Symmetry as a criterion of external beauty. Facial asymmetry: causes of pathological disorders and methods of their correction
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Date of:
2017-10-17
Views: 18 963 Grade: 5.0
Purpose of the training: correct facial asymmetry at 3 points (eyebrows, eyes, lips).
A person’s face is not symmetrical, just like the body, and this is not surprising.
However, there are cases when facial asymmetry is pronounced and causes you psychological discomfort. Let me make a reservation right away that not all types of asymmetry can be corrected with exercises.
Asymmetry cannot be corrected with exercises if:
- it is caused by bone deformities;
- pathological deformities;
- very “old” neuritis of the facial nerve;
- in some cases, the consequences of Botex injections are the so-called side effects.
Reasons for asymmetry
Also, facial asymmetry largely depends on the condition 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 occurs and its correction should begin with the heels!
But ASYMMETRY can be a consequence of excessive facial expressions, facial expressions and behavioral habits. All this becomes clear when you look closely at your face on video, for example.
Smile, talk, chew on one side only, or constantly raise one of your eyebrows. Remember the existence of muscle memory? And she remembers about you and pulls her active eyebrow upward all the time, and makes one eye look smaller.
How to measure asymmetry?
How to check facial symmetry? Need a photo! Move your hair away from your face and ask to take a photo of you. The photo is like a passport: we don’t smile, we don’t try to look great in the photo.
We take a ruler and draw a horizontal line along the eyes (pupils), eyebrows, and lips. Start with the eyes. After all, our internal spirit level (level) tends to the horizon just in the eye area, so that you can walk smoothly and not fall.
Now let’s 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 for 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 for asymmetry
Let's move on to the exercises. By the way, they can be combined with any of the complexes: , . Just add them to training program. For example, while doing, then do exercises to correct the asymmetry of the same zone.
In the example, I am considering the option of correcting one-sided facial asymmetry, 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 - such asymmetry is called ONE-SIDED.
Facial asymmetry can be diagonal or complex. In such cases, it is better to select exercises individually.
30 reps recommended, at the last count there is a static delay of 5 seconds. The training is based on performing the “BASE” - basic exercises with the addition special exercises to correct the asymmetry of a specific zone.
Forehead. Correction of eyebrow position
Exercise #1: Raising your eyebrows up
This basic exercise. When performing it, pay attention to your eyebrows? Which one rises worse? Which one do you feel less?
Place your fingers above your eyebrows. Push your eyebrows up with force, resist with your fingers. Make sure that during the exercise there are no horizontal wrinkles on the forehead, try to relax your shoulders and lower them down, tightly fix the skin above the eyebrows. After completing the exercise, tap your fingers on your forehead.
Let's move on to a set of exercises for correcting different eyebrow height positions:
Exercise No. 2: alternately raising eyebrows
Place your fingers on the forehead, above the eyebrows and lightly hold the skin with your phalanges so that it does not gather in folds. Now raise your eyebrows alternately: now the left, now the right.
Feel which of the eyebrows rises worse, or when raising one of their eyebrows, tension and discomfort arises. The eyebrow that rises worse needs to be extended by 2 counts: 1-raised, 2-extended. After completing the exercise, tap your fingers on your forehead.
Exercise No. 3: raising one eyebrow
After you have found an eyebrow that works worse and is located lower, you need to “train” it separately.
We fix the eyebrow that is located above with our hand, and lift the other one up, holding the skin above the eyebrow with the phalanges of our fingers so that it does not gather in folds. After completing the exercise, tap your fingers on your forehead.
Eyes
General video:
Exercise No. 1: to strengthen the upper eyelid
This is a basic exercise. While performing, monitor the sensations under index fingers, under one of the fingers the pulsation and trembling of the muscle will be less pronounced. When you close this eye, try to press the lower eyelid a little harder with your upper eyelid.
IMPORTANT! Do not press too hard with your fingers and do not stretch the skin in different directions!
We hold the corners of the eyes with our fingers and close our eyes with a little effort, pressing the upper eyelid onto the lower ones. Try to keep your eyebrows in place and not crawl down behind your upper eyelid, and relax your forehead. Then we open our eyes. After completing the exercise, blink your eyes.
Exercise No. 2: alternate work of the eyes Let's close our eyes one by one. Index and middle finger
We place it in the corners of the eyes, without pressing or pulling the skin. We close our eyes in turn: left, right, left.... When you close one eye, you need to keep the other open. Be sure to relax your forehead so that the eyebrow does not fall down along with the upper eyelid. After completing the exercise, blink your eyes.
