home › Transmission (gearbox) › Compasses of the golden section. golden compasses

Compasses of the golden section. golden compasses

Definition: “The ratio of the larger part to the smaller one is equal to the ratio of the entire value to its larger part” - generally breaks the brain completely for those who rarely use it. But this is very important concept. And the more you begin to study the Golden Section, the more you understand that this is the Truth, written in the form of a formula. And in fact, this formula is simple. This is the division of the whole into two parts - 62% and 38%, which can last indefinitely, while all parts are in absolute harmony with each other and the whole. It is amazing. And this is not a discovery. This is a common observation that people have been observing for many millennia. And observing, they began to use it in their lives, thereby making it divinely beautiful and correct.

You will be surprised, but everything that really tells us about the Truth fits into the Golden Section, which, we can say with confidence, is the revealer of the true and the false. Against the background of the Golden Section, you simply cannot say or do something contrary to the Truth. At least you won't be able to do it in front of people who are knowledgeable about the Golden Section. Therefore, I highly recommend that you watch this short film so that you can join this cosmic Knowledge and know what is true and what is not.

Fibonacci Compass

In the movie, I'm talking about a very useful tool that I called the "Fibonacci Compass", it is likely that it is called differently, but I decided to call it that. If you creative person, draw, draw, create, do something, then you simply need it. Yes, and even in ordinary life it is needed if, of course, you are interested in having things around you in golden harmony. This compass, for example, will allow you to choose the right house, which has a golden ratio, a carpet, a pool .. whatever. This is very right tool. In the film, I tell you how to measure them. And you can do it in just five minutes. I've attached the diagram below in the picture.

golden ratio- the universal principle of harmony

"Tastes do not argue" - how many times each of us has heard this formula, and even pronounce it. By agreeing with it, we are thereby ready to defend any disgrace that the human imagination can afford. A person who is deeply selfish, fussy, passionate, unaccustomed to listening to the world in big and small, simply has no reason to develop taste and comprehend harmony, and therefore he is able to generate the most monstrous aesthetics, while calling it beauty. "You can't forbid a beautiful life," the inhabitant spits out through greasy lips, defending his tastes and forbidding others to argue about them. "Of course, of course, we will not argue about tastes! Everyone is right in his own way, so long as he does not harm us," animals in the form of people echo, not understanding themselves deeper than bodily needs. And they are settled in squalid dwellings, they are stuffed with destructive music, they are school bench they feed wretchedness, serving it under the sauce of inevitability. The decline of aesthetics, the inattention to beauty is always the decline of humanity, which no longer wants to dream or strive for beauty. It is suffering and death.

It is difficult for an individual person to resist the whole system of vulgarity, and he is doomed to submit to it and perish if he does not have sufficient knowledge. I would like to believe that the feeling of beauty, the harmony of the world lives in every person - you just need to show it, learn how to use it.

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

Trusting the eye more than other sense organs, a person first of all learned to distinguish the objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden section is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature. This idea was shared and shared by many prominent modern scientists, proving in their studies that true beauty is always functional. Among them are aircraft designers. And architects, and anthropologists, and many others.

History of the golden ratio

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

The German professor G.E. Timerding, who wrote a book on the golden ratio in the first quarter of the twentieth century, states: "Among the Pythagoreans<...>the thought of mysterious forces and properties was associated with the regular pentagon, but these properties are revealed only when, next to the ordinary regular pentagon, that star is considered, which is obtained by sequentially connecting through one of all the vertices of an ordinary pentagon, composed by the diagonals of the pentagon "- and further notes: the pentagram played a big role in all magical sciences. five pointed star, as Timerding shows, is literally stuffed with the proportions of the golden ratio.

The Greeks were skilled geometers. They even taught arithmetic to their children with the help of geometric shapes. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.

