Right to “left” or Love as the highest degree of freedom. Its true purpose, its full and maximum realization! This is the stopping of the mental dialogue ... Because only when we stop the chatter of the mind, we begin to hear the voice of our Soul ...

I wrapped the boiled pork in parchment paper. I'll take it to Besta in the morning, watchdog in the parking lot. Such tasty she will swallow without chewing. And now I'll take a shower, drink tea with honey and go to bed. And wake up the same person. I have a lot of work. Firstly, we need to bring to mind all our developments in the case of the extortionist and transfer them to the investigator. Secondly, you need to negotiate with a local newspaper about advertising. And thirdly…

It was not possible to complete the creation of an action plan for the near future. Damn phone call! Is it Irina again? What else does she want to put on me? I spat out the half-chewed piece of meat out the window and, wiping my hands with a towel on the way, returned to the room. Before answering, he glanced at the display out of habit. This time the number was unfamiliar to me.

- Kirill Andreevich? A man's voice asked me. - I am the senior of the task force private security... Could you drive up to your office now?

Never before had I received a call from the private security. I could not even imagine why the senior of the task force needed it. Maybe Irina did something in her hearts? For example, I broke tinted glass in a black Zhiguli ...

- And what happened?

It might seem that this question turned out to be difficult for my interlocutor. He began to bellow, cough, then reluctantly said:

- You see, the alarm went off on the remote control, we arrived at the facility very quickly, but did not find anyone. Most likely, there was no unauthorized entry into the premises. Maybe you forgot to lock the window with the locks and it was opened by a draft?

"Irina!" - I thought vengefully. She left with a clear conscience, but forgot to close the windows. Or didn’t do it on purpose.

I had to urgently go to the office. At the entrance to the agency there was a police car, next to it two guys in uniform and with machine guns were bored. By their appearance it was possible to judge how much they had to go out on false calls and how tired they were of this occupation. They checked my documents.

- The front door and all the locks are intact, - the red-haired sergeant began to introduce me to the operational situation. He walked next to me, and the barrel of his short submachine gun all the time rested against my rib. - And here is the window - admire - ajar. I will stand here, and you and my colleague go inside through the door.

We entered the entrance and approached the agency's metal door. I opened the lock. The policeman politely dismissed me and entered the office first. For the thieves, our agency was of no value, they had nothing to profit from us, and therefore I was calm like an elephant.

- Nothing is missing? - asked the policeman when we went around one by one all the offices and looked into the lavatory.

The agency was in a familiar order. The smell of perfume has not yet disappeared from Irina's office. For the sake of form, I looked into my desk drawer and then into the refrigerator. He took out a half-bottle of vodka and poured it into plastic cups. The vodka stood in the refrigerator for a month, but it didn't seem to run out of steam. A second policeman joined us. We drank to the toast: "For the vigilance of the private security!" The redhead began to close the window, pressing down on the frame with all his might. The window creaked and squeaked. Every summer, when the heating was stopped, the window frames swelled with dampness. It became difficult to close and even more to open the window, and therefore I immediately dismissed the draft version. Perhaps these were the tricks of Irina. Wanting revenge, she decided to ruin my romantic evening. I put the office on the alarm, went outside and pushed the frame. The alarm went off in the police, and as a result I was dragged out of the house.

“The vodka is so-so,” said the redhead.

I took the hint and reached for my wallet. The militiamen ate some candy, which they found on Irina's table, wished me to be more attentive in the future and left.

It got dark outside the windows. The crimson gleam of the sunset clung to the wall. I was sitting at Irina's table. She took all her things that were always here before - a small mirror, a malachite cat, ballpoint pen that looks like a goose feather. How heavy it is in my soul! As if Irina had died. I felt sorry for her, her orphaned table exuded a kind, warm memory.

I pulled the phone over to me and dialed her cell phone number. I remembered it by heart.

“Listen, don’t call me again,” Irina answered. I heard muffled music and the hum of a car engine.

“You forgot to close the window,” I said with a slight reproach. I tried to behave as if nothing had happened: I am the boss, she is my subordinate, the best subordinate, the best, to whom I never raised my voice, and for minor flaws I only scolded affectionately.

- I closed everything! And I don't give a damn about your problems!

She cut the connection. In one hand I continued to hold the tube, with the other I began to massage my chest. I didn't expect our conversation to be so short. I wanted to invite her to meet and calmly discuss all our problems. Did not have time. Well, let him fail. I'm tired of her whims. I won't call her again. I'll write on my forehead: Don't call Irina!

I got up from the table, took hold of it and turned it around. The new employee will sit facing the door, and not with his back, as Irina did. And it will be a man. No women in my agency!

The telephone rang behind the partition. Who has nothing to do but call a private detective office late at night?

- I'm listening!

I tried to deceive myself, to pretend that I didn’t notice the hope that was pouring in on me - what if it’s Irina? Suddenly she came to her senses, calmed down and now she will say: “Okay, Kirill! Let's put up! "

There is a quiet rustle in the receiver, restrained breathing.

“If you want me to hear you, speak louder,” I asked.

Short beeps. No, this is not Irina. She won't call. And if he calls, he will not remain silent. It’s not in her spirit. Her actions are always whole, have a beginning and an exhaustive end. If she goes to a restaurant with me, she walks there until closing time, and sometimes I have to carry her in my arms. If he follows me into the mountains, he dies, but gets to the top. If he leaves, he does not return. And if he hates someone, then it is better for this person to strangle himself right away ...

Never before had my office evoked such a depressing feeling in me. Irina's ghost hovered between the walls. Her table, wardrobe, leaflets loose-leaf calendar, covered with her hand, and the light, subtle scent of her perfume remained the last link with that, living, real, but already an inaccessible woman... It was hard for me to imagine that tomorrow I would come to the agency and would not see her at my workplace, and Nikulin would no longer tell me with a sarcastic grin: "Come to Irisha, monster, she made you coffee."

