How does the mirror reflect?

Of course, we all know how reflects the mirror, but if only it is necessary to describe it for sure, there will undoubtedly have difficulties. As a rule, we are satisfied with themselves, if you represent yourself at least "in principle." And the details that physician teachers explained to us on the board with the help of chalk and ruler, every normal schoolboy and student try to forget, and, the sooner, the better.

Each child, filled with surprise in front of the world around, will certainly be interested in how the mirror reflects it. But adults are usually responsible in such cases: "Do not ask stupid questions!" A man is distinguished, begins to be shy, his surprise gradually fades, and he tries no longer manifest it until the end of his life (and sor!).

But in this book we will be surprised as much as possible, having remembered the words of Bertold Brecht: "There are no stupid questions, there are only stupid answers."

What is the path from the burning house to the parking lot in the shortest? "The angle of fall", under which the fire truck will reach the river, should be equal to the "corner of the reflection", under which it will seek the place of fire

Of course, people can be divided into fools and smart, on large and small, they differ in language, religion, worldview. You can imagine such a way of division:

1) people who are never surprised;

2) People who are surprised, but do not think about their surprising phenomenon;

3) People who, surprised, ask "why?";

4) People who, surprised, turn to date and least.

Depending on the living conditions, traditions, the degree of education and all possible "intermediate" steps are found. Thinkers of antiquity and the Middle Ages amazed to the world and thought about his secrets. But they were only occasionally stepped out the case to measure any phenomenon.

Only in the era of the Renaissance, that is, in the XVI century, people came to the conviction that the measurement is better than blind faith or scholastic reasoning. This was facilitated by economic interests, which could only be satisfied with the development of natural sciences, by quantitative measurements. (We see that, essentially, the exchange cost "was measured" by money.) For the XVI century. Optics was an ultramodern science. From a glass bowl filled with water, which was used as a focusing lens, a magnifying glass appeared, and from it a microscope and a pickle tube. The largest Netherlands required for the fleet of the Netherlands, the Netherlands needed for the fleet, so that there is a hazardous shore to consider the dangerous coast or in time to get away from the enemy. Optics ensured the success and reliability of navigation. Therefore, it was in the Netherlands that many scientists dealt with it. Hollandets Willebrord, Snell Wang Royen, who called himself Snellius (1580-1626), watched (which, however, saw many to it), as a thin light beam reflected in the mirror. He simply measured the angle of the fall and the reflection angle of the beam (which no one did not do it) and established the law: the angle of the fall is equal to the reflection angle.

Now, in rear, this law seems to us something for granted. But in those days he had a huge, one might say, the ideological importance, which was a philosophical thought, until the XIX century.

We will put the following mathematical task: in some house there was a fire. The fire team is caused, and the water for extinguishing has to take from the river. Where should it be recruiting it to file a burning house as soon as possible?

The answer reads: the place must be chosen with this calculation so that the angle of entrance to the river was equal to the corner of departure from it in a straight home to the burning house. In this case, the total length of the segments of the path will be minimal. (Such a minimum maximum principle was previously considered as a manifestation of the "Will of the Lord").

The law of reflection of Snellulus explains the phenomenon of the mirror reflection, it should only be added to it why it is characteristic only with brilliant and smooth surfaces. In fact, the rough surfaces also obey the reflection law. But as a result of roughness, they seem to consist of small mirrors, unsystematic directed in all directions. In addition, the material that we consider as a mirror should be very small to absorb light and not be transparent. Such qualities are distinguished, for example, polished metals, calm water over a dark bottom, some polished stones and primarily placed on an opaque substrate glass.

Each point of the subject corresponds to its reflection in the mirror, and therefore in it our right eye moves to the left side. As a consequence of the transfer of points, the objects located further in the mirror also seem reduced in accordance with the prospects. Technically, we can reconstruct the mirror image as if it is located behind the glass surface. But this is only apparent perception. It is not by chance that animals and little children often look at the mirror; They believe that the image is lighted from behind, as if the picture visible outside the window. The fact of rearrangement of the left and right is correctly realized only by adults.

Mirror with conveyor

In one of the Greek myths, Narcissue is narrated, which was lying on the shore of the lake, admiring his reflection in the water.

Be Narcissus is a wealthy man, he must think, would acquire a mirror of polished metal. In those days, to bring a piece of steel or bronze with a palm with a palm, it was not so simple. In addition, the surface of such a mirror was oxidized and it had to clean it daily. Latin spectrum in German turned into Spiegel ("Spiegel" - a mirror). Of which you can conclude that the mirrors brought in Germany.

Only in the XI century. There were mirrors known to us from glass. One of the first mention of them belongs to the French Menstrel of the San de Beauva. According to him, in such mirrors on the glass, Lead was imposed on the bottom. Obviously, comment, in which context, the Messenter mentions the mirror, too. And in 1773, in Nuremberg, there was already a workshop. Since that time, the manufacture of mirrors becomes an important branch of European crafts.

Venice was the first country (in those days she had the status of an independent state), which began to issue patents for inventions. In 1507, the brothers Danzalo Del Gallo received a patent for the manufacture of crystal mirrors. Today, venetian mirrors are jewelry on the antiques market. In those days, a thin tin foil was placed under the glass plate (tin easily rolled on the rolls). On Foil poured mercury, which formed with Tin Amalgam. Since mercury pairs are very poisonous, this method has long been banned and replaced by silver.

In a rectangular corner mirror (at the corner between 90 ° mirrors), the "right" and "left" position persists

For a long time, the receiving the protection of a thin metal layer with lacquer coating has been preserved. Today, the sheet glass moves through the conveyor, where it is sequentially applied to its surface with a solution of silver salt and a reducing agent, which precipitates a clean silver solution in a fine (colloidal) form; After that, the thin layer of silver is applied to a thin layer of silver, protecting the silver film, and in conclusion both metal is covered with varnish. The conveyor belt is moving at a speed of about 2.5 m / min. Monthly products of such an aggregate about 40,000 m 2 mirrors. If some kind of "smart" reader will be removed to scrape a silver with a large wall mirror on the decoration of his wife or a friend, then it is worth to know that the layer of silver on the mirror is so thin that the "heater is not worth it". On 1 m 2 of the surface of the mirror, less than 1 g of silver are deposited.

Production of glass was considered before large art. The story came that during the times of the Roman Emperor Tiberia (42 BC. E.) Someone opened the unbreakable glass. Tiberius ordered to execute this person so that his discovery did not lead to the depreciation of the glass. Today, inventors working in the field of the glass industry may not be afraid of a similar fate. On the contrary, all efforts are reduced to make glass is possible cheaper.

Among solids of inorganic origin (stone, metal) Glass occupies a special place. Strictly speaking, the individual properties of glass bring it closer to the liquid. Most substances in solid and liquid state behave differently. The easiest way to watch water and ice. Water is in drip-liquid form. Exactly at 0 ° C, clean water begins to crystallize. The apparatus temperature is saved by zero, until all the water turns into ice. Even in the Outolars in the frost - 50 ° C water under the ice retains the temperature of 0 ° C. Only when all the water disappears, the ice can be cooled further. Ice as a solid has a crystal structure. Inside its small sections, crystals, we find a distinct symmetry. This symmetry is recognized on X-rays (radiographs).

Another thing is glass. It does not find crystals in it. There is no sharp transition in it and a sharp transition at a certain temperature from a liquid state to solid (or back). The molten glass (glass mass) in a large temperature range remains solid. If we decide the viscosity of water for 1, the viscosity of the molten glass at 1400 ° C is 13,500. If you cool the glass up to 1000 ° C, it will become drum and 2 million times more viscous than water. (For example, a loaded glass tube or a sheet is bent over time.) With an even lower temperature, the glass turns into a liquid with an infinitely high viscosity.

The main component of glass - silicon dioxide, or silica - SiO 2. In its purest form, it is represented in nature with white quartz sand. Silicon dioxide is crystallized when moving from melt to a solid state is relatively gradually. Quartz melt can be cooled below its coherence temperature, and it will not be solid. There are a lot of other liquids and solutions that can also be percoolen. But only quartz can be supercooling so much that it loses the ability to form crystals. Silicon dioxide remains "free from crystals", that is, "liquid-shaped".

Recycled clean quartz would be too expensive, primarily due to its relatively high melting point. Therefore, technical glasses contain only 50 to 80% of silicon dioxide. To reduce the melting point, the additives of sodium oxide, alumina and lime are introduced into such glasses. The preparation of certain properties is achieved by additives of some more chemicals. The famous lead glass, which is thoroughly grinding in the manufacture of bowls or VAZ, is obliged to about 18% lead in it with its brilliance.

Glass for mirrors contains predominantly cheap components that reduce the melting point. In large baths (as they call the glass winds), which accommodate more than 1000 tons of glasses, first melts low-melting substances. The molten soda and other chemicals dissolve quartz (as the water is a mustache salt). In such a simple means, it is possible to translate silicon dioxide into a liquid state at a temperature of about 1000 ° C (although in its pure form it begins to melt only with much higher temperatures). Gas gases are distinguished to the large anniversary of glass winds from the glass mass. At 1000 ° C, the melt is still too knit for the free output of gas bubbles. For degassing, it should be brought to a temperature of 1400-1600 ° C. Such high temperatures are achieved in the so-called regenerative glass furnaces, invented in 1856 by Friedrich Siemens. In them, the exhaust gases are heated by preheating chambers lined with refractory materials. As soon as these chambers are quite split, they feed combustible gases and air necessary for their combustion. The gases arising during burning is evenly mixed by molten glass, otherwise it would be far from simply to mix a thousand tons of viscous melt.

Modern glass coating oven is a continuous oven. On the one hand, the starting materials are supplied to it, which, thanks to the light slope, is moving, gradually turning into the molten glass, to the opposite side (the distance between the furnace walls is about 50 m). There, accurately measured portion of the finished glass enters the cooled rolls. For the entire length of the stater cooling section, a glass tape is stretched in several meters wide. At the end of this section of the car cut it onto the sheets of the desired format and size for mirrors or window glass.

The hardness of the glass is known (in German, there is even an expression "solid as glass"). In the poem Pushkin "Eugene Onegin" in love Tatiana cuts on the window glass expensive the name of the diamor rings ( Apparently, the author is familiar with the work of Pushkin on translation. In the original Tatyana "Adorable finger wrote on a blurred glass." - approx, translation). Today, "diamonds" for cutting glass are made of synthetic stones or solid alloys. Glass distinguishes and pretty compressive strength. This property is used when creating stained glass windows, decorative partitions. In contrast, the strength of glass for stretching is negligible. The novelty today are high strength glasses. Along with other applications, they are used for pipelines in the chemical industry. Transparency is important for the mirror. Normal glass skips from 70 to 90% of visible light. Glass transparency remains an indispensable condition in the manufacture of good mirrors. For ultraviolet light (≈ 10 15 -10 16 Hz), glass is not transparent. In the first spring days, when it is still cold, but the sun begins to harvest, there are fanatical tan fans, which are sitting in the windows, substituting the face with the sunshine. But all their efforts are in vain if special glasses are inserted into the frame, transparent for ultraviolet rays.

Those who in the apartment several mirrors probably have to notice that they have different quality. First of all, a good mirror should not have a whisie distorting image. Similar fibers arise due to incomplete melting glass or uneven cooling.

The gloss of the mirror can be improved both due to the composition of the glass, and by careful surface treatment (grinding and polishing).

And yet it is amazing: both Narcissus in antiquity, lying on the shore of the lake, admired his reflection in the water, and we, modern people, look in the mirrors, which are essentially "liquid"!

However, in the future, the production of mirrors will most likely go along the path of using a plastic film, which is sprayed a thin layer of metal.

From toward the radar

Of course not: a sufficiently mirror image is repeatedly reflected in the mirror to see your true face. Often in the homes there are still so-called trolls. They have one large main mirror in the CRNTR and two smaller mirrors on the sides. Many people think that these side mirrors serve only in order to look at the curls behind the ears. But if such a lateral mirror put at right angles to the middle, then you can see yourself exactly in the form in which you see others. Look at the left eye, and your reflection in the second mirror will repeat your movement to the left eye. Before, you can choose whether you want to see yourself in a mirror or in a direct image.

The angular mirror with a straight angle between the components of it with mirrors is still some more interesting properties. If you make it from two small mirrors, you can make sure that in such a mirror with a rectangular solution (and now we are talking only about it) The reflected ray of light is always parallel with a falling beam. This is a very important property. But not the only one! When the angular mirror is rotated around the axis connecting the mirror (within certain limits), the reflected beam will not change its direction.

The technique usually does not make up the mirrors, and the rectangular prism is used, in which the corresponding edges provide a mirror course of the rays.

Rectangular prisms, as if "folding" the running of the ray "accordion", maintaining its necessary length given by the focal length of the lens, allow you to reduce the dimensions of optical instruments. In prismatic binoculars, the rays of light with such devices are 180 °.

On vintage patterns you can see captains and commander with exorbitantly long pylon pipes. Thanks to the angular mirrors, vintage pylon pipes turned into modern binoculars.

Players in billiards have long been familiar to the action of reflection. Their "mirrors" is the board of the playing field, and the role of the beam of light perform the trajectories of the balls. Having hitting the side near the corner, the ball rolls to the side located at a right angle, and, reflected from it, moves back parallel to the direction of the first blow.

The properties of the reflected beam to maintain the direction when the angular mirror turns around the axis is widely used in the technique. Thus, in a triangular mirror corner reflector, the beam retains a constant direction, despite the very strong swings of the mirror. In shape, such a mirror is a cube with a cut corner. And in this case, there are not three mirrors in practice, but an appropriate glass prism with mirror facilities.

An important area of \u200b\u200buse of a triangular mirror is the corner reflector (feline eye, potatoes) on bicycles, motorcycles, signal safety shields, limiters of the roadway. With whatever part of the light on such a reflector, the light reflex always retains the direction of the light source.

A large role triangular mirror corner reflectors are played in radar techniques. Airplanes and large steel ships reflect Radar beam. Despite the significant scattering of it, the small share of reflected radio waves, which returns to the radar, is usually enough to recognize the object.

The situation is worse with small shipments, signal floats and plastic sailing yachts. In small items, the reflection is too weak. Plastic yachts are also "transparent" for radio waves, on which the radar technique works like window glass for sunlight. Therefore, sailing yachts and signal bucches are equipped with metal corner reflectors. The length of the faces of such a "mirror" is only about 30 cm, but this is pretty to return a fairly powerful echo.

Let's go back again to the angular mirror of two connected mirrors. Swim his axis is right or left - our image will also lean towards. We can even put it if you place the axis of the mirror horizontally. But, tilting the mirror even further, we note that the image is "straightened." Of course, we are looking for an explanation. It fully meets the topic of this book.

The angular mirror has a plane of symmetry, which divides in half the space between both mirrors. With the appropriate form, it may have another plane perpendicular to the mirrors, but we will not consider it here. We are only interested in the plane of symmetry, passing between mirrors in which, so to speak, both mirrors are mutually reflected.

Each plane of symmetry changes, as we already know, the right to the left (and on the contrary). But this is a few simplified Breeding. If the plane of symmetry knew how to talk, she would say: "I do not change the right to the left or the top of the bottom. I generally do not know what it is. I only displays the point for the point everything is on one or the other side of me. If a person gets along with his longitudinal axis in parallel, soy axis, I change it to him the right and left side, but if the same person is located with his longitudinal axis perpendicular to my axis (for I always remain unchanged), then I will change what people call the riding and bottom " . As you can see, it all depends on the point of view.

But ultimately true what can be measured and count. Today we do not see a special achievement that Snellius measured the angles of falling and reflection of the beam. But we should not forget that scientists of the XVI century. More than twentieth tradition broke like such discoveries.

Among the secrets of television is a famous trick with a decrease in the artist, which, against the background of the entire environment "In full-size" looks like a small doll. Sometimes the viewer can see the actor at the same time on two scales: in the foreground in the usual value, and in the rear in the reduced.

To the one who is tempted in the photo, it is clear how such an effect is achieved. First, the reduced version is removed, and then the actor plays in front of the screen, which is projected by its reduced image.

Famous "Corrod" Iochen Musk in his fascinating book "Magic World of Magic" ( Zmeck J. Wunderwelt Magie. Berlin: Heuchel-Verlag, Kunst Und Gesellschaft, 1974) Describes how such miracles can be done without a photo. When the reduced object should appear in space itself, with the help of a concave mirror, its image is projected in such a way that it seemed to stand on the stand.

Illusionist Alexander Fürste built this trick as follows. The viewer saw a small scene with highly reduced artists. To predict them in this form on the screen, Fürst used an angular mirror in its construction. It was before him the artists moved. But the mirror turned over to 180 ° and thereby put "on the head", and already this image is a concave mirror, turning over again, discarded on a small scene. An indispensable effect of the effect was the perfect purity of all mirrors.

Of course, the "wizard" could demonstrate not only the emergence of some objects, but also their lightning disappearance, it was worth only to pronounce the magical "simsalabim" (and, of course, turn off the light source or unscrew the mirror). As challenges such a Tanagra Theater (so called similar spectacles), you can make sure to look into the inverted binoculars. Reduced, as if the concentrated world looks very interesting in it. The principle of operation and prismabbing binoculars, and the Tanagrian theater is the same. Only in one case are used lenses, and in the other - a concave mirror.

About left-handers and right-handers

Now, when we know how the mirrors work and how they are manufactured, we will think about what we see in the mirror in our daily life.

It can turn into a hobby: analyze every item from the point of view of symmetry. Recall that if you cut the object along its plane of symmetry and put one of the halves perpendicular to the mirror, then in the mirror, as it were, the second, "sliced" half. Therefore, whether we speak the mirror or about the symmetry plane, we are in essence, about the phenomena of one order.

In principle, all possible "magic" optical tricks are based on "seamless" image transition to its mirror reflection. The secret of the "cut in half of the ladies" and other similar focus you can easily comprehend and reproduce, using a trellling consisting of several mirrors. Turn one of the small mirrors inside so that it was clearly visible in the Big Mirror. Put your hand on the edge of a small mirror so that the middle finger will lie in parallel to the edge, and you will see in the mirror that your hand consists of two little fingers and two ring fingers. Hop the little finger, and two fingers move in the mirror. A little fantasy - and this "number" can be prepared for a demonstration on the home evening. The condition of success here, as in the variety or circus, is in the flawless purity of the mirror. A good and quite large mirror (so that its edges can not be seen) for the eyes not noticeable.

Muckers are always released with the calculation that they will be taking their right hand. But all the left-handed would prefer the bucket in the "Mirror" execution

After we mentally divide the planes of symmetry chairs, tables, vases, people, animals, houses and trees, we, of course, want to search asymmetric bodies.

We have already mentioned screw staircases and screw cutting. Perhaps, the properties of asymmetry should be clarified again: symmetry plane cannot be carried out through an asymmetrical subject ( The author here refers to symmetrical only those bodies that possess symmetry planes. In modern teaching on symmetry, symmetric bodies include all figures consisting of equal naturally repeated parts. In particular, both the shapes with the screw lines, considered as infinitely extended systems, have screw axes of symmetry, that is, considered symmetrical. - approx. Red). Therefore, it is impossible to reflect it in the mirror. Conversely: Each helix twists in the mirror "in the other direction." The left round becomes right. The left hand turns into the right. Maybe from here and went to the words "left-handed" and "right-handed"?

However, there may be objections: how can a person, creatures endowed with the plane of symmetry, can "change" in the mirror of the hands or ears?!

In order to figure out, imagine that only a hand is visible in the mirror, without its owner. You can try yourself, getting sideways to the mirror, put one arm in front of him. Or simply consider your gloves carefully. They relate to each other as the image and its mirror reflection. But if you can cut the middle of the cube, then do not distinguish the halves! They are combined (mentally) without any work.

At the cup, the surface is symmetrical: it can be drunk on the right and on the left. But our grandfathers used special cups for Usach. From above, such a cup had a visor that the proud mustache would not dip in coffee. The hole through which the cup was filled and drank, was on the one hand. Such a cup is no longer symmetrical. She was done either for the left or right.

Scissors, as a rule, are made for the right hand. You immediately amuse it, as soon as you try to crash on it, in the scissors in the left hand. Muckers are also always made for the right hand. Among the souvenir little things sometimes as curious corks for the left hand are for sale: after all, Lefters is very uncomfortable to open a bottle with a normal corkscrew. Asymmetrical, of course, such objects like a shut screw or aircraft. Before large hydroplanes had two propellers: pulling and pushing. It is easy to imagine how they rotated. Or take, for example, a pencil sharpener into the right hand, and the left rotate the griffel. You will immediately notice that asymmetry is manifested here.

Finally, look at guitars, violins and other string tools. They are symmetrical (if you do not take into account the thickness of the strings and the location of the rings). But the whole system of violin and a suit asymmetric. It would be curious to find out if there are left-handed among the violinists!

Charlie Chaplin and sea nodes

And great people have their own problems. Very important for public figure Question: Where to give your hands? In the film "The Great Dictator" unsurpassed Charlie Chaplin is trying to find a solution to this problem before seemingly to the people. It becomes in front of the mirror. Of course, it would be best to just put your hands in your pockets. But it is impossible to drop your dignity! And here Chaplin goes through all imaginable positions. In the end, he crosses his hands on his chest in a pose, in his opinion, the most impressive contemporaries.

Considering paintings, monuments or parade portraits, it is not difficult to note that there are only a few spectacular positions of the hands. But for us, only crossed hands are of interest. Not lazy to try it, you will find that there are two options. Your right hand lies so that her brush is hiding under the left forearm. Or vice versa: the right brush lies on the left forearm, and the left is hiding under the right hand.

Direct sea node is symmetrical. Asymmetric "Babi Knot"

Imagine that it is not a hand, but laces for boots. They can also be overlapping from left to right or right left.

In the language of seafarers, such a simple connection is called "half-hyshtyk". If you do not believe that you tied your limbs, a node, ask you to give you the tip of the rope into each of the crossed hands. Now remove the hands of the armpits - the KNOT "HRISE" will be on the rope.

To this "half" of the node should, naturally, add a second half to get a one-piece node. But if you try to do it, be careful! There are two possible options here. If you "correctly" put the ends of the rope, then you will get a node "flat bayonet". It is worth putting them "wrong", and you will have "Babi Knot", suggesting the disgust to every sailor. Babi Knot is tightened hard, and it is very difficult to untie it. The "flat bayonet" is also tightened firmly, but it is very simple to unleash it, it is only worth moving the corresponding ends towards each other. For us, in both cases there is another significant difference: "flat bayonet" is symmetrical, and the "babium node" is asymmetrical.

But let's go back to Charlie Chaplin. Both crossed hands (or rope ends) are essentially reproducing the screws of the screw and are deprived of symmetry. Therefore, the intertwing ends and it is impossible to mentally translate one to another. They relate as an image and its mirror reflection. And if you cavulus "halftty" in front of the mirror, your reflection in the mirror will tying it "on the contrary." In order for the right sea node after the second overweight, it should be tieted to a mirror in relation to the first one.

Ropes or cables can be suite from left to right or right left. There are ropes (and cables), twisted on the right to left in the letter Z and twisted from left to right by the letter S. This meant the long average element of the letter directed along the rope fibers. The location of these elements in the letters is mirrored with respect to each other, which also applies to the relevant ropes.

Do these young people know that they "tied up" in front of each other's hands with the left and right knot?

However, if you become looking at your lurking rope, it may turn out that it is not a sweater at all, but woven. Twisted ropes under load are stretched, and almost no woven. (Lowering rope, which is stretched when wet underwear hangs, not very comfortable!) Interesting, by the way, that the snail curls his house Z-shaped twist.

In a special book about the maritime assemblies, we find about 4,000 different tasks for the tie of the ropes. Many of these nodes are very attractive in appearance, but hopelessly asymmetrical.

In the pictures depicting vintage sailing ships, it can be seen how the sailors climbs on the masts on the rope stairs. In the sailors it is called "climbing the guys." The guys are long ropes or cables that stretch from the ships to the mast. Rope "crossbars" are attached to them. These short sections of the gear must be attached "tight" (in no case by the node "flat bayonet"!). What does this fix look like, shown in the picture. At first glance, it seems symmetrical, but it is not. The same impression produces all sorts of decorative nodes. They can be found in artistic products, and on military uniforms.

The node "flat bayonet" gives us another excellent example of symmetry. Here it is necessary to consider not only the symmetry of the form, but also the symmetry of the load. Our cross-knot can be tied (correctly!) In such a way that the ends of the rope are first binding to each other, which should later experience the load. But it is possible to tie it and so that the loaded end will be connected to the free, unloaded ("self-distribution" node). V. Ridded by both nodes are practically indistinguishable. However, if you load an incorrectly knotted knot, then it will not hold. As sailors say, the node "eats".

It is it that fockers and illusionists are used in their ideas. Previously, hammocks still existed on the ships, there were always helpful assistants to mount his hammock. Naturally, among the night, the gulling novice turned out to be on the floor.

Mathematics and engineers often have to engage in the nodes and solve the tasks associated with them. It is theoretically interesting to know which types of nodes exist. But practitioners are worried about another question: how to create a transport hub for the unimpeded movement of cars of cars or people. This kind of "nodes" can be seen on the topological scheme of terrestrial and underground transport of Berlin.

There are even patents on the nodes. There is, for example, an American patent based on a special node - Möbius tape. German mathematician Augustus Ferdinand Möbius (1790-1868), twisting once a flat tape at an angle of 180 °, bonded both of its end. This tape has an amazing property. If we, touching the finger of one of her sides (we note that), we will slide along the surface, you will find that this tape has only one surface (not twisted the tape, naturally, has two surfaces). On this property, a patent is based. When using the drive belt (according to the patent description), its inner side running over the leading and slave wheels, over time, is extended and becomes unsuitable. When using the Moebius tape essentially disappears the difference between the inner and outer surface and the belt wear, respectively, is much reduced. Actually, it was patented.

Self-disseminating node, which frequenses often use. If you pull the "desired" end, the node will dissolve

If mabius tape is transparent and put on it some icon, say the letter n, it will be found that the opposite figures correlate as an image and its mirror reflection. This is very curious, considering that the "straight" and "opposite" letters are on one side of the tape! After all, the tape is generally only one surface.

When designing complex intersections, it is important to know one property of the nodes that we derive with the help of an experiment. Draw any transport node. It can be confusing and wrong. Mark only every intersection of the letter, of course, in each case different. Now lead a pencil or finger according to your drawing in the direction, the opposite is what painted you. And every time, passing the intersection, write down the corresponding letter. In order for the result (which we strive to find) was visually, write down letters in two rows: either from left to right or top down. It is just important that you alternate the crossroads (depending on whether the street is held above or under the other). And does not play the role that you have taken the first intersection - upper or lower. When the sign is ready and you should check it out, you will find that each letter denoting the intersection is found in each of the rows one time.

Imagine that you should design a system of traffic lights regulating the passage of transport. All traffic lights included on the green light will be in one row, while all the traffic lights of another row should be included on red.

Focical fockers use knowledge of the theory of nodes for an elegant "experiment to read thoughts." You ask to draw a similar node and designate it with letters (not peeping), and then propose to drive around the obstacle, calling the letters (which the magician writes according to the already known scheme). In some place two intersections are "confused". And the magician, "reading" thoughts, calls the letters. How it is easy to check, the tumbled letters will fall apart in one row.

In conclusion of this section, another question: what happens if the Möbius ribbon is cut along? In the case of a simple, not an inverted ribbon, this is clear: two new tapes will turn out, which will be twice as early. What happens to the Moebius ribbon, which we previously twisted before glueing her ends, difficult and imagine! If one side already "disappeared" after one turn, then in this case you can wait for anything. We formulate a question somewhat differently: what happens if the owner of the patented belt transmission will reduce it along to get two belt transmission from savings? Experience tells us that two new tapes will not work. There will be a closed tape, twice as long as the length. She, although it is passing, but, like any normal tape, again has two sides.

Transportation of milk and floor in the bathroom

Please book several pages back and take a look at five platonic bodies. Only these five bodies (repeat it again) can be constructed from the same right flat figures - faces.

Tetrahedron We are familiar from everyday life. In packages-tetrahedra we buy dairy products. Some time ago, the question was discussed why it is a tetrahedron, and not hexahedron, that is, a cube. After all, the cube has the smallest (after a ball) surface with respect to the volume. Therefore, with such packaging, it would take less packing material for the same volume of milk than when packing into tetrahedra. However, if we look at the scan of both bodies, we will see that tetrahedra can be folded from a continuous moving tape. But Cuba from a simple ribbon will not work. Two squares will always hang around, so trimming will always remain much more than when gluing the tetrahedra packages.

This small example allows you to analyze a common occurrence. Often, in search of the optimal solution, we forget to accurately determine what exactly should be optimized. The Nizhnenenevskaya saying says: "What the owl is suitable, then no one for the nightingale." It sounds like this in the modern way: "If you create the optimal conditions for Solovyov, what will have to be owls!" (And vice versa!)

In our packaging task, you can put a lot of questions, depending on what exactly it should be optimal:

1. What gives the least packaging consumption with the same contents? (Ball, cubic.)

2. What body is easiest to get from a flat sheet by simple folding? (Five plato bodies, that is, not a ball!)

3. What body when assembling has a minimal connective strip, which can be glued, cook or connect another way? (Tetrahedron.)

4. When painting what body is the minimum of cropping? (Tetrahedra.)

5. What bodies can be folded most tightly, without cleansing? (Cube, tetrahedron.)

6. Which body is the least likelihood of "confusing" the edges in the event that it should lie down to a certain side upward (let's say that the marking is visible)? (Tetrahedron, he has the least faces.)

From the formulation of these six questions it is not difficult to understand how carefully should be clarified that we are going to optimize.

If we get the challenge to develop the form of packaging for cargo intended for shipment by the aircraft that determining optimization criteria will be paragraphs 1 (small packaging format) and 5 (dense styling without gaps), since with air transportation every gram cost additional money. But when choosing a packaging for the carriage of milk, the main role is played by paragraph 3 (the smallest length of the gluing line) and even more important - paragraph 4 (minimum waste). More advantages of paragraphs 5 (laying density) and 6 are added here and 6 (the smallest probability to lay packages are not the same side).

If you go around this "node" by arrow, then B.ukva will appear once in the "indirect" row and once - in direct

In front of futurologists, today there is a problem: we will buy milk in tetrahedra in 2000 or only in powder, and maybe we will have to mess with dairy bids again?

However, in this book we are primarily interested in issues closest to the topic.

Right, surprisingly, a polyhedron can also be built from pentagons. And why is it impossible from hexagons? Especially since the hexagon can be built out of six triangles?

Obviously, the matter is not only in the most source flat figure (triangle, square, pentagon), but also how these surfaces, adjoining, are connected to each other. If the hexagons put on the table, it becomes clear that they cover the plane without gaps. It is also characteristic of triangles and squares. But folded from hexagons without deforming them, the bulk body is impossible. If you still try with a light pressure to make such a polyhedron of hexagons, his face will be curved and the form will approach the spherical.

The ball design of a special kind is a soccer ball. Millions of people see this ball on the TV screen many times a week. Hundreds of thousands see it "in nature", at the stadium. Everyone knows that the tires of the ball consist of white and black figures. But, oddly enough, only a few can say with confidence, from which polygons it is made. Even football players fluctuate, remembering, from five or hexagons. This is a typical example of our inattention in everyday life.

Previously, the leather tire was made of two-spin points, similar to those that are torn on an orange peel. Most modern balls have a tire consisting of curved polygons. It weighs about 300 g when the ball circle is about 64 cm and is made up of 12 black and 20 white "fields". The edge of each polygon, regardless of the number of its corners, has a length of 4.3 cm. Around each black pentagon is six white hexagons.

As already mentioned, on the plane hexagon, surrounded by six other hexagons, forms the motive of a solid pattern. Pentagon, surrounded by five hexagons, does not fill the entire plane without gaps. But if with some effort to connect such polygons from the skin, it turns out (with a very good approach) the ball is our soccer ball. Spatially deformed hexagons are also applied in construction when constructing modern lightweight structures.

Thus, only five platoan bodies can be folded from non-deformed flat figures of one type and size.

Great features for combinations of flat figures are opened when drawing patterns from tiles (for example, on the floor in the bathroom). They are infinitely repeated by motives from equilateral triangles, squares and hexagons. But with pentagonal tiles, the tiler hardly could do something. They cannot be folded into a similar pattern.

Special properties of an equilateral or equifiable triangle (for the square consists of two isceived, and the hexagon of six equilateral triangles) are associated with the sum of its corners, which is 180 °. The sum of the angles of any N-parliament is (N - 2) 180 °. In a pentagon, it will (5-2) 180 ° \u003d 540 °. Sharing 540 to 5, we get for each angle of 108 °. At points where all tiles converge, the sum of all angles should be 360 \u200b\u200b°. But from angles of 108 °, it is impossible to compile a total angle of 360 °!

We have already said that the pattern of tiles can be made only if you take the right triangles, squares and hexagons. However, this is true only when the side is applied to the side and angle to the corner. But these three types of polygons will detect differences as soon as we choose another pattern of pattern for our floor. Squares and equilateral triangles will fill the entire plane and in the event that they do not adjoin the angle to the corner. In the motive lined with hexagons, gaps are formed between the adjacent angles and the parties. But these gaps themselves contribute to the creation of new delightful patterns. For hexagons there are four motive of their combination into a single pattern with triangles and squares.

In addition, two more combinations are known in which only squares and triangles are involved, and two, in which the eight, and twelve-broth are also used to ensure. Many mathematicians were fascinated by the creation of "patterns for the tile".

So, it is known that Johann Kepler was engaged in drawing up a pattern of hexagons surrounded by triangles. It is curious that this pattern (and only he) can have a mirror symmetric image. The remaining patterns in the mirror do not change. Only the Kepler pattern turns over.

Taking any deeds and not limited to special rules when they are connected, we can come up with a great set of mosaic patterns. The Russian crystallograph E. S. Fedorov in 1891 proved that at the same time 17 different groups of symmetry are distinguished. In practice, these groups have already been known to the Arabs and used them in the mosaics of Alhambra in Spain.

The person's eye is inclined to continue to crush seen patterns, especially if they are contrasted in color, like, for example, a chessboard. Let's start with a "chessboard" consisting of only two rows of two cells. (Instead of a chessboard, you can take four square tiles of the floor or walls.)

How can the pattern consisting of 2x2 tiles be divided in half? Answer this question, of course, is not difficult. Only one feature passing in the middle either from left to right or from top to bottom and separating two cells (left or from above).

A board consisting of 3x3 cells is not possible to divide in half (not overweight). In some games, however, the gaming fields 3x3, 5x5, etc., eliminate the middle of me so that when dividing the playing field, the whole number of cells obtained in half. But here we will not consider such already from those that fold from an integer cell number, the head can go around.

How many opportunities exist to divide the pattern made up of 4 x 4 cells in half without crossing them? At the same time, we will neglect the distinction of the top - the bottom and the left - right. (Such decisions can be translated into each other by a simple turn.) The one who should be faced with such a division will find, bad - poorly, 6 ways.

And if you try to divide the 6x6 cell field? English Master Puzzle Henry E. Dyudeni found 255 ways to divide such a field. For a chessboard with 64 cells (8x8), the computer calculated 92 263 options for divisions!

There are many similar tasks over which chess players and mathematics are fighting. The tasks of this kind remain favorite: how many queens (or elephants, or looses) can be put on one board so that they do not threaten each other? (For those who do not play chess, it should be noted that the queen has the right to walk in all directions, including diagonal, how much is far away.) Chess lovers identified that there may be 8 queens on the board.

The following question arises: how much is the options for their alignment? In 1850, Franz Science published the answer in the Leipzig "Illustrated newspaper": such main positions 12.

Since we talked a lot about the mirror planes, you need to hope, you, without thinking, will spend the plane of symmetry through a chessboard from top to bottom. This will be the first solution.

You can spend the next plane of the mirror reflection from left to right, two more planes will be diagonally. Thus, we found four more decisions. Now turn the field by 180 ° and again we will carry out two diagonal planes of the mirror reflection and one - from top to bottom. But here to hold the plane of symmetry from left to right we will no longer be able to: it will only give us the same picture that we have already seen.

Thus, by simple mirror reflection and rotation, we added to the main position of the figures seven more options. At one point, this operation is possible for all other basic provisions that found sciences. In the mentioned exceptional case, there are only three reflections. In total, the Ferhi can be simultaneously placed on a chessboard, not threatening each other, in 92 different positions.

This example teaches us how you can benefit from the presence of symmetry. Of course, you first needed to establish that only 8 queens can be on Iol. Then it was necessary to develop 12 main source positions, which, of course, was not easy. But the remaining 80 options could be found, by no means as a specialist in chess. It was enough to know how the mirror acts. On the other hand, it should be recognized that certainly there are many prominent chess players who have never heard of symmetry planes.

To the question of definitions

It is said that all the problem can be viewed from three points of view: with mine, with yours and from the point of view of facts.

Undoubtedly, there is something in this aphorism. A glass can be half empty or half full. In your pocket can be as much as 5 rubles or only 5 rubles! Passengers are experiencing a strong storm, and visiting the captain at the same time feels only a fresh breeze.

We define what a chessboard is. It can be said that these are 64 cells located in 8 longitudinal rows of 8 cells in each, so in general, together they form a square. But it can be expressed differently: this is a square divided by 64 equal square cells. (In both cases, it would be necessary to say about black and white fields, but because for our purposes this circumstance is insignificant, lower this part of the definition.) In the first case, we form a large square of small, in the second - division is large for small ones.

For curiosity, ask how many parts can be divided so that small, but the same squares arise? Obviously, the square shares at least 4 smaller squares. It is impossible to divide it for 2 or 3 squares. Next division, each of the four small squares will be divided into 4 even smaller, that is, the total will become 16 squares. The course of divisions we learned. Result Whether we get a multiplication by 4. Accordingly, with the next division of 16 squares, we will receive 64, that is, a chessboard. There are only two flat figures that can be divided into two equal parts, and these parts will be accurately reduced reproduction of large figures. Since we are accustomed to sharing in half, everything that occurs around, we only have to be surprised that only in two cases we can comply with the condition formulated above. These are such figures: a rectangular raw-headed triangle and parallelogram with the aspect ratio of 1: √ 2.

Such parallelograms in one particular case - in the form of a rectangle - plays a significant role in art and technique. Rectangle, the long side of which is more of its short side in √ 2 times (that is, 1.4142 times), perceived by us as proportionate. It is such or close to it the format of paintings prefer artists.

The photographs are widespread 7x10 formats (last 6x9) and 13x18. If you calculate the aspect ratio, it turns out 10: 7 ≈ 1.43, and 18:13 ≈ 1.38, that is, the numbers close to √ 2 \u003d 1.4142.

More accurately adhere to the relationship 1: √ 2 in the technique. It is based on paper format. Thus, when AO (841 x 1189 mm) format (841 x 1189 mm), the side attitude is 1.413 ≈ √ 2. If the sheet passes in half, for the most side, the format A1 (841x1189 / 2, i.e. 841x594 mm), where 841: 594 \u003d 1.415. Further the big side is folded again. It turns out the format A3. In the next folding, we obtain the known A4 format, in which 291: 210 \u003d 1.414. This division goes further to the A8 format (74:52).

The one who deals with paper knows that there are still two other rows - for superstries and other purposes. The row B begins with 1414: 1000 \u003d 1,414 and a series C - from 1297: 917 \u003d 1,414 ...

The book you read (and would like to hope, not without interest), has a format of 260x200 mm, and 260: 200 \u003d 1.3.

Of course, you noticed that the paper format here is noteworthy as accepted: not through the work of the parties, but through their attitude, but we allowed themselves for greater visibility.

We could say that the calculation of the format of paper corresponding to the standard is made by re-dividing the sheet with the aspect ratio of 1: √ 2, starting with 917x1297 mm format. But it will be more correct to another definition: the standard paper calculation is made by proportional to the increase in the sheet with the aspect ratio of 1: √ 2, sequentially starting from the format 52x74 mm. In both cases, it would be necessary to make a reservation that when dividing (or multiplying), a party is taken with a relative length √ 2.

Recall that the rectangle is only a particular case of a parallelogram and that parallelograms with the aspect ratio of 1: √ 2, as well as a rectangular anoscele triangle, can be divided into two reduced copies.

Pollogram, one of the sides of which is √ 3, can be divided into 3 diminished similar parts. In general form: parallelograms with the aspect ratio of 1: √ n can be divided into n identical similar parts.

There are still many figures with a variety of separation options. We will consider another motive, which was sometimes laid out on antique tiled floors in the corners. These are the trapezoids that the mirror reflection turns into a solid motive pattern. Here again arises "reflection". It means that in such patterns, a combination of flat figures are allowed, which cannot be combined or rotated to combine each other, that is, "left" and "right".

How to put bars or bricks so that the design does not have through "seams"

The drawing given here brings us to divisions without a breaking of continuity. If, with a decrease in paper size, the surface of the figure crossed the gap (fold or trait), then in our main pattern there are lines that do not continue, and rest in other lines. Sometimes it is especially advisable to completely avoid division with ruptures. Let's say, I would like the wall of a brick house to have a seam crossing the entire wall from above to donomis. Instructions for welding boilers and oil pipes of large diameter prohibit contact of two longitudinal and two transverse seams. In each transverse or circular seam, only one longitudinal seam of one direction can be dried. The longitudinal seam of another direction should be surely shifted to the side. Thanks to this, the ruptures in the longitudinal seam will be distributed only to the next transverse seam.

Now you probably have already guessed what task is offered to you: Collect from standard parts (bricks, parquetin or sheets of tin) the surface, which is not disturbing its continuity.

Legends of Rudokopov

In the old days, the rudocks were people purely practical. They did not score their heads to the names of all sorts of rocks, which were met in the gallery, and simply shared these breeds and minerals on useful and useless, unnecessary. They were removed from the subsoil, they melted copper, lead, silver and other metals, and unnecessary piled into the dumps.

For useful (in their opinion) minerals, they looked for visual and memorable names. You can never see a spell-shaped cchedan, but without much difficulty to imagine it by name. It is not more difficult to distinguish the red iron from the brown iron.

For useless stones (as already mentioned - in their opinion), the miners often found names in legends and legends. So, for example, the name of the ore cobalt shine occurred. Cobalt ores are similar to silver and at the extraction sometimes taken for them. When the silver could not be paid out of such ore, it was thought that it was enchanted by mountainous spirits - Koboldami.

When mineralogy has turned into science, the great many breeds and minerals were opened. And at the same time, it was increasingly difficult for the invention for them names. New minerals were often called at the location of the find (Ilmenit - in the Ilmen Mountains) or in honor of the famous person (gheet - in honor of Goethe) or gave him a Greek or Latin name.

Museums were replenished with ambitious collections of stones, which became unbarrous. Chemical tests were not too helped, because many substances of the same composition form sometimes crystals of a completely different appearance. It is enough to remember at least snowflakes.

In 1850, the French physicist Auguste Brave (1811-1863) put forward the geometric principle of the classification of crystals, based on their internal structure / according to Brav, the smallest, infinitely repeated pattern of pattern and is a decisive, decisive sign for the classification of crystalline substances. Brava imagined the crystalline tiny elementary particle of the crystal. Today, with school benches, we know that the world consists of the smallest particles - atoms and molecules. But Brave operated in his ideas a tiny "brick" crystal and explored, what could be the corners between the ribs and in what ratios of his parties could be among themselves ( For greater clarity, the author simplifies the leadership story of Brava. Brave's predecessor - French Crystalographer R. J. Gayui (1743-1822) - really imagined crystals folded from elementary "bricks". O. Brava replaced these "bricks" of their centers of their gravity and thus switched from the "brickwork" of GAYUI to the spatial grid. - approx. Red).

In Cuba, three ribs are always located at an angle of 90 ° to each other. All parties have equal length. In brick, the corners also make up 90 °. But his part of various lengths. In the snowflakes, on the contrary, we will not find an angle of 90 °, but only 60 or 120 °.

Brava found that there are 7 combinations of cells with the same or different sides (axes) and angles. For the corners, he received only two options: equal to 90 ° and not 90 °. Only one angle in its entire system in order of exception has 120 °. In the very bad case, all three axes and all corners of the cell are different in magnitude, while there are no angles in it in 90, no 120 °. All in it is space and crooked, and, you might think in the world of crystals there should be such a place. Meanwhile, it belongs to them, for example, copper sulfate (copper sulphate), whose blue crystals usually like everyone.

In some of these 7 spatial lattices, elementary "bricks" can be packaged differently. For us, who know today about the structure of an atom, it is easy to imagine and demonstrate with the help of ping-pong balls. But 125 years ago, the brilliant idea of \u200b\u200bBrave was innovative and opened up new paths in science very likely that Brava proceeded from the patterns of the tile or the motives of the chessboard.

If we divide the square fields with diagonals, then a new drawing arises from the squares on the corners. In three-dimensional CPOSTRANCE, this corresponds to Cuba laid down on six pyramids. Each such pyramid is half an octahedron.

Those who have ever grown crystals of the table salt, nate that salt can crystallize in the cubes, and maybe in octahedra. In other words, experimental observations coofer with theoretical considerations.

Having tried the possible packaging options for all seven axial systems, brave brought 14 lattices. We bring them here in our modern atomistic image.

Considering the lattices of Brava attentive and trying to mentally build crystals from them, you will probably see how you can spend the plane and axis of symmetry. These opportunities will immediately expand if we form new faces in one of the elementary cells. Take a cube (naturally, mentally!), We will put it an angle and cut (everything is also mentally) all the angles, then it forms completely new triangular facets. And octagons will arise from square faces: thereby new symmetry motifs will appear.

Analysis of symmetry elements in each of the axial systems of crystalline lattices leads to the emergence of 32 symmetry classes. All variety of minerals in nature is divided based on 32 symmetry classes. Armed with these knowledge, we think about the classification of the five bodies of Plato. The fact that the cube, with its three equal axes and three straight corners, belongs to the cubic axial system (Singonia), does not need proof. As part of a more detailed unit, it belongs to the Pentagon-tetrahedral symmetry class ( The cubic system includes 5 of 32 classes of crystallographic symmetry. They own 5 varieties of cube differing in symmetry. The most symmetric cube has 9 planes of symmetry, 3 quadruple, 4 triple and 6 double axes of symmetry. A symmetrical cube, which is in question in the text, has only three double and four triple axes of symmetry. - approx. Red). We will not give the names of other classes here because of their complexity. However, pay attention to the term "tetrahedral", since the tetrahedron is one of the platonic bodies.

And if you have good memory, you remember both the Pentagon-Cahedron, which is also included in this class of symmetry. On the picture, it can be clearly visible as a tetrahedron can be formed from Cuba. The remaining platonic bodies also belong to the cubic system. The ancient Greeks, it should be thought, would be terribly upset, they would know that such a prose mineral, like a sulfur colentan, has the same symmetry as their "perfect" bodies.

A person is able to see thanks to the light. Light quanta - photons possess properties and waves, and particles. Light sources are divided into primary and secondary. In primary - such as the sun, lamps, fire, electrical discharge - photons are born as a result of chemical, nuclear or thermonuclear reactions.

A secondary light source is any atom: absorbing the photon, it goes into an excited state and sooner or later returns to the main one, radiating a new photon. When the beam of light falls on an opaque item, all components of the photons are absorbed by atoms on the surface of the subject.

Excited atoms almost immediately return the absorbed energy in the form of secondary photons, which are evenly emitted in all directions.

If the surface is rough, then the atoms on it are randomly located, the wave properties of light do not appear and the total radiation intensity is equal to the algebraic amount of the radiation intensity of each re-infused atom. At the same time, regardless of the angle of observation, we see the same light flow, reflected from the surface, is such a reflection called diffuse. Otherwise, there is a reflection of light from a smooth surface, for example, mirrors, polished metal, glass.

In this case, the re-release lights atoms are ordered relative to each other, the light shows the wave properties, and the intensities of secondary waves depend on the differences of the phases of neighboring secondary light sources. As a result, the secondary waves compensate each other in all directions, with the exception of a single one, which is determined by a well-known law - the incidence of the incidence is equal to the reflection angle.

Photons seem to be bounced away from the mirror, so their trajectories go from items, as if being behind him, - they see them, looking into the mirror. True, the world of the Cascarlo differs from our: Texts are read right left, the arrows of the clock are spinning in the opposite direction, and if you raise your left hand, our twin in the mirror will raise the right, and he doesn't have the rings in that hand ... Unlike the movie screen, where all the audience See the same image in the reflection mirror for everyone different.

For example, a girl in a picture sees in the mirror not at all, and a photographer (since he sees her reflection). To see yourself, it is necessary to settle down opposite the mirror. Then photons coming from the face towards the view, falling on the mirror almost at right angles and return back.

When they reach the eye, you see your image on the side of the glass. Closer to the edge of the mirror of the eyes, photons reflected by them under some angle are caught. So, they also came at an angle, that is, from items located on the sides of you. This allows you to see yourself in the mirror along with the surrounding situation.

But the mirror is always reflected less than the light than falls, for two reasons: there are no perfectly smooth surfaces, and the light always heats the mirror slightly. From widespread materials, the light reflects the light polished silver (more than 95%).
From it made mirrors in antiquity. But in the open air, silver fumes due to oxidation, and polishing is damaged. In addition, the metal mirror is expensive and heavy.

Now the thin layer of metal is applied to the opposite side of the glass, protecting against damage by several layers of paint, and instead of silver, aluminum is often used for saving. Its reflection coefficient is about 90%, and the difference is invisible.

A person is able to see thanks to the light. Light quanta - photons possess properties and waves, and particles. Light sources are divided into primary and secondary. In primary - such as the sun, lamps, fire, electrical discharge - photons are born as a result of chemical, nuclear or thermonuclear reactions. A secondary light source is any atom: absorbing the photon, it goes into an excited state and sooner or later returns to the main one, radiating a new photon. When the beam of light falls on an opaque item, all components of the photons are absorbed by atoms on the surface of the subject. Excited atoms almost immediately return the absorbed energy in the form of secondary photons, which are evenly emitted in all directions. If the surface is rough, then the atoms on it are randomly located, the wave properties of light do not appear and the total radiation intensity is equal to the algebraic amount of the radiation intensity of each re-infused atom. At the same time, regardless of the angle of observation, we see the same light flow, reflected from the surface, is such a reflection called diffuse. Otherwise, there is a reflection of light from a smooth surface, for example, mirrors, polished metal, glass. In this case, the re-release lights atoms are ordered relative to each other, the light shows the wave properties, and the intensities of secondary waves depend on the differences of the phases of neighboring secondary light sources. As a result, the secondary waves compensate each other in all directions, with the exception of a single one, which is determined by a well-known law - the incidence of the incidence is equal to the reflection angle. Photons seem to be bounced away from the mirror, so their trajectories go from items, as if being behind him, - they see them, looking into the mirror. True, the world of the castorcall is different from our: Texts are read right left, the clock arrows are spinning in the opposite direction, and if you raise your left hand, our double in the mirror will raise the right, and he doesn't have the rings in that hand ... Unlike the movie screen, where All viewers see the same image in the reflection mirror for everyone different. For example, a girl in a picture sees in the mirror not at all, and a photographer (since he sees her reflection). To see yourself, it is necessary to settle down opposite the mirror. Then photons coming from the face towards the view, falling on the mirror almost at right angles and return back. When they reach the eye, you see your image on the side of the glass. Closer to the edge of the mirror of the eyes, photons reflected by them under some angle are caught. So, they also came at an angle, that is, from items located on the sides of you. This allows you to see yourself in the mirror along with the surrounding situation. But the mirror is always reflected less than the light than falls, for two reasons: there are no perfectly smooth surfaces, and the light always heats the mirror slightly. From widespread materials, the light reflects the light polished silver (more than 95%). From it made mirrors in antiquity. But in the open air, silver fumes due to oxidation, and polishing is damaged. In addition, the metal mirror is expensive and heavy. Now the thin layer of metal is applied to the opposite side of the glass, protecting against damage by several layers of paint, and instead of silver, aluminum is often used for saving. Its reflection coefficient is about 90%, and the difference is invisible.

I can say about the photo - it can how to display you as truthfully and change to be unrecognizable. A good photographer uses all the advantages of light, filters, optics, poses, angles, cropping and processing so that you get very beautifully in the photo. More beautiful than in ordinary life. The bad photographer clicks you with the wrong conditions, and the same light, the posture, angle, optics and cropping will make you much worse than you are usually.

And who then takes you truthfully? Are you yourself? No, the wrong answer. So, as we focus ourselves, no one else besides us and does not perceive. Just as in the mirror, we see ourselves only the eyes-in-eyes and with a special facial expression. The rest of people see us without special expressions and from all sides.

Well, who then? He who photographed not you. Or you, but you did not know about it. It should be a natural, reporting photo, not setting. The lighting is natural, the best solar (but not too bright), an angle - from the eye level (other people see you), the pose is relaxed, but not during active actions (for example, you are sitting or talking).

If you are not a photographer, how to understand, you turned out on the photo "How to eat" or conditions change your image too much? The easiest way, if the photo is group (not setting, or minimum of production). Look at the rest of the participants. Are they themselves on themselves? Do they not look much more worse than usual? A little better? Do they have skin color? Such faces? If everything is fine with the rest, then you are most likely in order.

Pay attention to the fact that you moved at the time of photograph. Movements frozen in the photo, almost always look strange. In rare cases, they look cool, but in any of the options, in reality, no one has seen this strange facial expression and poses, they flashed for a split second.

Pay attention to the shadows (light). Too dark shadows, too closely located the light source, the location of it is exactly from the top \\ exactly the side \\ smoothly in front give an implausible appearance. If you see failed dark socies or something like that - then it is not you, it is wrong light. If you see the spot of light on the forehead - consider it makes your face flatter and not so plausible.

In general, people see us in motion. So, probably, the video will be closest to the truth. Recommendations are the same - natural soft light, no posing and staging, shooting from the eye level, do not forget to move away from the shooting object so that there is no distortion, use high-quality equipment (the cheap phone does not suit, if there is nothing like the camera, take at least a dear telephone)

In matters of appearance, we focus, first of all, on our reflection in the mirror. However, it not only can not convey the whole truth, but can also be deceived.

To clarify the question of the truthfulness of the mirrors, you need to remember the lessons of history, physics and anatomy. The reflecting effect of modern mirrors is based on glass properties covered with a special metal layer. In antiquity, when the method of producing glass has not yet been open, the plates of precious metals used as a mirror, most often round shape.

To increase the reflective ability, metal discs were subjected to additional processing - grinding.
The glass mirrors appeared only in the XIII century, they were learned to make Romans, breaking the vessels with a frozen layer of tin inside. Sheet mirrors based on tin alloy and mercury began to produce for 300 years later.

The reflective part of the mirror Many in the old manner is called amalgam, although aluminum or silver is used in modern production (0.15-0.3 μm thick), covered with several protective layers.

How to choose a "truthful" mirror?

The reflecting properties of modern mirrors depend not only on the type of amalgam, but also from the level of surface and "purity" (transparency) of glass. The rays of light are sensitive even to such irregularities that are not visible to human eye.

Any glass defects arising in the process of its manufacture, and the structure of the reflective layer (waviness, porosity and other defects) affect the "truthfulness" of the future mirror.

The degree of permissible distortion displays the mirror marking, it is divided into 9 classes - from M0 to M8. The number of values \u200b\u200bof the mirror coating depends on the method of manufacturing the mirror.
The most accurate mirrors - class M0 and M1 produce the float method. The hot glass is poured into the surface of the hot metal, where it is evenly distributed and cooled. This method of casting allows you to get the most thin and smooth glass.

M2-M4 classes are made by less perfect technique - Fourco. The hot ribbon glass is pulled out of the furnace, passing between the rollers, and cooled. In this case, the final product has a surface with thickening, which cause the reflection distortion.
The perfect M0 mirror is rare, usually on sale the most "truthful" - M1. M4 Marking speaks of insignificant curvature, buy mirrors of subsequent classes, perhaps for the equipment of the room of laughter.

Experts consider the most accurate silver coating mirrors produced in Russia. Silver has a higher reflection coefficient, and domestic manufacturers do not use marking above M1. But in Chinese-made products, we buy mirrors M4, which cannot be accurate by definition. We must not forget about the light - the most realistic reflection provides bright uniform lighting of the object.

Reflection as a projection

Everything in childhood was visited by the so-called laughter room or looked a fairy tale about the kingdom of the curves of the mirrors, so no one needs to explain how the reflection on the convex or concave surface changes.

The effect of curvature is present in even, but very large mirrors (with a party ≥1 m). This is explained by the fact that their surface is deformed under its own weight, so large mirrors make a thickness of at least 8 mm thick.

But the perfect quality of the mirror is not the key to his "truthfulness" for a separate individual. The fact is that, even having an impeccably smooth mirror, which very accurately displays external objects, a person will be reflected with defects caused by its individual characteristics.

What we are accustomed to consider our reflection is in reality is not them - it is just a visual projection, which manifests itself in the cerebral cerebral, thanks to the work of a complex person's perception system.
In fact, perception largely depends on the function of the organs of view (the eye of a person, which looks in the mirror) and the work of the brain transforming the incoming signals into the image. How else can you explain the visual dependence of the distortion of reflection from the molding form?! After all, everyone knows that elongated (rectangular and oval) mirrors are slimming, and square and round visually fell. So the psychology of the perception of the human brain works, which analyzes the incoming information, tie it to familiar subjects and forms.

Mirror and photo - what is true?

Another strange fact is known: many people simplify the separation differences between their reflection in the mirror and their own image, which they see in the photo. Especially this concerns the representatives of the beautiful sex, who wanted to know only one in the old Russian tradition: "I am in the light of all beautiful?"

The phenomenon when a person does not recognize himself in the photo is quite common, because in his inner world he or she see himself otherwise - and largely due to the mirror. This paradox caused hundreds of scientific research. If all scholars are conclusing to a simple language, such differences are explained by the features of the optical device of two systems - the lens of the camera and organ of human vision.

1. The principle of action of the eyeball receptors is not at all like glass optics: the camera lens differs from the structure of the eye lens, and it can also be deformed due to the fatigue of the eye, age-related changes, etc.
2. The reality of the image affects the number of points of perception of the object and their location. In the camera, only one lens, so the image is flat. Vision organs in humans and brain shares, fixing the image, are paired, so we perceive the reflection in the volumetric mirror (three-dimensional).
3. The accuracy of the image fixation depends on the lighting. Photographers often use this feature, creating an interesting image in the photo, straightening from the real model. Considering itself in the mirror, people usually do not change the lighting as it makes a flash of the camera or sofa.
4. Another important aspect is the distance. People look at the mirror got used to the mirror close, then as is photographed more often from afar.
5. In addition, the time you need the camera for a snapshot is negligible, in the photo there is even a special term - excerpt. The photo lens snatches the fraction of a second, imprinting the expression of the face is sometimes impressive.

As you can see, each system has its own characteristics affecting the image distortion. Considering these nuances, it can be said that the photo more accurately fix our image, but only for a moment. The human brain perceives the image in a wider spectrum. And it's not only in volume, but also in non-verbal signals that people send constantly. Therefore, from the point of view of the perception of us with the surrounding people, the reflection in the mirror is more truthfully.