Plato (427...347 BC) also knew about the golden division. The Pythagorean Timaeus in Plato’s dialogue of the same name says: “It is impossible for two things to be perfectly connected without a third, since a thing must appear between them that would hold them together. This can best be done by proportion, because if three numbers have the property that the average so is to the lesser as the greater is to the middle, and vice versa, the lesser is to the mean as the mean is to the greater, then the last and the first will be the middle, and the middle the first and the last. since it will be the same, it will make a whole." earthly world Plato builds using triangles of two kinds: isosceles and non-isosceles. He considers the most beautiful right-angled triangle to be one in which the hypotenuse is twice the smaller of the legs (such a rectangle is half an equilateral, the main figure of the Babylonians, it has a ratio of 1: 3 1/2, which differs from the golden ratio by about 1/25, and is called Timerding "opponent of the golden ratio"). Using triangles, Plato builds four regular polyhedra, associating them with the four earthly elements (earth, water, air and fire). And only the last of the five existing regular polyhedra - the dodecahedron, all twelve faces of which are regular pentagons, claims to be a symbolic image of the heavenly world.

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

In the facade of the ancient Greek temple of the Parthenon there are golden proportions. During its excavations, compasses were found, which were used by architects and sculptors of the ancient world. The Pompeian compass (Museum in Naples) also contains the proportions of the golden division.

In what has come down to us ancient literature the golden division is first mentioned in Euclid's Elements. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (II century BC), Pappus (III century AD) and others were engaged in the study of the golden division. medieval Europe got acquainted with the golden division by Arabic translations"Began" Euclid. The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

In the Middle Ages, the pentagram was demonized (as, indeed, much that was considered divine in ancient paganism) and found shelter in the occult sciences. However, the Renaissance again brings to light both the pentagram and the golden ratio. So, a scheme describing the structure of the human body gained wide circulation in that period of the assertion of humanism:

Leonardo da Vinci also repeatedly resorted to such a picture, essentially reproducing a pentagram. Her interpretation: the human body has divine perfection, because the proportions inherent in it are the same as in the main celestial figure. Leonardo da Vinci, artist and scientist, saw that Italian artists empirical experience is great, but knowledge is small. He conceived and began to write a book on geometry, but at that time a book by the monk Luca Pacioli appeared, and Leonardo abandoned his idea. According to contemporaries and historians of science, Luca Pacioli was a real luminary, the greatest mathematician in Italy between Fibonacci and Galileo. Luca Pacioli was a student of the artist Piero della Francesca, who wrote two books, one of which was called On Perspective in Painting. He is considered the creator of descriptive geometry.

Luca Pacioli was well aware of the importance of science for art. In 1496, at the invitation of the Duke of Moreau, he came to Milan, where he lectured on mathematics. Leonardo da Vinci also worked at the Moro court in Milan at that time. In 1509, a book by Luca Pacioli was published in Venice "On Divine Proportion"(De divina proportione, 1497, published in Venice in 1509) with brilliantly executed illustrations, which is why they are believed to have been made by Leonardo da Vinci. The book was an enthusiastic hymn to the golden ratio. There is only one such proportion, and uniqueness is the highest attribute of God. It embodies the holy trinity. This proportion cannot be expressed by an accessible number, remains hidden and secret, and is called irrational by mathematicians themselves (so God can neither be defined nor explained by words). God never changes and represents everything in everything and everything in each of his parts, so the golden ratio for any continuous and definite quantity (regardless of whether it is large or small) is the same, cannot be changed or otherwise perceived by the mind. God called into being heavenly virtue, otherwise called the fifth substance, with its help four other simple bodies (four elements - earth, water, air, fire), and on their basis called into being every other thing in nature; so our sacred proportion, according to Plato in the Timaeus, gives formal being to the sky itself, for it is attributed to the form of a body called the dodecahedron, which cannot be built without the golden section. These are Pacioli's arguments.

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

At the same time, in northern Europe, in Germany, Albrecht Dürer was working on the same problems. He sketches an introduction to the first draft of a treatise on proportions. Durer writes. "It is necessary that the one who knows how to teach it to others who need it. This is what I set out to do."