I was startled by the phone call. A nefarious invention is the telephone. Someone imposes their will on me, breaks into my world, brazenly declares themselves, demands attention to themselves. And I don’t want to talk to anyone, I don’t even want to know that someone needs it. I looked at the yellow phone, shaking from the bells. It seemed that there was a little evil man sitting there, demanding to be released. How many times has a phone call interfered with my life tonight. And what did it eventually lead to? I feel almost unhappy. People should talk, looking into each other's eyes, feeling each other's breath, seeing facial expressions, gestures - all this is incomparable more information than just words. Would Irina, standing in front of me, allow me to say: "I don't care about your problems!"

I picked up the phone. Silence.

- What do you want from me? I asked.

And again short beeps. I took hold of the wire and pulled the plug out of the socket. That's better. Now go home! Tea with honey - and sleep! And it will be necessary to make repairs in the office in the next few days. And start with Irina's office. Rip off old wallpaper, throw away tables, chairs, flowers with pots, stupid calendars with kittens. All to the garbage dump! Irina is proud, and I am proud too. Chop so chop ...

I already called the security console from the street. What a warm and stuffy evening! Be a thunderstorm. Yes, there are flashes in the dark sky. While silent, dumb, like the reflections of a welding machine. But an hour or two will pass, and the elements will begin to rage over the Coast. Streams of water will rush over the city, streams will flow along the warm asphalt, and people will rush into cafes, under the eaves of houses, covering their heads with handbags, now with plastic bags, now with newspapers, and will shine with silver in the rays of lanterns wet leaves trees, and peals of thunder will drown out the music coming from the embankment.

I had not yet reached my car when I remembered the unfortunate window. So did I close it or not? First, one policeman dabbled with him, pressed on the frame, pulled the handles, then another. Maybe I safely forgot about it and left it ajar? I had to turn around and walk around the house, looking at the windows of the first floor occupied by my agency. Our yard is quiet and dark. Leave the sewer hatch open - you will never notice. Normal people try to bypass this place at dusk. Even the ubiquitous homeless people did not live in dense bushes. And, of course, no driver will leave his car here overnight, and if he does so, he has a lot of chances in the morning to find it “naked”. That is why I was surprised to notice in the dense darkness the "nine" pressed against the bushes. It was pitch black, plus tinted windows. In a word, a black cat in dark room... It seemed to me that the side window, which is next to the driver, is slightly lowered and somewhere in the impenetrable womb of the car, like Mars in the night sky, the bloody light of a cigarette flickers.

However, I immediately forgot about this car. After checking the windows and making sure they were locked, I returned to my Opel. He got behind the wheel, started the engine, but did not move for a while. I like to sit in the dark interior of the car and look at the devices glowing with a ghostly greenish light. Cozy and peaceful. Perhaps it reminded me of a distant childhood. In my bedroom was a bulky, old-fashioned receiver with a wide, upholstered panel. And on it green light the tuning indicator was on. If the wave left, a green ray would spread out on the indicator like a peacock's tail. Mom turned on the receiver in the evenings and found quiet music. And I, before falling asleep, looked for a long time at the glowing in the dark green eye how it changes, shimmers, as if winking at me.

I started slowly. I did not want to go home so much that even by willpower I could not bring myself to press the accelerator pedal harder. I came up with an excuse to go to the other end of the city, to a bakery, where you could buy hot lavash. I rolled along the Derekoiki embankment in the right lane at the speed of a funeral procession. Even the old half-dead "Zaporozhtsy" overtook me. The first raindrops appeared on the windshield. I turned on the radio and put on the Vivaldi cassette. Irina really liked his concerts, especially "The Four Seasons". When the minor tones of autumn sounded, tears rolled into her eyes. Subtle, vulnerable nature.

At the bus station square, I turned around and drove towards the embankment. At the closed gates of the central market, like shadows, wandered either cleaners or homeless people. They collected cardboard boxes, flattened them and stacked them in a pile. The unkempt, stooped man threw the cardboard under the bush, knelt down, then fell on his side, twisted into a ball like a chilled dog, and became motionless. Two girls voted on passing cars, coming out almost to the middle of the road. The cars slowed down, carefully drove around them. The girls jumped with impatience, smoked, sipped from tin cans.

- Hey man! One of them shouted loudly and bent down to get a better look at me through the windshield. - Life is too short to deny yourself small pleasures ...

I often gave Irina a lift home after work. The prostitutes, blinded by the headlights, did not immediately notice the girl next to me and rushed to the car, offering me their services. Irina reacted to this with the holiness and purity of a child raised in a monastery. She was wildly embarrassed, and the blush that appeared on her cheeks was noticeable even at dusk. In her silence, an inexplicable feeling of guilt in front of me was guessed, as if she wanted to say: perhaps I interfered with you? Perhaps you would like to invite them into the car?

I almost crushed the boldly exposed leg with the wheel and, having passed the priestesses of love, involuntarily looked in the rear-view mirror. The girls rushed to the car that followed, and only now I saw that it was the same black "nine" with tinted windows. In any case, like two drops of water similar to the one that stood in our yard. In response to the girls' proposal, a cigarette butt flew out of the window of the nine and, drawing a crimson arc, crashed into sparks on the asphalt.

Maybe this is the same car that Irina was talking about? I didn't get too deep into her words when she mentioned some "nine"; I listened more to the intonation, and I perceived the very warning about surveillance only as Irina's desire to spoil my mood. I was conquered by the words "your agency", and I could not think of anything else.