General video:
Corners of lips
Exercise No. 1: helps to raise drooping corners of the lips
This is a basic exercise. Use your fingers to fix the nasolabial area (from the corner of the mouth to the nostril). Raise the corners of your lips upward, as if you were smiling, apply resistance with your fingers, the movement of the corners of your lips goes up under your eyes, while the center of your lips is relaxed. Try not to let your fingers run across your face; when you lift, the corner of your lip rests on your fingers.
Exercise No. 2: alternately raising the corners of the lips
Use your fingers to fix the nasolabial area (from the corner of the mouth to the nostril). We lift the corners of our lips up IN ALTERNATION, as if we are smiling with one corner of our lips, we apply resistance with our fingers, the movement of the corners of our lips goes up under our eyes, while the center of our lips is relaxed. Try not to let your fingers run across your face; when you lift, the corner of your lip rests on your fingers.
Exercise No. 3: raising one corner of the lip
Using your fingers, we fix the nasolabial area (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 lift the corner of our lips upward, as if we are smiling with one corner of our lips, we apply resistance with our fingers, the movement of the corner of our lips goes up under the eye, while the center of our lips is relaxed. Try not to let your fingers run across your face; when you lift, the corner of your lip rests on your fingers. P.S. I'm Designing individual programs
The establishment of facial asymmetry has become something of a sensation, since asymmetry is rarely noticeable. It turned out that people differ in the degree of their asymmetry as much as in their facial features. This was confirmed not only by measurements, but also by comparison of portraits composed 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 from the front. They turn out perfect. 
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 on the child’s face. To assess the beauty of a face, what is important is the combination of features and slight asymmetry, which, by the way, is inherent in the faces of all people and does not at all detract from the merits of the portrait. Even in the sculptural images of the 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 face with indisputable strict symmetry of the right and left halves. This is probably why Claudius Galen wrote that “real beauty is expressed in perfection of purpose and that the first purpose of all parts is the expediency of the 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 frequent use of the corresponding muscles.”
Michelle Monaghan
So, we should recognize as a fact the asymmetry of the face, that is, the disparity 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 cause of asymmetry is in most cases the inequality of the structural elements of the skull bones. On the human face, increased asymmetry is determined by 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 taller, then it is also narrower. In this case, the eyebrow is located higher than on the opposite, wider half of the face, and the palpebral fissure is larger. The eye as a whole appears to be turned upward. 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, stands out more sharply, and expresses masculinity. The left half is generally softer, reflecting femininity.
Facial asymmetry has long been observed as a reflection of overall body asymmetry. Attempts were made to restore the face in the portrait using the exact half of the photograph and its mirror image. Right and left half gave different images. They did not coincide with the original version. Facial asymmetry, although layered on disproportions of the right and left halves facial skull, also has its own characteristics. Determined that neural regulation the right facial muscles are richer, movements of the head and eyes to the right are reproduced more readily. Even squinting the right eye turns out to be more habitual.
Candidate medical sciences, plastic surgeon " "
Back in the 15th century, Leonardo da Vinci created drawings depicting 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 in living nature there are no absolutely symmetrical objects: in any of them there is always a unity of symmetry and asymmetry.
| Rice. 1. | |
Throughout history, people have tried to “measure” beauty, 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 were personified in the shining images of gods and goddesses, immortalized in beautiful statues.
According to Greek sculptors, symmetry characterizes harmony, proportionality, and 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 architectural monuments, naturally repeating patterns of traditional ornaments, the amazing harmony of Greek vases (Fig. 2).
The fact of asymmetry of the human face and body was known to artists and sculptors of the ancient world and was used by them to add expressiveness and spirituality to the works they created.
A striking example of asymmetry is the face of the Venus de Milo (Fig. 3). Proponents of symmetry criticized the asymmetry of the forms of this generally accepted standard of female beauty, believing that Venus's face would be more beautiful if it were symmetrical. However, looking at compositional photographs, 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 humans. This symmetry is the main source of our aesthetic admiration for the well-proportioned human body.
This symmetry is not only beautiful, but also functional. Thus, symmetrical limbs allow you to easily move in space, the position of the eyes allows you to create the correct visual image, an even nasal septum ensures adequate breathing. However, the symmetry of living organisms does not manifest itself with mathematical precision due to the unevenness of development and function.