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

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

Kepler called the golden ratio continuing itself. “It is arranged in such a way,” he wrote, “that the two junior terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains until infinity".

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

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

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

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

Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in most detail. Greek vases, architectural structures of various eras, plants, animals, bird eggs, musical tones, poetic meters were subjected to research. Zeising defined the golden ratio, showed how it is expressed in line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they constituted a Fibonacci series, which could be continued indefinitely in one direction and the other. His next book was entitled "Golden division as the basic morphological law in nature and art." In 1876, a small book, almost a pamphlet, was published in Russia, outlining Zeising's work. The author took refuge under the initials Yu.F.V. Not a single painting is mentioned in this edition.

IN late XIX- early XX centuries. a lot of purely formalistic theories appeared about the use of the golden section in works of art and architecture. With the development of design and technical aesthetics, the law of the golden ratio extended to the design of cars, furniture, etc.

A bit of geometry

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

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

into two equal parts AB: AC = AB: BC;

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

so when AB: AC = AC: BC.

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

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

a:b = b:cor c: b = b: a.

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

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

Segments of the golden ratio are expressed by an infinite irrational fraction AE= 0.618... if AB take as a unit BE\u003d 0.382 ... For practical purposes, approximate values \u200b\u200bof 0.62 and 0.38 are often used. If the segment AB taken as 100 parts, then the largest part of the segment is 62, and the smaller is 38 parts.

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

x2 - x - 1 = 0.

Solution to this equation:

The second golden ratio

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

Such a proportion is found in architecture, and also takes place in the construction of compositions of images of an elongated horizontal format.

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

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

Golden Triangle

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

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

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

We draw a straight line AB. from point A set aside on it three times a segment O of arbitrary size, through the resulting point R draw a perpendicular to the line AB, on the perpendicular to the right and left of the point R set aside segments ABOUT. Received points d And d1 connect with a straight line A. Line segment dd1 put on the line Ad1, getting a point WITH. She split the line Ad1 in proportion to the golden ratio. lines Ad1 And dd1 used to build a "golden" rectangle.

Fibonacci series

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

Months

etc.

Pairs of rabbits

etc.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, is equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 \u003d 34, etc., and the ratio of adjacent numbers of the series approaches the ratio of the golden division. So, 21:34 = 0.617, and 34:55 = 0.618. This ratio is denoted by the symbol Ф. Only this ratio - 0.618: 0.382 - gives a continuous division of a straight line segment in the golden ratio, increasing it or decreasing it to infinity, when the smaller segment is related to the larger one as the larger one is to everything.

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

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

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

The facts confirming the existence of golden sections and their derivatives in nature are given by the Belarusian scientist E.M. Soroko in the book "Structural Harmony of Systems" (Minsk, "Science and Technology", 1984). It turns out, for example, that well-studied binary alloys have special, pronounced functional properties (thermally stable, hard, wear-resistant, oxidation-resistant, etc.) only if the specific gravities of the initial components are related to each other by one of golden proportions. This allowed the author to put forward the hypothesis that the golden sections are numerical constants for self-organizing systems. Confirmed experimentally, this hypothesis can be of fundamental importance for the development of synergetics - a new field of science that studies processes in self-organizing systems.

Principles of shaping in nature

Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds realization mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral.

The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral.

The shape of the spirally curled shell attracted the attention of Archimedes. He studied it and deduced the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. At present, the Archimedes spiral is widely used in engineering.

Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch (phylotaxis), sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself. The spider spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

Among the roadside herbs, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf.

Rice. 12. Chicory

The process makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of less force, releases a leaf of an even smaller size and ejection again. If the first outlier is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section.

Rice. 13.viviparous lizard

In a lizard, at first glance, proportions that are pleasant to our eyes are captured - the length of its tail relates to the length of the rest of the body as 62 to 38.

Both in the plant and animal world, the form-building tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth.

Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.