I pulled the car to the side of the road and stopped. "Nine" drove by, turned into some alley and disappeared around the corner of the house. No, no one is watching me. Nobody needs me. What am I to watch over me? I’m not a politician, not a banker, I don’t have a suitcase full of bucks with me. I am the head of a private detective agency, which is essentially an illegal branch of the local police department. We have no more than thirty orders a year, which brings us a very modest profit, but big troubles, where conflicts with the police are the most common thing.

A grocery store window glowed behind the line of palms. I need to take something for dinner. When the soul is hard, the stomach must be weighed down. For balance, so that in the morning it does not skew. I’ll arrange a nasty supper for myself. I’ll take a jar of olives, pickled mushrooms, sliced ​​uncooked sausage, one hundred grams of sturgeon, caviar, Korean carrots and vodka. And celebrate my loneliness Loneliness is the highest degree of freedom. The pale saleswoman, stupefied from the stuffiness, looked at me with poorly concealed hatred.

- Man, - stretching out the vowels, she said loudly, - we don't keep vodka in the freezer. All wine and vodka products are on the counter.

God be with her, let it be warm. While I set the table, the bottle will have time to become covered with frost in the freezer. Only now I felt that I was hungry. I had breakfast at the Beslan airport before boarding the plane - a cup of coffee and a thin loaf of cheese. And nothing else for the whole day. The weighty package had to be carried with both hands. I was about to kick open the glass door to leave the store when a tall, slender girl in a blood-red skirt obligingly opened it in front of me. It turns out there is still good people in the world! I thanked the girl for her outburst of altruism and left the store. The girl, like a hungry cat who guessed right in the package of sausages, was again next to me. Looking imploringly into my eyes, she babbled:

- I'm sorry! I urgently need to call a friend and say that I will not be able to come to her. Could you give me your cell phone for a second? I will only say two words.

Everything that you wanted to know about it, but did not know that you want it.

When talking about the biomechanics of the human body, the concept of degrees of freedom often arises. For example, it is difficult to do without this when speaking about the structure and classification of joints. At the same time, the method of calculating these degrees of freedom and the resulting numbers often remain in some kind of fog. This article is for those who felt some dissatisfaction and lack of clarity after such conversations.

First, let's explain it on the fingers.

On fingers

If we compare a steam locomotive traveling on rails and a steamboat sailing on the sea, what is the difference between their movement? The locomotive can only run on rails. He will not turn off them anywhere. Can only back up.

The steamer, by contrast, is free to sail in any direction. Especially if there is an endless ocean around it. The locomotive travels only along the line, and the steamer is already on the plane. (Well, okay - along the surface of a sphere. Or, more precisely, a geoid.) For now, let's say - loose and not very correct from the point of view of the accepted terminology - that the degree of freedom of a steamship is clearly greater than that of a steam locomotive.

Now let's take the plane. His degree of freedom turns out to be even greater. He can already take off. He can get to any point in space. If, of course, he is allowed by the dispatch service.

So far we have only looked at the movements of all these machines, or, as physicists usually say, bodies. But there are also twists and turns. The locomotive can neither lift its nose (lift), nor tilt to the side (tilt), nor stand sideways across the rails (turn). Yes, if the track makes a turn, the locomotive will turn along with the rails. But not himself. Therefore, such turns are not counted. They do not increase the locomotive's degree of freedom.

The steamer can already make a turn. Let the sea be calm and smooth as glass for now, so that it is easier to consider it as a plane. Then the ascent to the steamer is inaccessible as well as the steam locomotive. Tilting the steamer to one side is probably difficult. But if we take a small sailing yacht, then tilting it seems to be no problem. Judging by the photographs, they for the most part and they swim: skewed to the side, with the crew hanging overboard and striking their priests along the crests of the waves.

But the plane can do anything: raise / lower the nose, and tilt to the side, and turn. Especially if it is controlled by an ace from the "Russian Knights" squadron. They even fly their tails forward. And upside down. Which, however, no longer increases the degree of freedom of the aircraft - it is already maximum.

For contrast and generality, imagine a mechanism with a zero degree of freedom. He's just not going anywhere: he's broken. And it was not possible to start from the pusher.

Now we will gradually begin to impose scientific rigor.

One degree of freedom

Let's start expressing ourselves more correctly right away. Let's say "degrees of freedom" in the plural. There can be zero, one, two, and so on. It's just a number. Natural, that is, the whole is positive. Now you need to understand how they are considered.

Let's go back to the beginning - to the steam locomotive. Suppose we need to know how to accurately set its position on a straight section of the track near the station. He cannot collapse anywhere. He also cannot switch to another path: we prudently switched all the arrows so that it would not go anywhere. All he can do is drive a few hundred meters in one direction or the other. How do we set its position? Yes, just the distance from some point on the way. For example, from a point that is directly opposite the station entrance. If the locomotive traveled 100 meters from the station in the direction of St. Petersburg, then one number of 100 meters is enough for us to know where it is now. And if he drove the same 100 meters towards Moscow? This is another case. Then we will write a negative number: –100 m. And again we will know exactly where the locomotive is.

Rice. 1. A steam locomotive on a straight section of rails.

So what did we get? In order to know the exact location of the locomotive in our situation, we only need one number. This also means that a steam locomotive - within the framework of the situation we have invented - has one degree of freedom. And this number itself will be called the coordinate of the locomotive. The only coordinate we need to know. Or which we need to tell the driver so that he knows where to drive the locomotive.

Let now we have not a straight rail track, but a winding one. Does it change anything? Nothing, as long as the locomotive can't get away from this path. We can measure the distance along the rails in the same way, and in the same way we can set the position of the locomotive with one number - the distance from the station. He still has only one coordinate, only one degree of freedom.

We can come up with another coordinate system for it. Let it be not a real steam locomotive, but a toy one that runs in a circle. In this case, we can still take the distance from the toy station as the coordinate. 20 cm - the locomotive drove off clockwise. –20 cm - and this is counterclockwise.

Rice. 2. A locomotive on a rail circle. The coordinate is the distance.

But since we have a circle - more precisely, a circle - then it may seem more convenient for us to set the position of the locomotive with an angle. We mark the center of the circle, put a protractor there and measure the angle between the direction to the station - this will be zero - and the direction to the train. So he drove 90 ° clockwise - we consider that his coordinate is 90 °. But he drove 90 ° counterclockwise - then his coordinate will be -90 °.

Rice. 3. A locomotive on a rail circle. The coordinate is the angle.

But we still only need one coordinate. We went from distances to corners, but nothing has changed. The engine still has one degree of freedom.

We will even do this. Since we remember all the time hour hand, then we will use the clock. We put them in the center of the circle and mark the position of the engine in minutes on the dial. Or for hours - this is less accurate, but convenient. A locomotive for 3 hours or for 9 hours - what could be simpler? Again, it only has one coordinate. And one degree of freedom.

Rice. 4. Steam locomotive on a rail circle. The coordinate is the clock on the dial.

Let us generalize: if a body can only move along one line, no matter how curved, it has one degree of freedom. But this is if we are talking only about the location of the body and do not take into account its turns, tilts and rises. Why don't we take it into account? Maybe we don't care. Or maybe it cannot go anywhere, like a steam locomotive on rails.

Two degrees of freedom

So, what about the steamer that sails on the sea? How many coordinates do we need in this case? You can look at the GPS navigator and see: two coordinates. Longitude and latitude. How they are counted there is no longer important to us. As long as we are not interested in where the steamer turned its bow, but we are only interested in what point in the sea it is, we only need two coordinates, which are given to us by the GPS system.

Rice. 5. Steamer at sea. Coordinates: latitude and longitude.

We can come up with our own coordinate system. Suppose, for example, a steamer floats only within sight, and we have a compass and a rangefinder. Then, as coordinates, we can take the direction to the steamer (the angle determined by the compass) and the distance to it (by the rangefinder) from the lighthouse, the tower of which we climbed and which we assigned as the origin. In mathematics, such a coordinate system is called polar.

Rice. 6. Steamer in polar coordinates.

And again we get two coordinates. And two degrees of freedom for the steamer. And again a remark: we are only interested in the position of the steamer at sea. And we are not interested in where he turned his nose and how he bent down.

And if we do not have a ship on the sea, but a hiker in the mountains? It doesn't matter, the tourist also has a navigator and he sees the same two coordinates on it. That is, the surface does not have to be flat.

Let's generalize: if a body can only move on some surface, even if not flat, it has two degrees of freedom. Of course, if we are not interested in his turns and tilts.

Three degrees of freedom

Now it is already easy to deal with the plane. In addition to the two coordinates that the navigator will give us, we also need the flight altitude, which we will determine with the altimeter. (The GPS system also calculates the altitude, but rather approximately.) We get three coordinates and, accordingly, three degrees of freedom.

For an airplane, we can also enter polar coordinates, just a little more difficult. We need two angles: the direction to the plane horizontally (compass), the direction to the plane vertically (some kind of goniometer), and also one distance - from us to the plane (rangefinder). And again we get three coordinates.

Rice. 7. Airplane in polar coordinates.

Let's generalize: if a body can move anywhere in three-dimensional space, it has three degrees of freedom. Again, if we are not interested in how it turned at the same time and where it bent.

And if you are interested?

Rise, tilt, turn

Let's not go back to the steam locomotive, let's stay with the plane, consider the most difficult case.

If it is important for us, not only where the plane is flying, but also how it is located in the air (I think this is important for the pilot), then the three coordinates already available are not enough for us.

The plane can lift or lower its nose - we will call this lifting. It can tilt to the right or left - this is what we call a tilt. And it can turn right or left - it will be a turn. We get three corners - three new coordinates. There are six coordinates in total. And our plane has six degrees of freedom.

Rice. 8. Angular coordinates of the aircraft: rise, tilt and turn. The picture rotated is a top view.

To generalize: a body in space has six degrees of freedom. And six coordinates: three spatial and three angular.

With a steamer and a steam locomotive, you can probably figure it out yourself.

It is necessary, however, to make one important remark.

So three or six?

It turns out that the number of degrees of freedom of a body is not its invariable property. This is a conditional value that depends on what we need to know, on the conditions of our problem. You can see for yourself: first, we counted three degrees of freedom for the plane, and after changing the conditions of the problem, all six. Both the one and the other answer are correct. But for different questions.

For example, are you sure that twice two is always four? Multiply 2 meters - the length of a square room - by 2 meters - its width. We get 4 square meters - the area of ​​the room. Any realtor will agree with this. Now let's take 2 meters of a steel pipe and multiply it by another 2 meters of the same pipe. And where do you see the resulting 4 square meters? They don't exist in nature. The calculation was clearly pointless.

Let's check the addition. One plus one is two. Take one liter of alcohol and one liter of water and mix. Chemistry assures us that we will never get 2 liters of diluted alcohol. The properties of alcohol and water are such that the volume (namely the volume, not the mass!) Of the solution will always be less than the sum of the initial volumes. Likewise, if we mix a glass of water and a glass of salt, we don't get two glasses of brine. Chemistry will deceive us this time too.

Even arithmetic can be wrong. If you use it thoughtlessly.

For collection - zero degrees of freedom

Zero is understandable without words. A broken mechanism that does not go anywhere and that you cannot move from its place. No coordinates are needed - and it is so clear where he stands. Zero degrees of freedom.

Closer to the body

Now we will move towards biomechanics. Let's talk about mechanisms.

