Vinciwho widespread it in his works. Art people have noticed that the proportions of forms that are particularly pleasant to the eye for perception are at the heart of their "golden cross section".
So what is this number? The gold cross section is called the number FI (PHI) equal to 1,61803. The number is named after the Great Ancient Greek Sculpty Fidius (Phidius), which used it in his sculptures. How to clearly demonstrate the principle of "golden section"? We give a simple example. If you build a rectangle, one side of which is 1.618 times longer than the other, then the resulting aspect ratio is the "golden cross section". The most common "gold rectangles" in the modern world are credit cards. The human body is considered beautiful, and its proportions are ideal if the ratio between the smaller and most of the body is equal to the ratio between most and the whole, that is, equal to the number of fi.
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The most famous mathematical composition of the ancient science is the "beginning" Euclidea. It is from "Beginning" to us a geometric problem "On the division of a segment in extreme and middle relations" came to us. As is the "golden section".
The essence of the task is as follows:
We divide the segment of the AV point with in such a relationship so that most of the cutting of the CFA refers to a smaller part of the segment of the speakers, as a segment of AB to its most of the SV, i.e.
Denote by the proportion (1.1) through x. Then, considering that AV \u003d AC + SV, proportion (1.1) can be written in the following form:
Where does the following algebraic equation impliate to calculate the desired proportion x:
x *
\u003d x + 1. (1.2)
x * - squared
Of the "physical meaning" of the proportion (1.1), it follows that the desired solution of equation (1.2) should be a positive number, from where it follows that the solution of the division of segment in extreme and mid-average is the positive root of equation (1.2), which we denote through, i.e
The approximate value of the golden proportion is:
= 1,61803 39887 49894 84820 45868 34365 63811 77203…
Golden geometric figures
Based on the foregoing proportions in geometry, such concepts of gold geometric shapes are defined:
- golden rectangle (in which the ratio of the main side to a smaller equal to the gold proportion);
- Golden right triangle;
- gold ellipse;
- Golden Equal triangle.
The rectangular triangle with the parties 3: 4: 5 is called "perfect", "sacred" or "Egyptian".
The creators of the Egyptian pyramids were chosen as a "main geometric idea" for the Heops pyramid - a golden rectangular triangle, and for the pyramid of Hefren - "sacred" triangle.
Pentagon ("Pentagonon" - Greek.), Proper pentagon. If they are diagonally diagonally in the Pentagon, then as a result we will get a pentagonal star, called the Pentagram ("Pentagrammon" - Greek.: "Pente" - five and "Grammon" - line) or a pentakl.
The pentagram, called folk beliefs "Witchie", played a big role in all magic sciences and was considered as a means of protection against evil spirits.
Every eight years, the Venus planet describes an absolutely correct pentaunet for a large circle of heavenly sphere.
The Pentagon building, the US military, has the form of the Pentagon.
Pentagon and pentacle include a number of wonderful figures that are widely used in artwork. In the ancient art, the so-called gold bowl law is widely known, which used antique sculptors and gold maststers. The shaded part of the pentagon gives a schematic representation of the golden bowl.
Once in the Soviet Union there was a state mark of quality, in which the motives of the golden bowl are clearly visible.
In the wildlife is widespread forms based on pentagonal symmetry - starfish, marine hedgehogs, flowers ..
Harmony of the golden section
(short review art history)
The beauty of the human body, the sample of the harmonic physique of the population, and rightly, the great creations of Greek sculptors are considered: Fidiya, Polycttte, Mirone, Praxitor. In his creations, the Greek masters used the principle of golden proportion. One of higher achievements Classical Greek art can serve as a statue of Dorifora, logged in a polycette in the V century BC. e. This statue is considered the best example for analyzing the proportions of the perfect human body installed by antique Greek sculptors, and is directly related to the golden section. M \u003d 0,618 ...
Venus Milosian, statue of Goddess Aphrodite and the standard of female beauty, is one of him from the best monuments of Greek sculptural art.
Leonardo da Vinci used the proportions of the Golden section in many of its most famous works, and in particular, in the "Last Evening" and the famous "Joconde".
Researchers "Joconda" paintings found that the composite construction of Kar-Tina is based on two gold triangles turned to each other with their bases. Harmonic analysis of the picture shows that the pupil of the left eye through which the vertical axis of the canvase passes, is on the intersection of two bisector of the upper golden triangle, which on the one hand, divide the angles at the base of the golden triangle, and on the other hand, at the intersection points with gold hips The triangle divide them in the proportion of the golden section. Thus, Leonardo da Vinci used in his picture not only the principle of symmetry, but also a golden cross section.
The painting "Holy Family" Michelangelo is recognized as one of the masterpieces of Western European art of the Renaissance. Harmonic analysis showed that the composition of the picture is based on the pentacle.
The proportions of David statues (Michelangelo's work) are based on the golden section.
A vivid example of Baroque architecture, a Smolny Cathedral in St. Petersburg, produces an indelible impression. Its basic proportions also seen a golden cross section.
On the famous picture of Ivan Shishkin "Ship Grove" are viewed by the motifs of the golden section. Sun brightly lit by the sun (standing in the foreground) divides the picture with a golden cross section horizontally. To the right of the pine - the sun illuminated at the sun. He divides the picture with a golden cross section vertically. To the left of the main pine is a lot of pines - it is possible to continue the division of the golden cross section horizontally to the left side of the picture. The presence in the picture of bright verticals and horizontals dividing it in relation to the golden section gives it the nature of balance and tranquility.
The construction of the UN headquarters in New York was completed in 1943. The building attracted then general Attention Not only as a social structure created using the latest architectural funds, but also as a first example of using a solid sun-generating screen on one of the facades. In this building, the motives of the golden section are also viewed. In the composition of the building, the three gold rectangles delivered to each other are distinguished, which are its main architectural idea.
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. In the musical works of various composers, not one gold cross section is usually stated, but a whole series of such sections. The largest number of works in which there is a gold section, at the island (95%), Beethoven (97%), Haidna (97%), Mozart (91%), Scriabin (90%), 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 makes poetry a native sister of musical works. The golden cross section in poetry is primarily manifested as the presence of a certain point in the poem (climax, the semantic fracture, the main thought of the work) in the row, which falls on the dividing point of the total number of lines of poem in the gold proportion. So, if the poem contains 100 lines, then the first point of the golden 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 of the golden proportion! Works of Shota Rustaveli and M.Yu. Lermontov is also built on the principle of the golden section.
One of the modern types of art - cinema, - who has gained playwright actions, painting, music. In the outstanding works of cinema-making right-dimensional look for the manifestations of the Golden section. The first was the creator of the masterpiece of world cinema "Potemkin's armadapors" 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.
For many millennia, the golden cross section was the object of admiration and worship of outstanding scientists and thinkers: Pythagora, Plato, Euclidea, Luke Pacholi, Johann Kepler, Pavel Florensky ...
Currently, the golden cross section turns out to be a source of new fruitful ideas in mathematics and theoretical physics, biology and botanic, economics and computer science ...
The material is formed by the book "Da Vinci and Fibonacci Code" A. Stakhova, A. Schurkhenkova, I. Shcherbakova, 2007, publishing house "Peter".