Rice. 14. bird egg

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

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

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

Golden ratio and symmetry

The golden ratio cannot be considered in itself, separately, without connection with symmetry. The great Russian crystallographer G.V. Wulff (1863...1925) considered the golden ratio to be one of the manifestations of symmetry.

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

Observe and apply

Understanding and using the principle of the golden section should not be the lot of some elite - this is the most basic knowledge from which the infinitely complex laws of harmony and proportion begin. There are no limits to the meaningful application of these laws in the life of every day. The allocation of the main and secondary in relation to the whole can concern anything. This is the distribution of their time, and any creative process, including all kinds of art, literature, music, and the formation of one's own attitude to any processes and phenomena. This is the Golden, middle way, which the ancients spoke about.

Every artist, every director, every advertising specialist knows how to make an image pleasing to the eye, how to build it according to the laws of harmony and psychology. human perception. Sometimes the worst enemies of culture achieve significant victories using knowledge of the laws of Nature. Thus, under the guise of something pleasant and endearing, we often allow the strongest poisons to enter our hearts. So many people talk about freedom, while they themselves poison themselves voluntarily, wondering later where their illnesses and misfortunes come from.

There can be no freedom in ignorance. Roughness and illegibility of taste must be overcome. Let this be the concern of both individuals and communities and states.

Compiled by R. Annenkov

Having made this vintage instrument, you will be able to create great projects.

The "golden ratio" was used by the ancient Greeks and Egyptians in the calculation of buildings and as a model for achieving ideal proportions.

You, too, can use it in your projects, armed with a Fibonacci meter.

To have your own gauge, make a drawing of the tool to begin with according to the dimensions given in the figure.

Made of 1.6mm thick solid wood ( fit good thick veneer) cut the blanks and process the three arms A, B, C to the desired width and shape. (We used maple, but other woods are fine.)

Transfer the hole centers from the full size drawing to the gauge arms. Drill, where shown, a 5.5 mm diameter hole and finish each shoulder.

Assemble the parts by connecting them with clamp screws and adding glue so they don't loosen over time.

According to the magazine "Wood-Master"

Comfortable and beautiful bed sheets has a special magical power. And how nice it is to wake up every morning in an airy bed, not letting go of your arms. Even more pleasant when the linen is sewn
Give your meal an extra touch of charm with this nicely rounded salt and pepper shaker set. If you want to receive such a set today, then select the material (from
I offer a simple device that will help out when you need to drill a vertical hole in the end of a long part.
Why put blocks of wood on the workbench in order to put the workpiece on them, if necessary, when there are special stands? Assemble cabinet furniture on them using gaps for sponges
Make a dozen or two clamp-bushings, which are very fond of manufacturers musical instruments, and you will be able to evenly distribute the pressure on any curved edge.

The desire to give a fashionable shape to the nose or lips is rare, which cannot be said about the eyebrows, which are either plucked into a thin thread, or drawn daily or regularly tinted. Blindly following fashion trends is not always beneficial - thin eyebrows-threads often do not harmonize with the type of face at all, and those painted with a pencil look rather vulgar and almost always unnatural. But nature does not always take care of the harmony of facial features, therefore, if necessary, eyebrow correction has to be modeled. Since color and proportions are the basis of our visual perception, successful correction requires preliminary marking, for which Leonardo's eyebrow compasses are used.

What is Leonardo's compass

The Leonardo compass is a tool made of surgical steel that allows you to apply the principle of the Golden Ratio when modeling the shape of the eyebrows. Outwardly, in its upper part it resembles English letter W, as it has three legs. The design of the compass helps to measure the ratio between large and small distances (depending on the change in one of these distances, the other also changes) - the middle leg is involved in measuring both large and small distances.

The instrument owes its name to the great scientist and artist Leonardo da Vinci, who studied harmonious proportions and created his masterpieces using the principle of harmonic division.

The "golden ratio" is the proportion at which the ratio of one part to another is equal to the ratio of the whole to the first part.

Since the ideal shape of the eyebrows depends not so much on fashion as on the characteristics of a particular person (face shape, size and shape of the eyes), the master must take these features into account when “marking up”.