The mechanisms that we have talked about so far - steam locomotives, airplanes - we generally considered not as mechanisms, but as simply bodies. We didn't care about their internal structure. Now let's get down to the arrangement of mechanisms. But much simpler ones.

We will consider mechanical systems consisting of solid, inextensible and rigid links connected by hinges. To begin with, we will consider two types of hinges: cylindrical and ball.

A cylindrical hinge, or pivot hinge, is the connection of two links that allows them to rotate around a common axis. Or, if we consider one link fixedly fixed - a fixed support - then this hinge allows the second link to rotate around the axis of the hinge.

Rice. 9. Cylindrical joint: with two free links and one fixed link - support.

Where can we find such a connection in the human body? This is, for example, the elbow joint. Interphalangeal joints of the fingers. The knee joint, in a first approximation, is also suitable, although with it everything is somewhat more complicated: here the real biomechanics strongly deviates from the abstract mechanism.

Rice. 10. Elbow joint (cut). (According to Sinelnikov.)

The second type of joint is the ball joint, where the links rotate around a common point. Again, one link can be considered a fixed support, then the second link can rotate around some point of this hinge. Points, not axes.

Rice. 11. Ball joint: with two free links and one fixed link - support.

What joints in our body fit this model? Shoulder and hip.

Rice. 12. Shoulder joint (cut). (According to Sinelnikov.)

Rice. thirteen. Hip joint(cut). (According to Sinelnikov.)

Hinges and degrees of freedom

Take a cylindrical joint with one fixed link. In the figure, the fixed link is shown simply as a fixed support. A free link can move in only one way: rotate around the axis of the hinge, while remaining in one plane. In this case, its loose end moves only along one line - an arc of a circle centered on the axis of the hinge.

In our body, the analogue will be, for example, the elbow joint. We only need to commit humerus... To do this, simply rest your elbow on the table and try not to move your shoulder.

How can we set the position of the free link? How many coordinates do we need for this? Since we can only rotate it around the axis, we just need to set the rotation angle. This will be the only coordinate that we need. It's the same for the elbow joint.

Rice. 14. Cylindrical joint and its possible movements.

It turns out that both the cylindrical hinge and the elbow joint have the same degree of freedom.

Now consider the ball joint and its analog, the shoulder joint. Reattach one link of the hinge. To secure the shoulder joint, we just have to try not to move the shoulder blade.

The ball joint already allows many more different movements. The free link can swing in all directions. In addition, it can rotate around its own longitudinal axis, while remaining in place. Our shoulder can do the same. In this case, the loose end of the free link moves not along the line, but along the section of the sphere with the center at the hinge.

In order to uniquely set the position of the link, we need three angles. Two of them set the inclination of the link in space, and the third - the rotation of the link around its own axis. We get three coordinates and three degrees of freedom for the ball joint and shoulder joint.

Rice. 15. Ball joint and its possible movements.

One, three ... But where are the two?

You may have noticed that when we talked about hinges, we jumped from one degree of freedom at once to three. Are there hinges that have two degrees of freedom? There is no simple hinge, but there is a mechanism that actually consists of three links: the cardan transmission. Its free end, like in a ball joint, can tilt to any side, but cannot rotate around its own longitudinal axis. This is precisely the basis of the use of cardan shafts in rear-wheel drive cars.

Rice. 16. Cardan transmission.

Of course, there are no cardan transmissions in the human body, but joints with two degrees of freedom are encountered. This is, for example, the wrist joint. Having fixed the forearm, we can tilt the hand as we like, but we cannot rotate it around the longitudinal axis. If, while checking this, you were still able to rotate the hand, this means that you did not sufficiently fix the forearm and used its mobility. Grip yourself firmly just above the wrist with your other hand, do not let your forearm turn, and you will make sure that the hand does not turn. This joint has only two degrees of freedom.

Human joints are generally much more complex than simple hinges. Here are a couple more examples of joints that do not fit the simplest mechanical schemes.

Tree of life

The knee seems to fit well with the cylindrical hinge pattern. If we fix the hip - for example, sit on a table with our legs dangling - then the knee will swing, drawing an arc, just like the free link of the hinge. But, in fact, when the knee is bent, the lower leg can also rotate slightly around its longitudinal axis, adding another degree of freedom to the knee joint. When we bend the knee, the tension on some of the ligaments in the knee is relaxed, the attachment of the lower leg becomes looser and there is a possibility of rotation, which is not there when the knee is straightened. It turns out that the knee joint has one degree of freedom with an almost straightened knee and two with a bent one.

Rice. 17. Possible movements in the knee joint.

We also cited the elbow joint as an example of a cylindrical hinge. And it really fits this scheme if we consider the attachment of only the ulna. But when we talk about the wrist joint, we noticed that the forearm can rotate, providing a pronation / supination movement of the hand.

Rice. 18. Possible movements in the elbow joint and forearm.

This is possible due to the complex arrangement of the elbow joint, which actually consists of three separate joints. Three bones converge in it - the humerus, ulna and radius - and each pair of bones is connected by its own joint.

The ulna is attached to the shoulder joint with one degree of freedom, forming a cylindrical hinge. But the radial is connected to the shoulder is already a spherical joint - an analogue of a ball joint, with three degrees of freedom. The mobility of the radius relative to the ulna is limited by the two joints that hold them together: the elbow and the wrist.

All this complex device leads to the fact that the radius can rotate in some peculiar way around the ulna. The hand is attached precisely to the radius by the wrist joint and therefore can take advantage of its mobility. In this case, the ulna remains motionless. That is, to one degree of freedom that the elbow joint has, one more is added along the forearm.

Note that although the lower leg also consists of two bones - the tibia and the fibula - it lacks a mechanism similar to the forearm, and both of these bones move as one.

Further analysis and classification of the joints of the human body require a separate article. And here we will try to complicate our mechanisms. We will add more links to go from individual joints to whole limbs.

Links of one chain

Let's connect the three links now. The first will be, as usual, a fixed support. The second one is attached to it with a cylindrical hinge. And attach another link to the free end of the second link. Also a cylindrical hinge. For simplicity, we will make our mechanism flat: let the axes of both hinges be parallel, then all the links will move in the same plane.

Rice. 19. Flat mechanism of three links and two cylindrical hinges.

How many coordinates do we need to position the entire mechanism? The first link is motionless, its position is known. We can rotate the second link in the hinge at some angle. Not any: the angle of rotation is somehow limited by a fixed link, but this is not important to us. This angle alone is enough for us to set the position of the second link. Let's set this angle.

In this case, the far end of the first link will be at a well-defined point. We can calculate the position of this point for a given angle and length of this link. (We do not consider the length of the link as a coordinate, since it is constant.) At this point, there is a hinge that secures the third link. So, to set the position of this link, we just need to set the angle of its rotation (for example, relative to the second link) - in the same way as for the second link.

It turns out that by setting two coordinates - two angles - we set the position of our entire mechanism. This means that it has two degrees of freedom.

Rice. 20. Plane mechanism of three links with angular coordinates.

Note that by connecting the links with two hinges, each of which gives one degree of freedom, we get two degrees of freedom. That is, the degrees of freedom simply add up.

In the body, a similar mechanism can be found in the fingers of the hand: these are two successive phalangeal joints.

Rice. 21. The finger of the hand as an example of the previous scheme.

Now, in our three-link mechanism, we will replace the first joint with a ball joint, and leave the second cylindrical.

Rice. 22. Mechanism of three links, ball and cylindrical joints.

The analogy in our body is the connection of the forearm and shoulder with the torso. At the same time, we do not take into account the ability of the forearm to rotate the hand.

Rice. 23. Shoulder and forearm as an example of the previous diagram.

If you remember, the ball joint has three degrees of freedom. Adding to them one degree of freedom of the second, cylindrical hinge, we obtain four degrees of freedom. Indeed, we set the position of the second link (the first is fixed) by three angles. In this case, the position of the second joint and the direction of its axis are calculated. Therefore, to set the position of the third link, we need only one more angle of its rotation in the cylindrical hinge. This means that four angular coordinates are needed to set the exact position of the entire mechanism. And our mechanism really has four degrees of freedom.

My palm turned into a fist

Finally, let's calculate the degrees of freedom for the whole arm. We will not take into account the fingers: we will squeeze them into a fist. The body will be considered a fixed link. Then we have a chain of four links and three hinges: torso - shoulder joint - shoulder - elbow joint - forearm - wrist joint - hand.

Rice. 24. Degrees of freedom of the hand without taking into account the movements of the fingers.

Let's start adding the degrees of freedom. Shoulder joint - three degrees of freedom. The elbow joint is one degree of freedom. The forearm is not an ordinary link, but a whole mechanism that adds one more degree of freedom (pronation / supination of the hand). And the wrist joint gives two degrees of freedom. Adding up, we get seven. Thus, the human hand (excluding fingers) has seven degrees of freedom.

Let us explain again what this number means. We have chosen a certain mechanical model arms: fixed body, hand as a single link (fist). In this model, we need exactly seven coordinates to uniquely set the position of the entire arm. Place your palm on the table, thus fixing the position of the hand. With your torso and palm stationary, your hand can still move: your elbow makes an arc in the air. If we want to set the position of the entire arm, determining the position of the elbow, we need these seven coordinates and nothing else.

We wrote above that if a body moves on any surface, then it has two degrees of freedom. Three - if we want to take into account the rotation of the body in this plane as well. We will move our palm across the table. Here is a body that is moving on a surface. This means that the palm has three degrees of freedom. Where is seven?

But we also wrote that calculating the degrees of freedom depends on the model, on the problem. If we only care about the position of the palm on the table and no matter what is attached to it and what happens to it, then there are three degrees of freedom. If we want to know the position of the whole hand, then seven.

The reality is even more complicated. If we, sitting at the table, reached for bread, then we include additionally complex finger mechanics, as well as, possibly, the tilt and rotation of the body. If we consider such a more complex model, then the number of degrees of freedom for the entire system will be much larger. How our brain manages the calculation of such a heap of coordinates is again a topic that requires a separate article.

When knowledge passes not only through the logical mind, but begins to be experienced by the HEART .... this is a significant moment, it personifies a very specific stage of Freedom.

Now A person is really capable of NOT ONLY choosing, but also DOING, CREATING ... we achieve our INNER BALANCE.

Reaching balance

Imagine an ordinary person who does not think about the meaning of his Life, about causes and effects, about why he lives on this Earth. His life is like a monotonous and endlessly repeating video: work is at home, on weekends - beer in a nearby bar and TV in the evenings with the same programs, advertisements and boring jokes ... How do you think, how free such a person is in his life? Can he change anything in her? Does he have a chance of that?

- I think not ... His freedom will consist only in the fact that he can choose in a nearby supermarket - buy milk or kefir for him tonight ...

If you look at his LIFE LINE, then it will be clearly fixed, from one obligatory point to another - birth - childhood - study - work - marriage - children - Death ...

Such a route has been laid out for such every Immortal Soul before incarnation - with its rigidly fixed events, predetermined karmic knots of past incarnations, which are mandatory in any version.

If you represent this in the form of a graphical diagram, then this will be the only line with mandatory points - events.

Now imagine the same Person, but in slightly different conditions:

- at one point he feels that this gray monotony got him to such an extent that he says to himself - EVERYTHING! ENOUGH! I DO NOT WANT TO LIVE SO MORE! ...

And the question arises - ... how to change your Life?

-the answer is always there, it is always at our arm's length ... behind the left shoulder ...

- who still does not know - there is our Death ... having turned on the timer, we undoubtedly find the answer - and what do we want from our Life - concretely and clearly!

So, the goals were determined, ... but how to come to them? As a river that flows only in its own channel and will never flow backwards or uphill - we are all thrown into the same understanding - we need to change ourselves, change our old ideas about the World, about our capabilities, IN ORDER TO CHANGE OUR LIFE. Precisely because we are all MIRRORS! Our inner world invariably reflects our Reality.

It makes no sense to persist in a rigidly fixed idea of ​​oneself, if we want changes in our Life, this is the Law!

And, as always - there is, was and will be - our FREE CHOICE in this matter: we want - we change and change our Life, we want to persist - and everything remains as it is ...

At this moment, the question always arises - where to start?

- the answer is always trivial, Humanity has not yet come up with another way to its true Self, to its Soul, to the realization of its destiny -

This is the stopping of the mental dialogue ... Because only when we stop the chatter of the mind, we begin to hear the voice of our Soul ...

Then the stage of understanding and awareness of one's own fears begins. This is the little junk that our consciousness has gathered in the process of growth and the state of an “unconscious vegetable”. This is a natural stage, we all go through it, such is the human nature of consciousness. At this stage, the fears and limiting beliefs we received in early childhood and adolescence are worked through. Two wonderful tools from our arsenal help to understand this level perfectly: these are the books by Vladimir Zhikarentsev "The Path to Freedom or Karmic Causes of Problems" - this is for the head; and an initial wellness course by Mirzokarim Norbekov, which works with the body. To this, it is imperative to add the practice of "Zero Cycle" - and this is work at the level of Laws.

So, at this stage, the most important action begins - the Man removes the medal that has long been hung on it from his neck - on the one hand it says "SACRIFICE", and on the other "IMPORTANCE". This is not a one-time process; you will have to work with this topic in all the manifested hypostases of this dual pair. This is where the real path to Freedom begins.

At the most material level, you can very well help this process from one more side - since everyone already knows that our thoughts are reflected in our body, first of all - the body can and should be helped with purification. And it is necessary to start with cleaning the gastrointestinal tract.

Why?

-because the state of the Gastrointestinal tract is directly related to our ability to RECEIVE, ACCEPT - in this case - new thoughts, new knowledge.

For this purpose, I select a special program individually for each person. This program usually includes herbs and dietary supplements. Since this is selected by the wave resonance method, the program turns out to be purely individual, like a fingerprint ...

Let's look at the Life Path of a Man at this stage - he already begins to have FORKING at key points ... which means that in these situations a Man gets a choice. In some cases, it can do this, but in some other ways ...

HIS FREEDOM OF CHOICE INCREASES!

But that's only First stage getting Freedom. Then it becomes much more interesting - completely naturally having worked through the fears and limitations of this incarnation, EVERYONE WHO WALKED THIS WAY are faced with the understanding that the roots of some fears, attachments, life situations lie in the deeper layers of our memory. From this moment, work with karmic situations begins, and for this you need to look into your past incarnations. After all, only own experience and only our self-felt, CONSCIOUS past mistakes are taken into account. As a rule, by this moment there is already enough Personal Power for a person to easily and simply begin to work with these memories. Everything happens in a completely natural way, there would only be a desire to understand, and not close our eyes to situations that simply line up ... for awareness ...

It should be said about one regularity at this stage: A person, in a sense, begins to leave the egregor of a “common man in the street”, and the system resists this. Egregor is trying to return the Human to his place ...

FREEDOM ALWAYS NEEDS TO PAY... and this is a battle to the death, for the present ... and situations at this time are very real, not illusory ... And we pay by REFUSING our fears, attachments, old patterns of perception, all our limitations, which we part with and which did not allow us to move forward before !

When experience is gained in this area, a person at a subtle level changes the structure of his subtle budhial body. This is the body of Laws, all the Laws of the Universe are written in it. Before conscious work at this level, this body naturally has an imbalance, it can even be partially destroyed. Now it is, in fact, being restored. It is when knowledge passes not only through the logical mind, but begins to be experienced by the HEART. This is a momentous moment, it personifies a very specific stage of Freedom. Now

A person is really capable of NOT ONLY choosing, but also DOING, CREATING ... and we achieve our INNER BALANCE.

At this level, he already understands that he is able to give up past karma, because all the karmically inevitable situations of negative, he REALIZES, and does not LIVE! ...

AND NOW THE MOST IMPORTANT: there is an opportunity to GO ON THE WAY

ITS TRUE PURPOSE, ITS FULL AND MAXIMUM IMPLEMENTATION!

Let's look at the picture of the Life Path of such a Person:

Recently I again visited Nizhny Novgorod, with which I have so many memories. I was glad to be convinced that Jan Goland, my old friend, is in good health and continues to work successfully. It seemed to me that Nizhny again became a place of pilgrimage for homosexuals who dream of changing their fate. It would seem that it has become incomparably easier to get qualified psychotherapeutic help now than in the past, but this does not apply to this category of patients. “Why do you need to be treated? - the doctors tell them cheerfully. - How do you live, with whom you live - now nobody cares about this. Calm down, relax, be yourself and be happy! " So they have to travel around the country in search of a place where their problem will be taken seriously, although nowadays they have to sacrifice a lot for such a trip.

So should homosexuals be treated? And can they be treated? In Jan Goland's office, next to his archives collected over many decades, these questions seem ridiculous.

Goland recalls one of his first patients. Then, in the 60s, the principles of anonymous treatment were not applied in our country, but since for most people who needed help, this was the first condition for turning to medicine, the Doctor at his own peril and risk abandoned the names and surnames. Each patient has a nickname. Organizations that, on the other hand, needed lists, were furious. But the doctor managed to gain the upper hand in single combat. The patient about whom in question, taught at the institute, therefore the nickname Associate Professor was assigned to him.

This man, undoubtedly, had every right to say about himself - "this is how nature created me." The first signals she sent were in early preschool age: little boy took every opportunity to sneak into the potty and spy on what the other boys look like as they take off their panties. These deeply embedded prerequisites were apparently still fixed by a peculiar family education... Parents raised three sons. Perhaps they - especially the mother - lacked a daughter, a girl. The assistant professor said that he was treated very differently from the brothers. His mother was unusually gentle with him, took him with her to the women's bath. For the Christmas tree, the boy liked to dress up as a girl, tried to achieve a complete resemblance - he was praised for this.

Both brothers grew up to be tough, as they say now, peasants, their Don Juan exploits thundered throughout Gorky. And this further complicated the position of the Associate Professor. At the age of 16, he got his friend drunk and tried to have sexual intercourse with him. This could have the most sad consequences, but it was saved by the fact that the "object", having come to his senses, remembered little. The associate professor managed to successfully finish school and enter the university.

In everything related to study and work, he was an excellent student, but his personal life did not bring anything but torment. Girls did not interest him, they all seemed to him the same person. He was able to highlight in the crowd beautiful eyes or hair, but those were the analyst's dispassionate observations. But the youths - not younger than 17 and not older than 21 years old - brought him into excitement, which in no case could be detected.

Every summer the Associate Professor went to Sochi - there he allowed himself to relax a little, but he had to wait too long for this unfortunate vacation and he flew by too quickly. Over the years, this man has developed a kind of mask. He portrayed a great lover of alcohol. A drunk is forgiven for everything - including inappropriate caresses, hugs, kisses, especially if then he makes a confused face and says that he does not remember anything.

In our conversation, Goland mentioned another characteristic feature. Before starting treatment, he had to spend a lot of time to create the necessary conceptual apparatus in the patient, to "oliteraturize" homosexual experiences - without this their contact was impossible. The candidate of sciences, a teacher who was fluent in literary speech, was unable to describe (and, therefore, to a large extent and understand) what was happening to him.

On the basis of the newly acquired language, broad communication between patients should have begun - so that each, having learned to understand others, could better understand himself. But that was not safe. The most murderous thing could have been added to all the accusations - under the guise of treatment, a homosexual club was created. And Goland did the only thing that was left for him - he called on the technician for help. The way it was at that time - primitive, cumbersome, unreliable, with very poor image and sound recording quality. But main task assigned to her, this technique nevertheless performed. Jan Goland's audiovisual archive is truly inexhaustible.

Psychoanalysts, I remember, were reproached for the fact that the state of the patient "at the entrance" and "at the exit" is assessed by the doctor himself, who may turn out to be not sufficiently perspicacious and too subjective. Time passes, the influence of the doctor weakens, and people return to their old way of life. And now it is enough to press the button of the reproducing device, so that a significant part of the doubts will be dispelled by itself. The patient talks about himself. He shares his experiences, discusses, analyzes, notes the changes - frankly, without hiding, without underestimating the drama of the struggle taking place in him. He is full of respect and gratitude to the doctor, but he assigns more of an auxiliary role to him: “I cured myself, and the doctor only helped me with this, created the necessary conditions". He is convinced that he accepted the right decision, and is ready to prove it in ten, and in twenty, and in thirty years.

When the Associate Professor has a desire to see a doctor, he changes his appearance, puts on a wig and glasses. Many people in the city know him, why unnecessary conversations? His fate was happy. He married very successfully, which has been confirmed for decades by nothing overshadowed family life, raised a wonderful daughter. The girl turned out to be extremely talented. We will not specify in which particular area, so as not to make the portrait too recognizable, especially since the daughter herself has no idea about her father's past. I just wanted to emphasize that in addition to all the other joys of life, which he was initially deliberately denied, our hero was given to experience an incomparable feeling of paternal pride.

Another of Goland's long-term patients lives in Riga. It is not so easy to communicate across the state border, but the connection, nevertheless, is not interrupted. For this man, too, everything turned out the way he planned for himself 20 years ago. A good relationship with his wife, who have stood the test of time, two beloved children. Memories of the past are neither tempting nor painful. "Was it with me or not with me?"

Do you often have to deal with relapses of homo sexual attraction? I asked. Over the years, there were three such episodes, and each time the impatience of the patients became the reason. It seemed to them that they had already grabbed God by the beard, although the doctor insisted on continuing the work. Fortunately, in all three of these cases, the failure was only temporary.

Has the medical method changed in more than three decades? Jan Goland considered the answer for a long time. V general outline, he said, the principles of treatment remained the same as those proposed by Professor Ivanov, but the forms of influence became much more sophisticated. More confident than Soviet years, the doctor now applies the elements of psychoanalysis. Time and experience have perfected self-hypnosis formulas. And, of course, the archive has been enormously enriched. Any patient can now find a "double" - so that not only age and contours, destinies, but also appearance, hair and eye color ...

Coming home from Nizhny Novgorod I thought about what a monstrous mistake medicine can make if it really refuses to deal with homosexuality.

A society that does not pay lip service to human rights, naturally comes to the idea of ​​the inadmissibility of the slightest discrimination against a sexual minority. But if one cannot put pressure on a person, forbidding him to follow the dictates of his nature or forcing it to change, then in the same way, in no way can one hinder his desire to change. This is also a kind of discrimination, restriction of freedom, an attempt to remove a person from solving the most cardinal issue of life - choosing his own destiny.