Facial symmetry and beauty standards
Over time, beauty standards have changed, but the principles and parameters that determine the relationships and proportions of the face, and, accordingly, its attractiveness, have been preserved since ancient times. In order for a 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 real 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 are only approximate for its aesthetics due to several reasons:
- Firstly, facial proportions change depending on age, gender, physical development person and are largely determined individual characteristics buildings
- Secondly, the assessment of proportionality becomes more difficult 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 eye slits and eyebrows, and the position of the corners of the mouth. The two sides of a face do not produce the same mirror image, even if the face is perceived by us as perfectly correct.
Thus, the fact of facial asymmetry, expressed by the disparity of the right and left halves, one of which, as a rule, is wider and higher, the other narrower and lower, is generally accepted today.
From the photographs presented in Fig. 4, it is clear that absolutely symmetrical faces clearly differ from the original face image with natural asymmetry. In our opinion, “synthetic” symmetrical faces seem less attractive, as in the original photographs, although we selected the faces of actors whose appearance was rated most highly to create composite portraits. Moreover, it is these faces that are distinguished by 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 actually 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 symmetry does not introduce disharmony, but only favorably highlights individuality.
The majority of patients who contacted plastic surgeon, do not notice the asymmetry of the proportions of their face and body. Therefore, one of the surgeon’s important tasks during the consultation is to draw the patient’s attention to the features of his proportions and 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 can serve as an indication for plastic surgery. However, slight asymmetry of the face only gives it attractiveness and individuality, 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 assess the proportions and symmetry of their face and body. This is exactly what we will talk about.
Can a long nose without spoiling a person’s appearance at all? Definitely yes. If the nose is proportional to his face.
To assess the proportions of your face, you need to go to the mirror and measure three distances:
from the hairline on the forehead to the bridge of the nose
from the bridge of the nose to the upper lip
from the upper lip to the chin.
If they are equal, you are the 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 contain a certain attractiveness and originality of the face, and secondly, the proportions can be changed.
Increasing or decreasing the first distance can be achieved using a hairstyle, as well as giving a certain shape to the eyebrows. The second distance is almost always corrected by changing the length of the nose. The third distance can be visually influenced by correctly selected lipstick or a more durable measure - lip augmentation.
Facial symmetry is also easy to assess. You need to pay attention to the location and shape of the steam rooms anatomical formations: eyebrows, eyes, ears, nasolabial folds.
If they are located at 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. Its sudden disruption is important diagnostic sign for a number of serious neurological diseases.
The easiest way to judge body proportions is by its volume: chest, waist and hips.
In a proportionally built man, chest volume predominates. Geometrically ideal male figure- 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 your waist size should be 1/3 less than these two volumes. Suffice it to recall the well-known standard: 90cm -60cm-90cm. However, the ratio 120cm-80cm-120cm is no less proportional. The geometric expression of the ideal is the hourglass shape.
Visually the desired proportions are achieved by clothing, corsetry, certain exercise. However, there are problem areas that are quite difficult to correct, for example, the notorious “breeches” - top part lateral surfaces of the thighs. Properly performed liposuction can help effectively here.
Body symmetry is also assessed by paired formations. The collarbones, nipples, shoulder blades, anterior superior spines should be at the same level iliac bones, gluteal folds.
It's worth knowing that visible violation body symmetry is always a reason for thorough examination musculoskeletal system.
In general, when assessing your appearance by any parameter, be it proportionality, symmetry or something else, you don’t need to be overly picky.
Certain features, imperfections, disproportions are what distinguish us from each other, and therefore make us unique.
Let’s not figure out for now whether an absolutely symmetrical person actually exists. 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 for most people. Yet these are only minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same! Stop. It's worth stopping here. If our hands were really exactly the same, we could change them at any time. It would be possible, say, by transplantation to transplant left palm on 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 in front of you is dedicated 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 driven by the desire to follow nature as closely as possible in their works. The canons of prodortius compiled by Albrecht Dürer and Leonardo da Vinci are well known. 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 searching 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 to be such a measure).
IN modern schools In 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 is eight times the size of the head. At first glance this seems strange. But we must not forget that the majority tall people They are distinguished by an elongated skull and, on the contrary, 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 size 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 resemblance or likeness in a few pages.) However, our proportions are only approximately consistent, and therefore people are only similar, but not the same. In any case, we are all symmetrical! In addition, some artists especially emphasize this symmetry in their works.
PERFECT SYMMETRY IS BORING
And in clothing, a person, as a rule, also tries to maintain the impression of symmetry: the right sleeve corresponds to the left, the right trouser leg corresponds to the left.
The buttons on the jacket and on the shirt sit exactly in the middle, and if they move away from it, then at symmetrical distances. Only rarely does a woman have enough courage to wear a truly asymmetrical dress (we will see further how strong deviations from symmetry are permissible).
But against the background of this general symmetry, in small details we deliberately allow asymmetry, for example, combing our hair in a side parting - on the left or on the right. Or, say, placing an asymmetrical pocket on the chest on a suit, often emphasized by a scarf. Or putting a ring on ring finger only one hand. Orders and badges are worn on only one side of the chest (usually on the left).
Complete flawless symmetry would look unbearably boring. It is small deviations from it that give characteristic, individual features. The famous self-portrait of Albrecht Durer at first glance seems absolutely symmetrical. But, taking a closer look, you will notice a small asymmetrical detail, which gives the picture liveliness and vitality: a strand of hair near the parting.
And at the same time, sometimes a person tries to emphasize and strengthen the difference between left and right. In the Middle Ages, men at one time sported trousers with trousers different colors(for example, one red and the other black or white). And these days, jeans with bright patches or colored stains were popular. But such fashion is always short-lived. Only tactful, modest deviations from symmetry remain for a long time.
WHAT IS SIMILARITY?
We often say that 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 understand what is meant by similarity or resemblance in mathematics. For 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 part of the nose (for example, the bridge of the nose) must be proportional to all the others.
This law of similarity is sometimes fraught with a catch. For example, in a problem 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 top of tower A through the top of tower B, then they will meet, respectively, 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 to solve this problem it is enough to pick up a compass and a ruler. But it immediately turns out that there will be an infinite number of answers. In other words, there cannot be a clear answer to the question about the value of X.
In this book, you will often encounter problems that require thinking. This has a certain pedagogical meaning. Problems of this kind, even if they do not have a solution, such as the one proposed above, relate to some problem that lies within the limits of our knowledge. For the most part, these are the very limits before which the famous “ common sense", and only strictly mathematical logical thinking coupled with natural science knowledge can lead to the right 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 part of the body, be it the nose or mouth, are different, the proportions of similar individuals are the same.
A striking example of similarity is the visual estimation of distance using the thumb. In this way, military men and sailors estimate the distance between two points on the ground or at sea, comparing them with the width of a finger or fist. In the simplest case, close one eye and look with an open eye on 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" about 6°
If you open it first closed eye(and close the second one), finger on apparent distance will move to the side. In degree terms, this distance is 6°. And, moreover, the magnitude of this “leap” (within the permissible error) is the same for all people! So, the right-flank company, a guy two meters tall, and the smallest one - the left flank, 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 that govern our vision.
The “fist rule” is also known - in the most literal sense of the word - for a rough estimate of the size of the angle. If we look with one eye at the fist of an 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 will be 3°. Fist and protruding to the side thumb will be 15°. By combining these measurements, you can approximately measure all the angles on the ground.
And finally, another angular measure of our body, which can be useful for household work. Angle between thumb and the little finger of the outstretched palm is 90°. This seems unlikely, but you can immediately check everything yourself by placing the outstretched fingers of your palm on the corner of our book. Place your little finger exactly parallel to one edge and move your hand down along it until your thumb also rests on the bottom edge. Are you sure?
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 back at different distances. But for the first test, which allows you to decide whether the measured angle deviates significantly from the right angle, this method is quite suitable.
LINELAND AND FLATLAND
People endowed with imagination have long noticed that the laws of congruence, so strict for two-dimensional space, when applied in practice often require the use of a third dimension.
When setting the table for a formal reception, napkins are usually folded into a triangle. But as soon as you collect these triangles in a stack, one on top of the other, it turns out that there are two types of these triangles: some immediately “fit” each other, while others have to be turned over “on right side" A similar problem occurs when stamping small parts when someone tries to fold finished products in stacks.
Poets and writers tend to fantasize around more or less probable situations. Thus, there are works in which life is depicted in two-dimensional space (where you can’t turn the “napkin” over).
Some authors go even further and try to imagine life in one-dimensional space, in the Straight Country - Lineland. Lineland is inhabited only by thin wooden sticks, which in the simplest case are no different from each other. However, as soon as you give them heads (matches immediately come to mind!), they immediately have two possibilities.
Or all the matches have their heads facing the same direction - then combining them does not cause any difficulties. Or some of the matches lie with their heads to the left, and some with their heads to the right. The mathematician from Lineland has no practical ability to convert “left” matches into “right” ones. But a mathematician from the Land of Flatness - 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 simple. Let's imagine that the inhabitants of this country are small rectangles with an eye (and they only have one eye) in one of the corners. Such a rectangle can, of course, only be seen in a plane, and he 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. The houses of the Flatlanders would be similar to those 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 supports would be needed to prevent the wall of the house from collapsing when its occupants wanted to open the door. And two Flatlanders could only look at each other 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 picture everything possible consequences 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 topic arose in England.
In 1880 English teacher Edwin Ebony Abbott wrote a book about Flatland and its inhabitants ( Abbott E. E. Flatland. In the book: Abbott E. E. Flatland. Burger D. Sferlandia. -M.: Mir, 1976). Abbott's Flatlander, having found himself in Lineland in a dream, tries in vain to convince the inhabitants there of the existence of a plane.
In the course of the action, one of the Flatlanders manages to understand three-dimensional space, for which he is recognized as “the craziest of the crazed.”
More than twenty years later, in 1907, C. G. Hinton published the novel The Flatland Incident. In it, two Flatland people wage war. Since all the Flatlanders face the same direction, one of the people is always at a hopeless disadvantage: he cannot turn and strike back in the right direction - the hated enemy is constantly sitting on his neck. But in the end, good wins. Some kind of clever mind notices that Flatland is located on a ball and, therefore, you can run around it and get behind the enemy’s rear.
The author of the novel builds his story on the tacit assumption that the Flatlanders can only move along certain general directions that exclude sideways detours, and that it is impossible for them to overthrow the enemy.
As can be seen, the most sophisticated theories have been put forward regarding life in two-dimensional space, but they have never found application. Presumably, both these books and their authors would have been forgotten long ago if Lineland and Flatland had not been so needed to explain the theory of mirror reflection and if the compilers of intelligence problems had not had to turn to Flatland again and again 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, Flatlanders transport goods by rolling platforms onto circles. Every time the load passes the circle, the transport officer there rolls the circle forward and places it in front of the platform.
Many interesting problems arise 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 not a single wheel (more precisely, not a single wheel axle) can move faster than the entire car. But on a flatland vehicle the wheel is not rigidly connected to the load. Upon reflection, it is not difficult to realize that the load here is involved in two movements.
Firstly, it moves along with the axis of rotation of the wheel (this is the same as a car). And besides, the load still rolls around 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 because the wheels always remain behind and have to be constantly moved forward.
Some readers will think: “The problem is really interesting, but so what?”
However, the operating principle of flatland transport finds its place in our technology. Thus, a designer, designing a door in a small room (for example, near a small elevator), is forced to abandon hinges. He divides the door into two halves (if, of course, he comes up with such a trick!), which run parallel to each other. One half of the door is fixedly attached to the roller axis, and the second moves along the circumference of this roller. While one half moves half the width of the door, the other manages to run across the entire width of the doorway (at double speed).
Let's not look down on Flatland and writers' fantasies. Let's assume that the Flatlanders actually live on the surface of the ball. This surface is so large that residents 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 studied in schools. Children there memorize, for example, the following definition: two parallel lines intersect at a finite distance. Or: the sum of the angles of a triangle exceeds 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, you can also see that two meridians form an angle of 90° with the equator. At the point of intersection at the pole, another angle appears. And the sum of all three angles is in any case greater than 180°. 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 whether we humans were also in the position of the Flatlanders. Perhaps, Gauss thought, we also live in a non-Euclidean world, but we just don’t notice it. If this were so, space would be curved (which, of course, we 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 Hohe Hagen, but found no significant deviation from 180°. This, of course, could not serve as indisputable proof, since the triangle could still be too small.
However, one cannot simply compare the non-Euclidean space that was discussed with the space in the theory of relativity. You and I, the Flatlanders and Gauss, are talking about a purely geometric, spatial problem and whether certain axioms are true (for example, 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 corresponding points are equal.
At school we study theorems on the congence of triangles. It has been established, for example, that the areas of triangles are equal if they have one side and two adjacent angles coincide. This means that, although you can use a side and two adjacent angles to construct triangles, the triangles must match in all their parts.
In colloquial speech (which is what we use in this book), we can say that congruent planes are exactly superimposed on each other, or, conversely, if one plane figure is exactly superimposed on 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. They are all congruent. Obviously, both triangles placed on the left will fit if you simply move them. But the triangle placed on the right, although congruent with the two left ones, we cannot combine it with them only by moving it in the plane. No matter how we rotate it in the plane, it will never align with any of the left triangles. To achieve this, you need to lift the triangle above the plane, rotate it in space and put it back on the plane. But if we compare the relative positions 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 we talk about identical congruence. If, when rotated in space, both upper surfaces of the paper are aligned, the flat figures are called mirror-congruent.
Congruent are flat figures that we perceive as equal and that can be combined with each other by shifting in a plane or rotating in space.
CONGRUENCE OF TRIANGLES
Congruence is the property of geometric flat figures to coincide with each other in size and shape.
Identically congruent figures are those that can be combined with each other by rotation and/or shift.
Mirror-congruent figures are those whose combination requires an additional operation of mirror reflection.
There are four signs of triangle congruence. 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 the larger one of one triangle are equal to two sides and the angle opposite the larger one of 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 the other triangle (W, S, W).
SIMILARITY
The coincidence of flat 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 such figures, the corresponding segments are proportional.
By shifting, rotating and (or) mirroring, two similar figures can be brought into a position of homothety. In this position, the corresponding sides of both figures are parallel to each other.
AXIAL SYMMETRY
Let the 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 the points of this half-plane will coincide with the points of the other half-plane.
The straight line s is called the axis of symmetry.
Due to the fact that the points on the inverted half-plane are in a mirror position with respect to their original position, this inversion is also called a specular reflection. If you draw lines on one half-plane indicating certain directions of rotation, 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. A figure reflected in this way is identically congruent.
Subsequent rotation operations will not affect the identity of the figures in any way. At a rotation angle of 180°, we speak of central symmetry.
DICE TRICK
Teachers say that playing with blocks develops spatial imagination. And so parents buy their offspring boxes with bright cubes covered with fragments of pictures from popular fairy tales. Putting these cubes together in the right way, you will see Little Red Riding Hood with Gray Wolf or Snow White and the Seven Dwarfs.
In fact, these kinds of cubes and puzzles develop spatial imagination not only in children, but in everyone - from young to old. Sometimes we have to fold a cube from various shapes Churbachkov.
Upon closer inspection of these individual elements, it appears that at least two of them have the same shape and size, but are related to each other like a left and right glove. The creators of these types of puzzles obviously hope that players will not immediately notice this difference. If we remember how many times we confused the right and left gloves, we will have to admit that such hopes are not without foundation.
It is almost impossible to combine these elements. It should be noted that when we use the expression “practically possible” here (or somewhere below), we mean the implementation of such a task in practice.
But there are also mathematical or physical methods, allowing to combine elements at least theoretically or according to external signs - this will be the subject of further consideration. And since we were talking about combining one element with another, 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 rotating them in space. In Lineland, in the same way, just one more dimension would be needed: one rotation in the plane, and the segments become compatible.
But we can only rotate spatial buildings in space! And since the fourth dimension, despite all Gauss’s reasoning, is closed to us, it is difficult to even imagine how practically (!) we can deploy our “bricks” somewhere other than three-dimensional space so that they fit together!
IN Everyday life We very often have to solve similar puzzles (I emphasize: solve them practically, not play!), for example, when packing various items. Or, for example, imagine central heating radiators. Some of them have the adjustment valve on the left, while others have it on the right. How to connect several radiators into one battery?
Refrigerators, stoves and other household items are usually designed with right- and left-handed handles, keys, and taps. The fantastic ability to rotate such objects in the fourth dimension would greatly please everyone who deals with their transportation and installation.
LOOK IN THE DICTIONARY!
At the beginning of the book we called man a symmetrical creature. Subsequently, 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 encountered the phenomenon of symmetry. These elements corresponded to 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 is understood as “proportionality, complete correspondence in the arrangement of parts of the whole relative to the midline, center... such an arrangement of points relative to the point (center of symmetry), straight line ( axis of symmetry) or 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 often happens with foreign words, the word “symmetry” has many meanings. This is precisely the advantage of such expressions: they can be used in cases where one does not want to give an unambiguous definition or simply does not know a clear difference between two objects.
We use the term “proportionate” in relation to a person, a picture or any object when minor inconsistencies do not allow us to use the word “symmetrical”.
Since we're rummaging through reference books, let's take a look at the Encyclopedic Dictionary ( Soviet encyclopedic Dictionary- M.: Soviet Encyclopedia, 1980, p. 1219-1220). We will find here six articles starting with the word "symmetry". In addition, this word appears in many other articles.
In mathematics, the word “symmetry” has at least seven meanings (among them symmetric polynomials, symmetric matrices). There are symmetrical relationships in logic. Important role symmetry plays a role in crystallography (you will read something about this in this book). The concept of symmetry in biology is interpreted interestingly. It describes six different types of symmetry. We learn, for example, that ctenophores are disymmetrical, while snapdragon flowers are bilaterally symmetrical. We will find that symmetry exists in music and choreography (dance). It depends here on the alternation of beats. It turns out that many folk songs and dances are built symmetrically.
So, we need to agree on what kind of symmetry we will be talking about. Regardless of the nature of the objects under consideration, the main interest for us will be mirror symmetry - the symmetry of left and right. We will see that this apparent limitation will take us far into the world of science and technology and will allow us to test the abilities of our brain (since it is the one programmed for symmetry) from time to time.
GAME OF DOTS AND LINES
We have not yet left Lineland and Flatland. And there is a special reason for this. Even if there are no inhabitants there, then the straight lines and planes themselves are quite real!
Let's think about how things stand with symmetry on a straight line. With the help of two matches we can very simply imagine two possible cases. (We have already examined some aspects of this situation earlier.) Matches can lie with their heads in one direction. Then they fit together easily. Or with the heads (or tips) facing each other. In this case, there is a point on the straight line at which the mirror can be placed in such a way that the match appears to align with its reflection. In other words, there is a center of symmetry on the straight line. We will have to imagine that the mirror fits at one point and half a straight line segment is reflected in it. In mathematical reasoning this is quite possible.
Flat figures are “reflected” in the axes of symmetry
When constructing on a plane, our mirror may still remain a point, or perhaps a straight line. It would probably be more correct to say reverse order: the mirror will be a straight line or a point. After all, if there is a straight line somewhere, then a point center of symmetry is possible on it.
Mirror reflections of halves of planes look the same as real planes: by rotating the plane around a straight line - the mirror - it can be combined with the reflection, hence the expression “axis of symmetry”.
A circle has an infinite number of axes of symmetry. "Clover Leaf" - only one
So, we now know what a center of symmetry and an axis of symmetry are, and also that an object (let's 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. For other figures, the number of axes of symmetry is finite, but still all axes (two or more of them) pass through the center of symmetry. This means that we can rotate the figure by a certain angle (maximum 180°) and it will again lie in exactly the same place as before the rotation.
Let's continue our reasoning about mirror symmetry. It is easy to establish that every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly correct figure, like an oblique parallelogram, is asymmetrical. At first it seems that an axis of symmetry could run parallel to one of its sides. But as soon as you mentally try to use it, you immediately become convinced that this is not so. The spiral is also asymmetrical.
Oddly enough, such a “symmetrical” figure, like a parallelogram, does not have not only axes of symmetry, but also mirror symmetry in general
While symmetrical figures completely correspond to their reflection, asymmetrical ones are different from it: from a spiral twisting from right to left, in the mirror you will get a spiral twisting from left to right. This property is often used in mass games and television competitions. The players are asked to look in the mirror and draw some asymmetrical figure, such as a spiral. And then draw the “exactly same” spiral again, but without the mirror. A 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 is practical life brings a lot of difficulties not only to children, but also to adults. Children often write some letters inside out. Latin N looks like I, instead of S and Z it turns out S and Z. If we look carefully at the letters of the Latin alphabet (and these are also, in essence, flat figures!), we will see among them symmetrical and asymmetrical. Letters such as N, S, Z do not have a single axis of symmetry (as do F, G, J, L, P, Q and R). But N, S and Z are especially easy to write “in reverse” ( They have a center of symmetry. - Approx. edit). The rest capital letters there is at least one axis of symmetry. The letters A, M, T, U, V, W and Y can be divided in half along the longitudinal axis of symmetry. The letters B, C, D, E, I, K - the transverse axis of symmetry. The letters H, O and X each have two mutually perpendicular axes of symmetry.
If you place the letters in front of a mirror, placing it parallel to the line, you will notice that those whose axis of symmetry runs horizontally can also be read in the mirror. But those whose axis is vertical or absent altogether become “unreadable”.
The question of why letters with a longitudinal axis behave differently than those with a transverse axis is quite interesting. Perhaps you will think about it too. We will discuss the reason for this phenomenon later.
There are children who write with their left hand, and all their letters come out in a mirrored, reflected form. The diaries of Leonardo da Vinci are written in “mirror script”. There is probably no compelling reason that forces us to write letters the way we do. It is unlikely that mirror font is more difficult to master than our regular one.
Spelling would not be any easier, and some words, such as OTTO, would not change at all. There are languages in which the outline of characters is based on the presence of symmetry. So, in Chinese writing, the hieroglyph means the true middle.
In architecture, axes of symmetry are used as means of expressing architectural design. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship.
OUR WORLD IN THE MIRROR
From Lineland we got the idea of a center of symmetry, and from Flatland we got the idea of an axis of symmetry. In the three-dimensional world of spatial bodies, where we live, there are correspondingly planes of symmetry. 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 many exactly is not always easy to decide.
Let's put a ball in front of the mirror and start rotating it slowly: the image in the mirror will not differ in any way from the original, of course, if the ball does not have any distinctive features on its surface. A ping pong ball exhibits countless planes of symmetry. Let's take a knife, cut off half of the ball and place it in front of the mirror. The mirror image will again complete 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, a figure with countless axes of symmetry was a circle. Therefore, we should not be surprised that in space similar properties are inherent in the ball. But if a circle is one of a kind, then in the three-dimensional world there is whole line bodies with 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 a segment of a ball. Or let's take examples from life: a cigarette, a cigar, a glass, a cone-shaped pound cake with ice cream, a piece of wire, a pipe.
If we take a closer look at these bodies, we will notice that all of them in one way or another consist of a circle, through an infinite number of symmetry axes there are countless 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.
For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality 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, naturally, as the most perfect flat figure.
In general, these ideas are quite acceptable to this day. Further, Greek philosophers concluded that the Universe must undoubtedly be built on the model of a mathematical ideal. Errors arose from this conclusion, the consequences of which we will discuss later. It is clear that the ancient Greeks did not yet have ice cream puffs! Otherwise, such a prosaic object, which has countless planes of symmetry, could disrupt their harmonious system.
If we look at a cube for comparison, we will see that it has nine planes of symmetry. Three of them bisect its faces, and six pass through the vertices. Compared to a ball, this is, of course, not enough.
Are there bodies that, in terms of the number of planes, occupy an intermediate position between a sphere and a cube? Without a doubt - yes. One has only to remember that a circle, in essence, consists of polygons. We went through this in school when calculating the number π. If we erect an n-gonal pyramid over each n-gon, we can draw n planes of symmetry through it.
It would be possible to come up with a 32-sided cigar that would have the appropriate symmetry!
But if we nevertheless perceive the cube as more symmetrical object than the notorious ice cream pound, this is due to the structure of the surface. The ball has only one surface. The cube has six of them - according to the number of faces, and each face is represented by a square. An ice cream funnel consists of two surfaces: a circle and a cone-shaped shell.
For more than two millennia (probably due to direct perception), preference has traditionally been given to “commensurate” geometric bodies. The Greek philosopher Plato (427-347 BC) discovered that only five three-dimensional bodies can be constructed from regular congruent flat 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 a three-dimensional geometric body be obtained. You can only make one of the squares three-dimensional figure- hexahedron (hexahedron), and from equilateral pentagons - 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 more asymmetrical things and objects in the life and technology around us. As a rule, the screw has a right-hand thread. But sometimes the left one is also found. Thus, for greater safety, propane cylinders are equipped with a left-hand thread so that a reducer valve intended, for example, for a cylinder with another gas cannot be screwed onto them. In everyday life, this means that when camping, before cooking on a camp stove, you should always try which way the cylinder is unscrewed.
Between the ball and the cube, on the one hand, and the spiral staircase, on the other, there are still many degrees of symmetry. We can gradually subtract planes of symmetry, axes and center from the cube until we reach a state of complete asymmetry.
Almost at the end of this series of symmetry we stand, we humans, with only a single plane of symmetry dividing our body into left and right half. Our degree of symmetry is the same as, for example, that of ordinary feldspar (a mineral that forms gneiss or granite together with mica and quartz).
FIVE PLATONIAN SOLIDS
For regular polyhedra the following statements are true:
1. In any polyhedron (including regular ones), the sum of all angles between edges converging at one vertex is always less than 360°.
2. By Euler’s theorem for convex polyhedra
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 equilateral triangles with an angle of 60°. Six such triangles already give 60° X 6 = 360° and, therefore, cannot limit a polyhedral angle.
Three squares (90° X 3 = 270°), 3 regular pentagons (108° X 3 = 324°), 3 regular hexagons (120° X 3 = 360°) define a polyhedral angle.
From Euler's theorem and the shape of the faces it follows that there are only 5 regular polyhedra:
| Face shapes | Number | Platonic solids | |||
| faces at 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)