In order to give the eyebrows a shape that will not be a dissonant note in the overall harmony of the face, makeup artists have to make “marking”, based not on subjective aesthetic perception, but on precise geometric constructions.

Create a verified and correct form in the shortest possible time, the makeup artist is helped by an eyebrow compass.

What proportions does Leonardo's compass help determine?

Only those eyebrows that have a wide and narrow part look natural. However, in order to create a beautiful, harmonious shape, the makeup artist needs to determine:

Where should the eyebrow start? They do not always start with the client where they are supposed to start according to harmonious proportions, so it is impossible to focus on the natural growth of hairs or intuitive perception.
Where should the brow end. This point can be felt in the place where the frontal bone ends (a small depression is felt under the finger). Of course, during the correction procedure, it is inconvenient to probe this place every time, in addition, without accurate measurement, the eyebrows can turn out to be asymmetrical.

Where should the wide end meet the narrow end (highest point). The location of this point depends on the school - in the Russian school it is located parallel to the pupil (you can see how such an eyebrow looks like in the photo of Lyubov Orlova), in the French school it is above the upper edge of the iris, and in Hollywood it goes to the outer edge of the eye.
What should be the distance in the bridge of the nose.
What should be the distance between the eye and the eyebrow (with a small vertical distance, the eyebrows seem to hang over).

Tips to help you in using the Leonardo eyebrow compass:

Why use Leonardo's compass

The location of the eyes visually changes depending on the slope of the base of the eyebrow - if this line is tilted towards the nose, the eyes become closer, and if this line is tilted away from the nose, the distance between the eyes seems wider. In this way, too wide or narrowly set eyes can be corrected.

The bridge of the nose will look more even in combination with a straight line at the base of the eyebrows.

The width of the eyebrows is adjusted depending on the proportions of the face (its widest part should correspond in width to half the iris and not exceed 1/3 of the length of the entire eyebrow).

There are a sufficient number of such recommendations, involving the removal of excess hairs or tattooing where there are not enough hairs. However, without the use of accurate measurements and the rule of the "golden section", one has to completely trust the experience and taste of the cosmetologist, and the taste of the client and the makeup artist may not coincide.

The use of Leonardo's compass allows you to create perfect shape eyebrows for a specific face and demonstrate to the client the advantage of the form chosen by the makeup artist.

How to work with Leonardo's compass

In order to build the correct lines as symmetrically as possible with the help of Leonardo's compass, it is important to know how to use the compass for marking. Marking with a compass is applied in a supine position.

The construction of a sketch begins with the definition of a central point - a “reference point”. To do this, between the eyebrows, slightly above the bridge of the nose, it is necessary to determine the center of the forehead and mark this point. vertical line. The nose cannot serve as a guideline for a symmetrical construction, since so many people have a slight deformation of the nose, which, although not conspicuous, will affect the symmetry during correction.
The second point necessary for construction is the point of the beginning of the eyebrow. In order to determine its location, Leonardo's compasses are taken, and the ends that determine long distances are placed on the lacrimal canals. The resulting small distance shows the distance between the eyebrows. Lines are drawn at the location of the points indicating the beginning.
The third point is the end of the eyebrow, its "tail". To determine it, the compass is applied like a ruler - from the point of the edge of the nose (in the place where it touches the cheek) through the point of the edge of the eye to the end of the eyebrow. A vertical line is also drawn at the third point.

The fourth important point is the highest point. It is necessary to determine this point regardless of the shape of the bend chosen by the client (this point can be either pronounced, “corner”, or smoothed, almost imperceptible). To determine this point, the extreme legs of the compass are placed at the end and beginning of the eyebrow. In this case, the middle leg of the compass should be directed towards the temple, and not towards the forehead. The location of the middle leg will be the highest point.
After applying these points, the width of the eyebrows is determined and the upper and lower lines are adjusted. To do this, all the marked points are connected. The result should be a clear outline, with which the master will work in the future.

