Symmetry as a criterion of external beauty. Facial asymmetry: causes of pathological disorders and methods for their correction
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The date:
2017-10-17
Views: 18 963 Grade: 5.0
Purpose of training: correct the asymmetry of the face at 3 points (eyebrows, eyes, lips).
The human face is not symmetrical, just like the body, and there is nothing surprising in this.
However, there are cases when the asymmetry of the face is pronounced and gives you psychological discomfort. I’ll make a reservation right away that not all types of asymmetry can be corrected with the help of exercises.
Asymmetry cannot be corrected by exercises if:
- it is caused by bone deformities;
- pathological deformities;
- very "old" neuritis of the facial nerve;
- in some cases, the consequences of Botox injections, the so-called side effect.
Causes of asymmetry
Also, the asymmetry of the face largely depends on the state of your body. About the relationship between face and body.
In a nutshell, with scoliosis, lordosis, pelvic distortions and other changes in the musculoskeletal system, asymmetry takes place and its correction should start from the heels!
But ASYMMETRY can be the result of excessive facial expressions, facial antics and behavioral habits. All this is revealed by closely looking at your face in the video, for example.
Smiling, speaking, chewing only on one side, or constantly lifting one of the eyebrows. Remember the existence of muscle memory? And she remembers about you and pulls the active eyebrow up all the time, and one eye makes less visually.
How to measure asymmetry?
How to check the symmetry of the face? Need a photo! Move your hair away from your face, ask you to take a picture. A photo is like a passport: we don’t smile, we don’t try to look cool in the picture.
We take a ruler and draw a horizontal line over the eyes (in the pupils), over the eyebrows, over the lips. Start with the eyes. After all, our internal spirit level (level) tends to the horizon just in the area of \u200b\u200bthe eyes, so that you can walk smoothly and not fall.
And now we look at the 3 resulting lines. Perhaps one eyebrow will be higher and the other lower, the corners of the lips may not be on the same line.
Remember that there are acceptable values of asymmetry and this is absolutely natural and does not require adjustment.
Where there are deviations from the horizon, you need to work with the muscles, and for some it will be enough to correct behavioral stereotypes and everything will fall into place on the face.
Exercises for the face with asymmetry
Let's move on to the exercises. By the way, they can be combined with any of the complexes:,. Just add them to your training program. For example, performing, then do exercises to correct the asymmetry of the same zone.
In the example, I consider the option of correcting one-sided asymmetry of the face, when the part of the face located lower relative to its half works worse, you feel it less! For example, the left eyebrow, left eye, left corner of the lip are located lower than on the right side of the face - this asymmetry is called ONE-SIDE.
Facial asymmetry can be diagonal, complex. In such cases, it is better to select exercises individually.
30 reps recommended, on the last account static delay 5 seconds. The training is based on the implementation of "BASE" - basic exercises with the addition of special exercises to correct the asymmetry of a particular zone.
Forehead. Eyebrow correction
Exercise number 1: Raising the eyebrows up
This is the basic exercise. When doing it, pay attention to the eyebrows? Which one rises worse? Which do you feel less?
Place your fingers over your eyebrows. Push your eyebrows up with effort, resist with your fingers. Make sure that during the exercise there are no horizontal wrinkles on the forehead, try to relax and lower your shoulders, tightly fix the skin above the eyebrows. After completing the exercise, tap your forehead with your fingers.
Let's move on to a set of exercises for correcting different positions of the height of the eyebrows:
Exercise number 2: alternately raising the eyebrows
On the forehead, above the eyebrows, place your fingers and phalanges lightly hold the skin so that it does not gather in folds. Now raise your eyebrows alternately: then the left, then the right.
Feel which of the eyebrows rises worse, or when raising one of their eyebrows, tension and discomfort arise. The eyebrow that rises worse must be pulled out by 2 counts: 1-raised, 2-stretched. After completing the exercise, tap your forehead with your fingers.
Exercise number 3: raising one eyebrow
Once you have found an eyebrow that works worse and is located lower, it needs to be "trained" separately.
We fix the eyebrow, which is located above, with a hand, and raise the other up, holding the skin over the eyebrow with the phalanges of the fingers so that it does not gather in folds. After completing the exercise, tap your forehead with your fingers.
Eyes
General video:
Exercise number 1: to strengthen the upper eyelid
This is the basic exercise. During execution, track the sensations under the index fingers, under one of the fingers there is a pulsation, the trembling of the muscle will be less pronounced. When you close this eye, try to push the lower eyelid a little harder with the upper eyelid. IMPORTANT! Do not press hard with your fingers and do not stretch the skin in different directions!
We hold the corners of the eyes with our fingers and with a little effort close our eyes, pressing the upper eyelid on the lower ones. Try to keep your eyebrows in place and not creep down behind the upper eyelid, relax your forehead. Then we open our eyes. After doing the exercise, blink your eyes.
Exercise number 2: alternate eye work
Let's close our eyes one by one. We put the index and middle fingers in the corners of the eyes, do not press or pull the skin. We close our eyes in turn: left, right, left .... When you close one eye, the other must be kept open. Be sure to relax the forehead so that the eyebrow does not fall down along with the upper eyelid. After doing the exercise, blink your eyes.
Lip corners
General video:
Exercise number 1: helps to lift the drooping corners of the lips
This is the basic exercise. Fingers fix the nasolabial zone (from the corner of the mouth to the nostril). We lift the corners of the lips up, as if smiling, with our fingers we resist, the movement of the corners of the lips goes up under the eyes, while the center of the lips is relaxed. Try not to “ride” your fingers across your face; when lifting, the corner of the lip rests on your fingers.
Exercise number 2 alternately raising the corners of the lips
Fingers fix the nasolabial zone (from the corner of the mouth to the nostril). We lift the corners of the lips up IN TURN, as if smiling with one corner of the lip, we resist with our fingers, the movement of the corners of the lips goes up under the eyes, while the center of the lips is relaxed. Try not to “ride” your fingers across your face; when lifting, the corner of the lip rests on your fingers.
Exercise number 3 raising one corner of the lip
With fingers we fix the nasolabial zone (from the corner of the mouth to the nostril) from the side of the corner of the lip, which is located below. We simply fix the opposite corner of the mouth with our hand so that it does not get involved in the work. We raise the corner of the lips up, as if smiling with one corner of the lip, we resist with our fingers, the movement of the corner of the lip goes up under the eye, while the center of the lips is relaxed. Try not to “ride” your fingers across your face; when lifting, the corner of the lip rests on your fingers.
P.S. I develop individual training programs for Facebook building, I conduct classes via Skype. If you are interested -
The establishment of facial asymmetry has become a kind of sensation, since asymmetry is rarely evident. It turned out that people differ in the degree of their asymmetry as much as in facial features. This was confirmed not only by measurements, but also by a comparison of portraits made up of photographs of the right and left halves (one of them must be turned upside down when printing) with an ordinary portrait of a person taken exactly in front. You get completely different faces. 
There is no perfect symmetry in the world. It is a mistake to consider facial symmetry an indispensable condition for its beauty. The mixture of hereditary traits cannot but be reflected in the child's face. To assess the beauty of a face, a combination of features and a slight asymmetry are important, which, incidentally, is inherent in the faces of all people and does not at all detract from the merits of the portrait. Even in the sculptures of Venus de Milo and Apollo Belvedere, their faces do not have complete symmetry. With good reason, we can say that there is not a single person with indisputable strict symmetry of the right and left halves. This is probably why Claudius Galen wrote that "real beauty is expressed in the perfection of purpose and that the first goal of all parts is the expediency of structure." Undoubtedly, P.F. Lesgaft was right when he wrote that “with the harmonious development of all muscles and muscle groups, the face would lose its definite expression. The individuality of facial features is acquired through the frequent use of the corresponding muscles.
Michelle Monaghan
So, it should be recognized as a fact the asymmetry of the face, i.e., the unequalness of its right and left halves: one of them, as a rule, is wider, the other is narrower, one is higher, the other is lower. The reason for the asymmetry is in most cases the unevenness of the structural elements of the skull bones. On the face of a person, the increase in asymmetry is due to the specificity of facial expressions (physiological asymmetry).
Naomi Watts
There are scientific works in which scientists identify the following patterns of facial asymmetry. If one half of the face is higher, then it is also narrower. In this case, the eyebrow is located higher than on the opposite, wider half of the face, the palpebral fissure is larger. The eye as a whole appears to be turned upwards. The left half of the face is usually higher than the right. Many authors still believe that the right half of the face is larger than the left, protrudes more sharply, and expresses masculinity. The left half is generally softer, reflecting the features of femininity. 
Kate Bosworth
Facial asymmetry has long been observed as a reflection of general body asymmetry. Attempts were made to restore the face in the portrait from the exact half of the photograph and its mirror image. The right and left halves gave different images. They did not match the original. Mimic asymmetry, although superimposed on the disproportions of the right and left halves of the facial skull, also has its own characteristics. It has been established that the nervous regulation of the right mimic muscles is richer, the movements of the head and eyes to the right are reproduced more readily. Even the squinting of the right eye is more habitual.
Candidate of Medical Sciences, plastic surgeon ""
Back in the 15th century, Leonardo da Vinci created drawings that depict the “divine” proportions of the human face and body, which are still the standard (Fig. 1). However, these proportions do not take into account the fact that absolutely symmetrical objects do not exist in living nature: in any of them there is always a unity of symmetry and asymmetry.
| Rice. one. | |
Throughout history, people have tried to "measure" beauty, to describe it using mathematical formulas or geometric proportions, thereby making it possible to recreate it. So, in ancient Greece, the order and harmony observed in nature was personified in the shining images of gods and goddesses, immortalized in beautiful statues.
According to Greek sculptors, symmetry characterizes the harmony, proportionality, harmony of natural bodies and the human body. Therefore, the concepts of symmetry and beauty are identical. Suffice it to recall the strictly symmetrical construction of architectural monuments, regularly repeating patterns of traditional ornaments, the amazing harmony of Greek vases (Fig. 2).
The fact of the asymmetry of the face and body of a person was known to the artists and sculptors of the ancient world and was used by them to give expressiveness and spirituality to the created works.
A striking example of asymmetry is the face of Venus de Milo (Fig. 3). Supporters of symmetry criticized the asymmetry of the forms of this universally recognized standard of female beauty, believing that the face of Venus would be more beautiful if it were symmetrical. However, looking at the composite shots, we see that this is not the case.
The very concept of "symmetry" is directly related to harmony. It comes from the ancient Greek word συμμετρία (proportionality) and means something harmonious and proportional in an object. The concept of "mirror" symmetry is applicable to a person. This symmetry is the main source of our aesthetic admiration for the well-proportioned human body.
Such symmetry is not only beautiful, but also functional. So, symmetrical limbs make it easy to move in space, the location of the eyes - to create the correct visual image, a flat nasal septum provides adequate breathing. However, the symmetry of living organisms is not manifested with mathematical accuracy due to uneven development and function.
Facial symmetry and beauty standards
Over time, beauty standards have changed, but the principles and parameters that determine the ratios and proportions of the face, and, accordingly, its attractiveness, have been preserved since ancient times. In order for the face to be harmonious, its various parts must be related in a certain proportion, with the help of which an overall balance is achieved. No part of the face exists or functions in isolation from the others. Any change in any particular part of the face will have a true or apparent effect on the perception of other parts and the face as a whole.
It is natural that all proportions of the human face have only an approximate value for its aesthetics due to several reasons:
- Firstly, the proportions of the face vary depending on the age, gender, physical development of a person and are largely determined by individual structural features.
- Secondly, the assessment of proportionality becomes more complicated depending on the position of the head.
- The third difficulty lies in the asymmetry of the human face, which often manifests itself in the shape of the nose, the position of the palpebral fissures and eyebrows, and the position of the corners of the mouth. The two sides of the face do not give the same mirror image, even if the face is perceived by us as perfectly correct.
Thus, the fact of facial asymmetry, expressed by the unequal right and left halves, one of which, as a rule, is wider and higher, the other is narrower and lower, is generally recognized today.
From the photographs presented in Fig. 4, it can be seen that absolutely symmetrical faces are clearly different from the original image of a face with natural asymmetry. In our opinion, "synthetic" symmetrical faces are not so attractive, as in the original photographs, although we selected for the creation of composite portraits the faces of the actors whose appearance is rated the most highly. Moreover, it is these faces that are more pronounced symmetry than is observed in most people, but slight asymmetry only emphasizes their attractiveness.
Beauty in asymmetry?
So, is the asymmetry inherent in all of us really beautiful or not? It is quite obvious that we do not consider significant violations of symmetry in the structure of the face attractive. However, small deviations from symmetry do not introduce disharmony, but only favorably set off individuality.
Most patients who turn to a plastic surgeon do not notice the asymmetry of the proportions of their face and body. Therefore, one of the important tasks of the surgeon during the consultation is to draw the patient's attention to the features of his proportions, to describe in detail the upcoming changes as a result of the operation. Correction of facial asymmetry is greatly facilitated by the use of minimally invasive methods, such as and.
So, pronounced asymmetry is usually considered unaesthetic, and in such cases, the desire to achieve a more symmetrical appearance is quite natural and may serve as an indication for plastic surgery. However, a slight asymmetry of the face only makes it attractive and individual, and therefore you should not strive for absolute symmetry.
Symmetry and proportionality are important components of a person's external beauty, and in some cases, indicators of health. But not everyone knows how to evaluate the proportions and symmetry of their face and body. This is exactly what will be discussed.
Can a long nose not spoil a person's appearance at all? Definitely yes. If the nose is in proportion to his face.
To assess the proportions of your face, you need to go to the mirror and measure three distances:
from the border of hair growth on the forehead to the bridge of the nose
from bridge of nose to upper lip
from upper lip to chin.
If they are equal, you are a happy owner of a proportional face.
If not, then there is a disproportion, which is not at all a reason for despondency. Firstly, this may be a certain attractiveness and originality of the face, and secondly, the proportions can be changed.
An increase or decrease in the first distance can be achieved with the help of hairstyles, as well as giving a certain shape to the eyebrows. The second distance is almost always corrected by changing the length of the nose. A properly selected lipstick or a more durable measure - lip augmentation - can visually affect the third distance.
Facial symmetry is also easy to assess. It is necessary to pay attention to the location and shape of paired anatomical structures: eyebrows, eyes, ears, nasolabial folds.
If they are located on the same level and have the same shape, then the face is symmetrical. Facial symmetry is very important not only from an aesthetic point of view. Sudden violation of it is an important diagnostic sign in a number of serious neurological diseases.
It is easiest to judge the proportions of the body by its volumes: the volume of the chest, waist and hips.
In a proportionately folded man, chest volume predominates. Geometrically, the ideal of a male figure is an isosceles triangle turned upside down.
In a proportional female figure, the volumes of the chest and hips are approximately equal to each other. And the waist should be 1/3 less than these two volumes. Suffice it to recall the well-known standard: 90 cm -60 cm -90 cm. However, the ratio of 120cm-80cm-120cm is no less proportional. The geometric expression of the ideal is the shape of an hourglass.
Visually the desired proportions are achieved by clothing, corset underwear, certain physical exercises. However, there are problem areas that are quite difficult to correct, for example, the notorious "breeches" - the upper part of the lateral surfaces of the thighs. This is where liposuction can help.
The symmetry of the body is also evaluated by paired formations. Collarbones, nipples, shoulder blades, anterior superior iliac spines, gluteal folds should be at the same level.
It is worth knowing that a visible violation of the symmetry of the body is always a reason for a thorough examination of the musculoskeletal system.
In general, when evaluating your appearance according to any parameter, whether it be proportionality, symmetry, or something else, you do not need to be overly picky.
Certain features, imperfections, disproportions - this is what distinguishes us from each other, and therefore makes us unique.
We will not yet understand whether there really is an absolutely symmetrical person. Everyone, of course, will have a mole, a strand of hair, or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least in most people. Still, these are just minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right hand and both hands are exactly the same! Stop. It's worth stopping here. If our hands really were exactly the same, we could change them at any time. It would be possible, say, by transplantation, to transplant the left hand to the right hand, or, more simply, the left glove would then fit the right hand, but in fact this is not the case.
Well, of course, everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. The book before you is devoted to the issues of symmetry and mirror reflection.
Many artists paid close attention to the symmetry and proportions of the human body, at least until they were guided by the desire to follow nature as closely as possible in their works. Known are the canons of prodorces compiled by Albrecht Dürer and Leonardo da Vinci. According to these canons, the human body is not only symmetrical, but also proportional. Leonardo discovered that the body fits into a circle and a square. Dürer was looking for a single measure that would be in a certain ratio with the length of the torso or leg (he considered the length of the arm to the elbow as such a measure).
In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body exceeds the size of the head by eight times. At first glance, this seems strange. But we must not forget that most tall people are distinguished by an elongated skull and, conversely, it is rare to find a short fat man with an elongated head.
The size of the head is proportional not only to the length of the body, but also to the dimensions of other parts of the body. All people are built on this principle, which is why we are generally similar to each other. (We will return to similarity or similarity in a few pages.) However, our proportions agree only approximately, and therefore people are only similar, but not the same. Anyway, we are all symmetrical! In addition, some artists in their works especially emphasize this symmetry.
PERFECT SYMMETRY IS BORING
And in clothes, a person also, as a rule, tries to maintain the impression of symmetry: the right sleeve corresponds to the left, the right leg corresponds to the left.
The buttons on the jacket and on the shirt sit exactly in the middle, and if they recede from it, then at symmetrical distances. Only rarely does a woman have the courage to wear a truly asymmetrical dress (we'll see later how much deviation from symmetry is acceptable).
But against the background of this general symmetry in small details, we deliberately allow asymmetry, for example, combing our hair in a side part - on the left or right. Or, say, placing an asymmetrical pocket on the chest on the suit, often underlined with a handkerchief. Or putting a ring on the ring finger of only one hand. Orders and badges are worn only on one side of the chest (more often on the left).
Complete perfect symmetry would look unbearably boring. It is small deviations from it that give characteristic, individual features. The famous self-portrait of Albrecht Dürer at first glance seems to be absolutely symmetrical. But, looking more closely, you will notice a small asymmetrical detail that gives the picture liveliness and vitality: a strand of hair near the parting.
And at the same time, sometimes a person tries to emphasize, to strengthen the difference between left and right. In the Middle Ages, men at one time flaunted pantaloons with legs of different colors (for example, one red and the other black or white). And these days, jeans with bright patches or color stains were popular. But such fashion is always short-lived. Only tactful, modest deviations from symmetry remain for a long time.
WHAT IS A SIMILARITY?
Often we say that some two people are similar to each other. Children usually look like their parents (at least according to their grandmothers). Similar but not the same!
Let's try to figure out what is meant by similarity or similarity in mathematics. In similar figures, the corresponding segments are proportional to each other. In our case, we can formulate this situation as follows: similar noses have the same shape, but may differ in size. In this case, each individual section of the nose (for example, the bridge of the nose) should be proportional to all the others.
This law of similarity is sometimes fraught with a catch. For example, in a task like this:
The height of tower A is 10 m. At some distance X from it there is a six-meter tower B. If we draw straight lines from the foot and from the top of tower A through the top of tower B, then they will meet, respectively, with the foot and top of tower C, which has a height of 15 m. What is the distance from tower A to tower B?
It would seem that for the solution it is enough to pick up a compass and a ruler. But then it turns out that there will be an infinite number of answers. In other words, there can be no unambiguous answer to the question about the value of X.
In this book, you will often encounter problems that require reflection. This has a certain pedagogical meaning. Problems of this kind, even if they have no solution, such as the one proposed above, concern some problem that lies at the limits of our knowledge. For the most part, these are the very limits before which the famous “common sense” yields, and only strictly mathematical logical thinking, coupled with natural science knowledge, can lead to the correct decision.
Let us turn again to man: when comparing living beings, the similarity is clearly felt if their proportions coincide. Therefore, children and adults can be similar. Although the mass and size of any of the parts of the body, be it the nose or the mouth, are different, but the proportions of similar individuals are the same.
A striking example of similarity is the visual estimation of distance with the help of the thumb. In this way, the military and sailors estimate the distance between two points on the ground or at sea, comparing them with the width of a finger or a fist. In the simplest case, they close one eye and look with an open eye at the finger of an outstretched hand, using it as a sight.
When sighting with the thumb of an outstretched hand (once with the left eye, and the other with the right), the finger "bounces" by about 6 °
If you open the previously closed eye (and close the second one), the finger will move to the side by a visible distance. In degrees, this distance is 6°. And besides, the magnitude of this "jump" (within the margin of error) is the same for all people! So, the right-flank company, a guy of two meters in height, and the smallest - left-flank one, only sixty meters tall, comparing these "jumps" of the finger, will receive the same value.
The reason for this phenomenon ultimately lies in the similarity of people and, of course, in the laws of optics, which our vision obeys.
The “rule of the fist” is also known - in the most direct sense of the word - for a rough estimate of the magnitude of the angle. If we look with one eye at the fist of the outstretched hand (this time with the same eye), then the width of the fist will be 10 °, and the distance between the two bones of the phalanges 3 °. The fist and thumb protruding to the side will be 15 °. By combining these measurements, you can approximately measure all the angles on the ground.
And finally, one more angular measure of our body, which can be useful for homework. The angle between the thumb and little finger of the spread palm is 90°. It seems unlikely, but you can immediately check everything for yourself by placing the outstretched fingers of your palm against the corner of our book. Place your little finger strictly parallel to one edge and move your hand down along it until the thumb also lies on the bottom edge. Convinced?
Of course, here the error sometimes turns out to be relatively large, since, depending on the age and development of the hand, the thumb can be set aside at different distances. But for the first test, which allows you to decide whether the measured angle deviates significantly from a straight line, this method is quite suitable.
LINELAND AND FLATLAND
Imaginatives have long noticed that the laws of congruence, so strict for two dimensions, often require the use of a third dimension when applied in practice.
When a table is set for a grand reception, napkins are usually folded into a triangle. But it is worth collecting these triangles in a pile, one on top of the other, as it turns out that these triangles are of two types: some immediately “fit” each other, while others have to be turned over “on the right side”. A similar problem arises in stamping small parts, when someone tries to stack finished products.
It is common for poets and writers to fantasize around more or less probable situations. So, there are works in which life is depicted in a two-dimensional space (where you can’t turn the “napkin” over in any way).
Some authors go even further and try to imagine life in a one-dimensional space, in the Land of the Line - Lineland. Lineland is inhabited only by thin wooden sticks, which in the simplest case do not differ from each other. However, it is worth giving them heads (matches immediately come to mind!), And they immediately have two possibilities.
Or all matches are turned heads in one direction - then their combination does not cause difficulties. Or some of the matches lie with their heads to the left, and some of them lie with their heads to the right. The Linelandian mathematician has no practical way of translating "left" matches into "right" ones. But a mathematician from the Land of the Plane - Flatland, who has one more dimension, will immediately find a simple solution: he will turn the match in the plane.
However, according to some writers, life in Flatland is not so easy either. Imagine that the inhabitants of this country are small rectangles with an eye (and they have only one eye) in one of the corners. It can, of course, only see such a rectangle in a plane, and it never manages to look at this plane from above. So no Flatlander will ever be able to imagine what he really looks like: this already requires a view from three-dimensional space. Flatlanders' houses would have been about the same as in children's drawings. With the difference that the doors would be on the side and would open only in the same plane. But the door hinges would have to be made outside the plane, above or below it. In addition, a complex system of props would be needed to prevent the wall of the house from collapsing when its inhabitants wanted to open the door. And two Flatlanders would be able to look at each other only if one of them managed to stand on his head.
The situation would be even more complicated if Flatland were inhabited by two peoples. Let's say left- and right-handed Flatlanders. It takes a lot of imagination to paint all the possible consequences of such a situation, especially considering that we are used to thinking in three dimensions!
Since both Lineland and Flatland were presented to writers in a humorous light, it is not surprising that literature on this subject arose in England.
In 1880 English educator Edwin Ebony Abbott wrote a book about Flatland and its inhabitants ( Abbott E. E. Flatland. In: Abbott E. E. Flatland. Burger D. Sferlandia. -M.: Mir, 1976). Flatlander Abbott, having fallen into Lineland in a dream, tries in vain to convince the inhabitants there of the existence of the plane.
In the course of action, one of the Flatlanders manages to cognize three-dimensional space, for which he is recognized as "the maddest of the mad."
More than twenty years later, in 1907, C. G. Hinton published The Incident in Flatland. In it, two Flatland peoples are at war. Since all Flatlanders face the same direction, one of the Folk is always hopelessly lost: he cannot turn around and strike back in the right direction - a hated enemy is constantly sitting on his neck. But in the end good wins. Some smart head notices that Flatland is located on a ball and, therefore, it is possible, by running around it, to go behind enemy lines.
The author of the novel builds his story on the tacit assumption that the Flatlanders can move only along certain general directions, excluding sideways detours, and it is impossible for them to overturn the enemy over their heads.
As you can see, the most sophisticated theories have been put forward about life in two-dimensional space, but they have never found application. One must think that both these books and their authors would have been forgotten long ago if Lineland and Flatland were not so needed to explain the theory of mirror reflection and if the compilers of quick wit problems did not have to turn again and again to Flatland to extract ideas from its two-dimensionality (by the way, not so long ago a cartoon was created in Hungary about the journey of the schoolboy Adoljar to Flatland).
Among other things, the Flatlanders transport goods by rolling platforms into circles. Whenever a load passes the circle, the local transport officer rolls the circle forward and places it in front of the platform.
There are many interesting problems here. But we are only interested in one thing: if the wheel axle moves at a speed of 10 m per minute, at what speed does the load move?
We know about our earthly car that no wheel (more precisely, no wheel axle) can move faster than the whole car. But in a flatland car, the wheel is not rigidly connected to the load. Thinking about it, it is not difficult to figure out that the load here is involved in two movements.
First, it moves along with the axis of rotation of the wheel (this is the same as with a car). And besides, the load is still rolling along the circumference of the wheel, and at the same time at a speed also equal to the speed of rotation of the axle. Therefore, in general, the load rolls at twice the speed of the wheel. Of course, the load must move faster, if only because the wheels are always left behind and have to be constantly moved forward.
Some readers will think: “The problem is really interesting, but so what?”
However, the principle of flatland transport finds its place in our technology. So, the designer, designing a door in a small room (for example, near a small elevator), is forced to abandon the hinges. He divides the door into two halves (if, of course, he thinks of such a trick!), Which run parallel to each other. One half of the door is fixedly fastened to the axis of the roller, and the second moves along the circumference of this roller. While one half moves half the width of the door, the other has time to run across the entire width of the doorway (at twice the speed).
Let's not look down on Flatland and writer's fantasies. Let's assume that the Flatlanders do live on the surface of the globe. This surface is so large that the inhabitants may not notice its curvature. Naturally, they think that they live on a plane, since they cannot imagine a sphere: after all, the third dimension is, in principle, unfamiliar to them. Therefore, Flatland professors develop Flatland mathematics, which is taught in schools. Children there memorize, for example, such a definition: two parallel lines intersect at a finite distance. Or: the sum of the angles of a triangle is greater than 180°. We, people of three-dimensional space, know that a spherical surface is a two-dimensional non-Euclidean space that does not fit into the usual Euclidean geometry.
Looking at the globe, we see that two meridians, parallel at the equator, intersect at the pole. Looking at the globe, one can also be convinced that two meridians form an angle of 90 ° with the equator. At the point of intersection at the pole, another angle arises. And the sum of all three angles is greater than 180° anyway. But the poor Flatlanders, of course, cannot even imagine all this. They are sure that they live on a plane.
One skeptical mathematician, Carl Friedrich Gauss (1777-1855), seriously wondered if we humans were in the same position as the Flatlanders. Perhaps, thought Gauss, we also live in a non-Euclidean world, but we just do not notice it. If this were the case, space would be curved (which we certainly could not imagine), and a sufficiently large triangle would have a sum of angles other than 180°. Gauss measured the triangle between Brocken, Inselberg, and High Hagen, but found no significant deviation from 180°. This, of course, could not serve as indisputable evidence, since the triangle could still be too small.
However, one cannot simply compare the non-Euclidean space in question with the space in the theory of relativity. We Flatlanders and Gauss are talking about a purely geometric, spatial problem and about whether certain axioms are true (for example, about the intersection of two parallel lines at infinity). Adherents of the theory of relativity introduce time as the fourth spatial coordinate.
ABOUT CONGRUENCE
Two plane figures are congruent if all their angles and line segments between the corresponding points are equal.
At school, we study theorems on the congruence of triangles. It has been established, for example, that the areas of triangles are equal if they have one side and two angles adjacent to it coincide. This means that although you can use a side and two corners adjacent to it to build triangles, the triangles must match with all their parts.
In colloquial speech (which we use in this book), we can say that congruent planes exactly overlap each other, or, conversely, if one plane figure exactly overlaps another, then they are congruent. The same is true for three-dimensional bodies: if they can be combined, then they are congruent.
Look at the triangles shown in the picture. All of them are congruent. Obviously, both triangles placed on the left will be aligned if they are simply moved. And here is the triangle placed on the right, although it is congruent with the two left ones, but we cannot combine it with them only by moving in the plane. No matter how we rotate it in the plane, it will never fit with any of the left triangles. To achieve this, you need to raise the triangle above the plane, rotate it in space and put it back on the plane. But if we compare the mutual arrangement of triangles combined by shifting and inverting, we will see that in both cases their different sides coincide. When sheared, the bottom surface of one paper triangle overlaps the top surface of the second triangle. The spatial orientation of the surface of the paper sheet has not changed. In this case, one speaks of identical congruence. If, when rotated in space, both upper surfaces of the paper are combined, flat figures are called mirror-congruent.
Plane figures are called congruent, which we perceive as equal and which can be combined with each other by shifting in a plane or rotating in space.
CONGRUENCE OF TRIANGLES
Congruence - the property of geometric plane figures to coincide with each other in size and shape.
Shapes that can be combined with each other by rotation and (or) shift are identically congruent.
Mirror-congruent are figures, for the combination of which an additional operation of mirror reflection is necessary.
There are four signs of congruence of triangles. Triangles are congruent if:
1) three sides of one triangle are equal to three sides of another (S, S, S);
2) two sides and the interior angle of one triangle enclosed between them are equal to two sides and the interior angle of another triangle enclosed between them (S, W, S);
3) two sides and the interior angle opposite to the largest of them in one triangle are equal to two sides and the angle opposite to the largest of them in the other triangle (S, S, W);
4) the side and both interior angles adjacent to it of one triangle are equal to the side and both interior angles adjacent to it of another triangle (W, S, W).
SIMILARITY
The coincidence of plane figures in shape, but not in size, is called similarity.
Each angle of one of the figures corresponds to an equal angle of a similar figure.
In similar figures, the corresponding segments are proportional.
By shifting, rotating and (or) mirroring, two similar figures can be brought into the position of homothety. In this position, the corresponding sides of both figures are parallel to each other.
AXIAL SYMMETRY
Let a plane be divided by a straight line s into two half-planes. If we now rotate one half-plane around line 5 by 180°, then all points of this half-plane will coincide with points of the other half-plane.
The line s is called the axis of symmetry.
Since the points on the inverted half-plane are in a mirror position with respect to their original position, this flip is also called a mirror image. If lines indicating some directions of rotation are applied to one half-plane, then after mirror reflection this direction will change to the opposite. Therefore, a single mirroring operation produces mirror-congruent figures. Two such operations lead to identically congruent figures. They correspond to a shift, or rotation.
RADIAL SYMMETRY
Radially symmetrical figures can be aligned with each other by rotating around the point S. This point is called the center of symmetry.
When rotating, the corresponding points of the figures are combined. The direction of rotation does not change. The figure reflected in this way is identically congruent.
Subsequent rotation operations will not affect the identity of the figures in any way. With a rotation angle of 180°, one speaks of central symmetry.
DICE TRICK
Teachers say that playing with blocks develops spatial imagination. And now parents buy their offspring boxes with bright cubes pasted over with fragments of pictures from popular fairy tales. Putting these cubes in the right way, you will see Little Red Riding Hood with a Gray Wolf or Snow White with seven dwarfs.
In fact, this kind of cubes and puzzles develop spatial imagination not only in children, but in everyone - from small to large. Sometimes we have to fold a cube from various shapes of logs.
Upon closer inspection of these individual elements, it turns out that at least two of them have the same shape and size, but relate to each other like a left and right glove. The creators of puzzles of this kind obviously hope that the players will not immediately catch this distinction. If we recall how many times we have confused right and left gloves, we will have to admit that such hopes are not unfounded.
It is almost impossible to combine these elements. It should be noted that, using here (or somewhere below) the expression "practically possible", we mean the implementation of such a task in practice.
But there are also mathematical or physical methods that make it possible to combine elements at least theoretically or according to external signs - this will be the subject of further consideration. And since the combination of one element with another was discussed here, one important circumstance should be especially noted. In Flatland, it would be possible to combine flat figures by taking them out of the plane and turning them in space. In Lineland, in the same way, it would take just one dimension more: one rotation in the plane, and the segments become compatible.
But spatial constructions we can rotate only in space! And since the fourth dimension, despite all the reasoning of Gauss, is closed to us, it is even difficult to imagine how practically (!) our “bricks” can be deployed somewhere other than three-dimensional space so that they are aligned with each other!
In everyday life, we very often have to solve such puzzles (I emphasize: to solve practically, and not to play!), For example, when packing various objects. Or, for example, imagine central heating radiators. For some of them, the valve for adjustment is on the left, for others - on the right. How to connect several radiators into one battery?
Refrigerators, stoves and other household items are usually made with right-hand and left-hand handles, keys, taps. The fantastic possibility of turning such objects in the fourth dimension would greatly please everyone who deals with their transportation and installation.
LOOK AT THE DICTIONARY!
At the beginning of the book, we called man a symmetrical being. In the future, the term "symmetry" was no longer used. However, you have probably already noticed that in all cases when line segments, flat figures or spatial bodies were similar, but without additional actions it was impossible, “practically” impossible to combine them, we met with the phenomenon of symmetry. These elements matched each other, like a painting and its mirror image. Like left and right hand. If we take the trouble to look into the Dictionary of Foreign Words, we will find that symmetry means “proportionality, full correspondence in the arrangement of parts of the whole relative to the midline, center ... such an arrangement of points relative to a point (center of symmetry), a straight line ( axis of symmetry) or a plane (plane of symmetry), in which every two corresponding points lying on the same straight line passing through the center of symmetry, on the same perpendicular to the axis or plane of symmetry, are at the same distance from them ... "( Dictionary of foreign words: Ed. 7th, revised. -M.; Russian language 1980, p. 465)
And that's not all, as is often the case with foreign words, there are many meanings for the word "symmetry". This is the advantage of such expressions, that they can be used when they do not want to give an unambiguous definition, or simply do not know a clear difference between two objects.
We use the term "proportional" in relation to a person, a picture or any object, when minor inconsistencies do not allow us to use the word "symmetrical".
Since we are rummaging through reference books, let's look at the Encyclopedic Dictionary ( Soviet Encyclopedic Dictionary - M.: Soviet Encyclopedia, 1980, p. 1219-1220). We find here six articles beginning with the word "symmetry". In addition, this word is found in many other articles.
In mathematics, the word "symmetry" has at least seven meanings (among them are symmetric polynomials, symmetric matrices). There are symmetrical relationships in logic. Symmetry plays an important role in crystallography (you will read something about this later in this book). The concept of symmetry in biology is interestingly interpreted. It describes six different kinds of symmetry. We learn, for example, that ctenophores are asymmetrical, while snapdragon flowers are bilaterally symmetrical. We will find that symmetry exists in music and choreography (in dance). It depends here on the alternation of cycles. It turns out that many folk songs and dances are built symmetrically.
So, we need to agree on what kind of symmetry we will talk about. Regardless of the nature of the objects under consideration, the main interest for us will be mirror symmetry - the symmetry of the left and right. We will see that this seeming limitation will take us far into the world of science and technology and will allow us to test the abilities of our brain from time to time (since it is it that is programmed for symmetry).
GAME OF DOTS AND LINES
We haven't left Lineland and Flatland yet. And there is a special reason for that. Even if there are no inhabitants there, then the straight lines and planes themselves are quite real!
Let's think about the situation with symmetry on a straight line. With the help of two matches, we can very simply imagine two possible cases. (We have already considered some aspects of this situation earlier.) Matches can lie with their heads in one direction. Then they fit together easily. Or heads (or tips) to each other. In this case, there is a point on the line at which the mirror can be placed in such a way that the match appears to coincide with its reflection. In other words, there is a center of symmetry on the line. We will have to imagine that the mirror fits in one point and it reflects a half line segment. In mathematical reasoning, this is quite possible.
Plane figures are "reflected" in the axes of symmetry
When constructing on a plane, our mirror may still remain a point, or it may be a straight line. It is probably more correct to say it in reverse order: a straight line or a point will serve as a mirror. After all, if somewhere there is a straight line, then a point center of symmetry is possible on it.
Mirror reflections of the halves of the planes look the same as real planes: by rotating the plane around a straight line - a mirror - it can be combined with a reflection, hence the expression "axis of symmetry" arose.
A circle has an infinite number of axes of symmetry. "Clover leaf" - only one
So, we now know what the center of symmetry and the axis of symmetry are, and also that some object (take this neutral word) is symmetrical if one half of it is related to the other, like an image and its mirror image.
A circle has an infinite number of axes of symmetry, and they all pass through a common center of symmetry. Other figures have a finite number of axes of symmetry, but all the same, all axes (two or more of them) pass through the center of symmetry. This means that we can turn the shape to a certain angle (maximum 180°) and it will again lie exactly in the same place as before the rotation.
Let's continue our reasoning about mirror symmetry. It is easy to establish that each symmetrical flat figure can be combined with itself with the help of a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As follows from the number of axes, they are distinguished precisely by their high symmetry. And vice versa: it is not so easy to understand why such a seemingly regular figure, like an oblique parallelogram, is not symmetrical. At first it seems that an axis of symmetry could run parallel to one of its sides. But it is worth mentally trying to use it, as you are immediately convinced that this is not so. Asymmetric and spiral.
Oddly enough, such a "symmetrical"-looking figure, like a parallelogram, does not have not only axes of symmetry, but also mirror symmetry in general.
While symmetrical figures fully correspond to their reflection, non-symmetrical ones are different from it: from a spiral twisting from right to left, a spiral twisting from left to right will turn out in a mirror. This property is often used in mass games and competitions held by television. The players are invited, looking in the mirror, to draw some kind of asymmetrical figure, such as a spiral. And then once again draw the “exactly the same” spiral, but without a mirror. Comparison of both drawings shows that the spirals turned out to be different: one twists from left to right, the other from right to left.
But what looks like a joke here, in practical life causes a lot of difficulties not only for children, but also for adults. Often children write some letters "inside out". Their Latin N looks like And, instead of S and Z, they get S and Z. If we look closely at the letters of the Latin alphabet (and these are, in fact, also flat figures!), We will see symmetrical and asymmetrical ones among them. Letters such as N, S, Z have no axis of symmetry (nor do F, G, J, L, P, Q, and R). But N, S, and Z are particularly easy to spell "the other way around" ( They have a center of symmetry. - Approx. ed). The rest of the capital letters have at least one axis of symmetry. The letters A, M, T, U, V, W and Y can be divided in half by the longitudinal axis of symmetry. The letters B, C, D, E, I, K - the transverse axis of symmetry. The letters H, O and X have two mutually perpendicular axes of symmetry.
If you place the letters in front of the mirror, parallel to the line, you will notice that those with a horizontal axis of symmetry can also be read in the mirror. But those in which the axis is located vertically or is completely absent become “unreadable”.
The question why letters with a longitudinal axis behave differently than with a transverse one is quite interesting. Perhaps you will think about it. The reason for this phenomenon will be discussed later.
There are children who write with their left hand, and they get all the letters in a mirrored, reflected form. The diaries of Leonardo da Vinci are written in mirror type. There is probably no good reason why we should write letters the way we do. It is unlikely that a mirror font is more difficult to master than our usual one.
It wouldn't make spelling any easier, and some words, like OTTO, wouldn't change at all. There are languages in which the inscription of signs is based on the presence of symmetry. So, in Chinese writing, the hieroglyph means exactly the true middle.
In architecture, axes of symmetry are used as a means of expressing architectural intent. In engineering, axes of symmetry are most clearly indicated where deviation from zero is required, such as on the steering wheel of a truck or on the steering wheel of a ship.
OUR WORLD IN THE MIRROR
From Lineland we took out the concept of the center of symmetry, and from Flatland - about the axis of symmetry. In the three-dimensional world of spatial bodies, where we live, there are planes of symmetry, respectively. A "mirror" always has one dimension less than the world it reflects. When looking at round bodies, it is immediately clear that they have planes of symmetry, but how much exactly is not always easy to decide.
Let's put a ball in front of the mirror and start to slowly rotate it: the image in the mirror will not differ from the original in any way, of course, if the ball does not have any distinguishing features on its surface. The ping pong ball reveals countless planes of symmetry. Take a knife, cut off half of the ball and place it in front of the mirror. Mirror reflection will again complement this half to a whole ball.
But if we take a globe and consider its symmetry, taking into account the geographical contours marked on it, then we will not find a single plane of symmetry.
In Flatland, the figure with countless axes of symmetry was the circle. Therefore, we should not be surprised that in space similar properties are inherent in the ball. But if the circle is the only one of its kind, then in the three-dimensional world there are a number of bodies that have an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular or hemispherical base, a ball or segment of a ball. Or let's take examples from life: a cigarette, a cigar, a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.
If we take a closer look at these bodies, we will notice that they all somehow consist of a circle, through an infinite number of symmetry axes of which there are an infinite number of symmetry planes. Most of these bodies (they are called bodies of revolution) also have, of course, a center of symmetry (the center of a circle) through which at least one axis of symmetry passes.
Clearly visible, for example, is the axis of the ice cream cone. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funky cone. We perceive the set of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, of course, as the most perfect flat figure.
In general, these ideas are quite acceptable to this day. Further, the Greek philosophers concluded that the universe, of course, must be built on the model of a mathematical ideal. This conclusion resulted in errors, the consequences of which we will describe later. It is clear that the ancient Greeks did not yet have ice cream pods! Otherwise, such a prosaic object, having an innumerable number of planes of symmetry, could violate their harmonious system.
If for comparison we consider a cube, we will see that it has nine planes of symmetry. Three of them bisect its faces, and six pass through the vertices. Compared to the ball, this, of course, is not enough.
But are there bodies that occupy an intermediate position between a ball and a cube in terms of the number of planes? Without a doubt, yes. One has only to remember that the circle, in essence, seems to consist of polygons. We went through this in school when calculating the number pi. If we erect an n-gonal pyramid over each n-gon, then we can draw n planes of symmetry through it.
One could come up with a 32-sided cigar that would have the appropriate symmetry!
But if we nevertheless perceive the cube as a more symmetrical object than the notorious ice cream pound, then this is due to the structure of the surface. A sphere has only one surface. The cube has six of them - according to the number of faces, and each face is represented by a square. Funtik with ice cream consists of two surfaces: a circle and a cone-shaped shell.
For more than two millennia (probably due to direct perception), "proportional" geometric bodies have traditionally been preferred. The Greek philosopher Plato (427-347 BC) discovered that only five volumetric bodies can be built from regular congruent plane figures.
From four regular (equilateral) triangles, a tetrahedron (tetrahedron) is obtained. From eight regular triangles, you can build an octahedron (octahedron) and, finally, from twenty regular triangles - an icosahedron. And only from four, eight or twenty identical triangles can you get a three-dimensional geometric body. From squares, you can make only one three-dimensional figure - a hexahedron (hexahedron), and from equilateral pentagons - a dodecahedron (dodecahedron).
And what in our three-dimensional world is completely devoid of mirror symmetry?
If in Flatland it was a flat spiral, then in our world it will certainly be a spiral staircase or a spiral drill. In addition, there are thousands of asymmetric things and objects in the life and technology around us. As a rule, the screw has a right-hand thread. But sometimes there is also the left. So, for greater safety, propane cylinders are equipped with a left-hand thread so that it is impossible to screw a valve-reducer designed, for example, for a cylinder with another gas, to them. In everyday life, this means that in camping, before cooking on a camping stove, you should always try which way the bottle unscrews.
Between the ball and the cube, on the one hand, and the spiral staircase, on the other, there are still many degrees of symmetry. From the cube, you can gradually take away the planes of symmetry, axes and the center, until we come to a state of complete asymmetry.
Almost at the end of this row of symmetry we stand, we humans, with only a single plane of symmetry dividing our body into left and right halves. The degree of symmetry we have is the same as, for example, that of ordinary feldspar (a mineral that forms gneiss or granite together with mica and quartz).
FIVE PLATONS
For regular polyhedra, the following statements are true:
1. In any polyhedron (including a regular one), the sum of all angles between edges converging at one vertex is always less than 360°.
2. By the Euler theorem for convex polytopes
where e is the number of vertices, ƒ is the number of faces and k is the number of edges.
The faces of regular polyhedra can only be the following regular polygons:
3, 4 or 5 60° equilateral triangles. Six such triangles already give 60° X 6 = 360° and, therefore, cannot limit the polyhedral angle.
Three squares (90° X 3 = 270°), 3 regular pentagons (108° X 3 = 324°), 3 regular hexagons (120° X 3 = 360°) limit the polyhedral angle.
It follows from Euler's theorem and the shape of the faces that there are only 5 regular polyhedra:
| Face shapes | Number | Platonic Solids | |||
| faces in one vertex | peaks | faces | ribs | ||
| Equilateral triangles | 3 | 4 | 4 | 6 | Tetrahedron |
| Same | 4 | 6 | 8 | 12 | Octahedron |
| Same | 5 | 12 | 20 | 30 | icosahedron |
| squares | 3 | 8 | 6 | 12 | Hexahedron (cube) |
| Correct pentagons | 3 | 20 | 12 | 20 | Pentagon dodecahedron |
(Any face of the Pentagon-dodecahedron is a pentagonal figure, in which four sides are equal to each other, but different from the fifth. - Approx. translation)