In the process of work, points are applied simultaneously on each half of the face.
How correctly the markings are applied should be checked in a sitting position. Symmetry is checked using a compass - the distance of each eyebrow from highest point before its beginning and end must match. It is also important to check whether the central point is correctly marked (the distance from this point to the beginning of the eyebrow should be the same on both sides).
Eyebrows should lie on the same line. To check, the compass is used as a ruler, which is placed between the lower starting points. Similarly, the relationship between the upper starting points is checked.

All hairs that go beyond the intended lines are removed.

The use of Leonardo's eyebrow compass is recommended for beginners, since this method of marking is more convenient than using a flexible ruler.

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once acquainted with the golden rule, humanity no longer cheated on it.

Definition

The most capacious definition of the golden ratio says that the smaller part is related to the larger one, as the larger one is to the whole. Its approximate value is 1.6180339887. In a rounded percentage, the proportions of the parts of the whole will correlate as 62% by 38%. This ratio operates in the forms of space and time. The ancients saw the golden section as a reflection of the cosmic order, and Johannes Kepler called it one of the treasures of geometry. modern science considers the golden ratio as "asymmetric symmetry", calling it in a broad sense a universal rule reflecting the structure and order of our world order.

Story

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

The Greeks were skilled geometers. Even arithmetic was taught to their children with the help of geometric figures. The square of Pythagoras and the diagonal of this square were the basis for constructing dynamic rectangles.

Plato(427...347 BC) also knew about the golden division. His dialogue "Timaeus" is devoted to the mathematical and aesthetic views of the school of Pythagoras and, in particular, to the issues of the golden division.

Rice. Antique compasses golden ratio

In the ancient literature that has come down to us, the golden division is first mentioned in the "Beginnings" Euclid. In the 2nd book of the "Beginnings" the geometric construction of the golden division is given. After Euclid, Hypsicles (2nd century BC), Pappus (3rd century AD) and others studied the golden division. In medieval Europe, they got acquainted with the golden division from Arabic translations of Euclid's "Beginnings". The translator J. Campano from Navarre (3rd century) commented on the translation. The secrets of the golden division were jealously guarded, kept in strict secrecy. They were known only to the initiates.

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

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

The name of the Italian mathematician is directly connected with the golden section rule. Leonardo Fibonacci. As a result of solving one of the problems, the scientist came up with a sequence of numbers, now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden ratio: “It is arranged in such a way that the two lower terms of this infinite proportion add up to the third term, and any two last terms, if added together, give the next term, and the same proportion remains indefinitely. ". Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.

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

Over time, the golden section rule has become an academic routine, and only a philosopher Adolf Zeising in 1855 gave it back a second life. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his "mathematical aestheticism" caused a lot of criticism.

Nature

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

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

Rice. Building a scale of segments of the golden ratio

Rice. Chicory

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

Rice. viviparous lizard

Rice. bird egg

The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with proportions of the golden section. According to him, one of the most interesting shapes it's a spiral.

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

Human

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

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

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

The Art of Spatial Forms

The artist Vasily Surikov said that “there is an immutable law in the composition, when nothing can be removed or added to the picture, even an extra point cannot be put, this is real mathematics.” For a long time artists follow this law intuitively, but after Leonardo da Vinci, the process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Dürer to determine the points of the golden section, he used a proportional compass invented by him.

The art critic F. V. Kovalev, having studied in detail the painting by Nikolai Ge “Alexander Sergeevich Pushkin in the village of Mikhailovsky”, notes that every detail of the canvas, whether it be a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions. Researchers of the golden section tirelessly study and measure the masterpieces of architecture, claiming that they became such because they were created according to the golden canons: in their list are the Great Pyramids of Giza, the Cathedral Notre Dame of Paris, Basil's Cathedral, Parthenon.

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

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

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

Golden ratio and symmetry

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

Word, sound and film

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

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

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

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

We invite you to discuss the topic in our group -

Popular in category: