Svyatoslav Fyodorov: the sky is the highest degree of freedom. What does Pearson's Chi-square distribution have to do with it? You always have to pay for freedom ... and this is a battle to the death, for real ... and situations at this time are the most real, and not illusory ... but we pay by refusing

THE SKY IS THE HIGHEST DEGREE OF FREEDOM

TOof course, the genes of life came to our planet from space. Otherwise, it is difficult to explain why, barely getting to their feet, people began to construct wings and tried to take off.
As a sixteen-year-old boy, I entered the Yerevan Artillery Special School in 1943. For a whole year I studied guns, trained in 72-caliber shooting, but ... All this did not bother my soul. I dreamed only about the fate of a fighter pilot. And I was lucky. My uncle returned from the front and became the head of the Directorate of Military Educational Institutions of the North Caucasus District in Rostov-on-Don. I wrote a report, and after a couple of months I was actually transferred to the Rostov 10th Special Air Force School.
My new comrades turned out to be completely different people from the cadets-artillerymen. The head of the flight school, Hanishek, was a wonderful person, in my opinion, and only selected romantics. Dreams of future flights and new aircraft created an unusual atmosphere of camaraderie and general friendship.
A foot loss accident put an end to my flight training. True, when I was in the hospital, I read "The Story of a Real Man" and also dreamed that they would be allowed to fly with a prosthesis: after all, Maresyev did not have two feet, and I had only one ...
But medicine became my profession. She also turned out to be quite romantic for me. After all, invention is a breathtaking thing. Artificial lenses, correcting any myopia and hyperopia, new methods of treating glaucoma, lasers for treating many diseases - all this reminded me of that flying mood. The surgeon, like the pilot, must make quick decisions. And, like the pilot, he wanted to get rid of the old and often unnecessary ground control.
In 1986, they managed to escape the shelling of the command system. Freedom in labor resembled the free flight of a pilot, and the steering wheel was finally in our hands.
Of course, as soon as the opportunity arose, we invested the large funds we earned, not only in medical equipment, but also in small aircraft. We bought a helicopter, a hangar, a radio station, a petrol station, and built a runway.
What a blessing - 52 years later to sit at the wheel again! It was Aviatika 890U. Sixty to seventy meters of run - and take off. How beautiful everything is from above, how beautiful it is for a man with wings! He defeats gravity, which always pulls everyone down. What happiness it is to pierce a cumulus cloud and hover over green fields and silver rivers! After that, I argue that the ecology of vision is more important than even our endless social ills. Maybe it’s the eyes that pull us up.
I turned 70 years old, and my friends presented me with the YAK-18T monoplane. Almost new, you just need to fix the chassis.
Now it's not a problem to fly to our branch even for 600-800 km.

Human freedom is, of course, both his economic well-being and the security of his life. But also freedom of movement. No wonder, when punishing a person, he is deprived of the opportunity to move in space. Therefore, aviation - both large with its speeds and small with its ability to land anywhere - gives those sensations without which it is difficult for a person to live.
Today, small aircraft, including paragliders, motor hang-gliders, gliders, have conquered space in the area of \u200b\u200bthree hundred meters from the surface of the earth. But we, as always, are lagging behind. Aviation bureaucrats want to control this space, explaining their intervention by caring for our lives. So I want to ask them: give us the opportunity to think about our own safety. Including in the sky.
February 1999

Recently I again visited Nizhny Novgorod, with which I have so many memories. I was glad to see 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 a change in 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 treatment? - the doctors tell them cheerfully. - How do you live, who do you live with - 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 risk gave up the names and surnames. Each patient has a nickname. Organizations that, on the contrary, needed lists were furious. But the doctor managed to gain the upper hand in single combat. The patient 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: a little boy took every opportunity to enter the potty and spy on what other boys looked like as they took off their panties. These deeply embedded prerequisites were apparently still reinforced by a kind of 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. 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 complete resemblance - he was praised for this.

Both brothers grew up to be tough, as they say now, peasants, their Don Juan exploits resounded 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 that 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 distinguish beautiful eyes or hair from the crowd, but those were the analyst's dispassionate observations. But the young men - not younger than 17 and not older than 21 - made him feel agitated, 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 everything - including inappropriate caresses, hugs, kisses, especially if he then makes a confused face and says that he does not remember anything.

In our conversation, Goland mentioned another characteristic feature. Before starting the 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 had an excellent command of literary speech, was unable to describe (and, therefore, to a large extent and understand) what was happening to him.

On the basis of the newfound language, broad communication between patients should have begun - so that everyone, having learned to understand others, could better understand himself. But that was not safe. The most damning 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 remained for him - he called on the technician for help. The way it was at that time - primitive, cumbersome, unreliable, with very poor quality of image and sound recording. But the main task assigned to it, this technique nevertheless fulfilled. 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 far-sighted enough 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 playback device to dispel a significant part of the doubts 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 in this, created the necessary conditions." He is convinced that he made the right decision, and is ready to prove it in ten, twenty, and thirty years later.

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 got married very successfully, which is confirmed by decades of unclouded 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 unnecessarily 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. Communication across the state border is not so easy, but the connection, however, is not interrupted. For this man, too, everything turned out the way he planned for himself 20 years ago. Good relationship with his wife, withstood the test of time, two beloved children. Memories of the past are neither tempting nor painful. "Was it with me or not with me?"

How often do you have to deal with relapses of homosexual 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 technique changed in more than three decades? Jan Goland considered the answer for a long time. In general terms, 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 confidently than in the Soviet years, the doctor is now applying the elements of psychoanalysis. Time and experience have refined the formulas for self-hypnosis. And, of course, the archive has become unusually rich. Any patient can now find a "double" - so that not only age and contours, fate, but also appearance, hair and eye color ...

Returning home from Nizhniy Novgorod, I thought about what a monstrous mistake medicine can make if it really refuses to deal with the problems of homosexuality.

A society that not in words, in fact respects human rights, naturally comes to the idea of \u200b\u200binadmissibility 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.

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 fog. This article is for those who felt some dissatisfaction and lack of clarity after such conversations.

First, let's explain 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 - on 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 steamer is clearly greater than that of a steam locomotive.

Now let's take the plane. His degree of freedom is 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 mostly swim: skewed to the side, with the crew hanging overboard and striking their priests along the crests of the waves.

But an airplane can do anything: raise / lower its nose, and tilt to the side, and turn. Especially if it is controlled by an ace from the "Russian Knights" squad. They even fly 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 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. It 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 a 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 towards St. Petersburg, then one number of 100 meters is enough for us to know where he 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.

Figure: 1. A steam locomotive on a straight section of rails.

So what did we get? 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 track, but a winding one. Does it change anything? Nothing, as long as the locomotive can't get out of 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 not be 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.

Figure: 2. Steam 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 engine 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 °.

Figure: 3. Steam 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 the hour hand all the time, we will use the clock. We put them in the center of the circle and mark the position of the locomotive in minutes on the dial. Or for hours - it's less accurate, but convenient. A locomotive for 3 hours or for 9 hours - what could be easier? Again, it only has one coordinate. And one degree of freedom.

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

Let's generalize: if a body can move only 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 lifts. 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 considered 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 need only two coordinates given to us by the GPS system.

Figure: 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 the origin. In mathematics, this coordinate system is called polar.

Figure: 6. Steamer in polar coordinates.

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 complicated. 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.

Figure: 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 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 now, 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.

An airplane 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 - this 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.

Figure: 8. Angular coordinates of the aircraft: rise, tilt and turn. The picture with a rotation 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 \u200b\u200bthe room. Any realtor will agree with this. Now let's take 2 meters of 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, by mixing 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 it 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 more simple.

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.

Figure: 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, as a first approximation, is also suitable, although with it everything is somewhat more complicated: here the real biomechanics strongly departs from the abstract mechanism.

Figure: 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.

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

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

Figure: 12. Shoulder joint (cut). (According to Sinelnikov.)

Figure: 13. 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 only move in 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 fix the humerus. To do this, just 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. For the elbow joint it is the same.

Figure: 14. Cylindrical joint and its possible movements.

It turns out that both the cylindrical joint 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.

Figure: 15. Ball joint and its possible movements.

One, three ... Where are the two?

You may have noticed that when talking about hinges, we jumped from one degree of freedom at once to three. Are there any 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.

Figure: 16. Cardan transmission.

Of course, there are no cardan gears in the human body, but joints with two degrees of freedom are found. 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, with a bent knee, 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 released, the attachment of the lower leg becomes looser and there is a possibility of rotation, which is not there when the knee is extended. It turns out that the knee joint has one degree of freedom with an almost straightened knee and two with a bent one.

Figure: 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.

Figure: 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 with which they are held together: the elbow and the wrist.

All this complex device leads to the fact that the radius can turn 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 be adding 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 we attach another link to the free end of the second link. Also a cylindrical joint. 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.

Figure: 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.

Figure: 20. Plane mechanism of three links with angular coordinates.

Notice 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.

Figure: 21. Hand finger as an example of the previous scheme.

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

Figure: 22. Mechanism of three links, ball and cylindrical joints.

The analogy in our body is the connection of the forearm and shoulder with the trunk. We do not take into account the ability of the forearm to rotate the hand.

Figure: 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 hinge, cylindrical, 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 joint. 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.

Figure: 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's explain again what this number means. We have chosen a certain mechanical model of the hand: a motionlessly fixed body, a 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 arm 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.

Above we wrote 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, 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.

The problem of human activity in society is closely related to the understanding of freedom. Much attention is paid to this in philosophy. There are two extreme points of view on freedom:

· freedom as an opportunity to do what you want

· freedom as a conscious need

The political definition of freedom states that a person has the right to do whatever he wants, except that which may interfere with the freedom of another person. Freedom is one of the main conditions for the formation of personality, full use of individual talents and abilities of a person. This is one of the greatest human valuesfor which people fought and died. The condition for personal freedom is opportunity to choose in accordance with their value orientation, needs and interests. Having made a choice of the desired, a person thereby removes the contradiction between freedom and necessity. The highest degree of human freedom is the coincidence of his aspiration, will, desire and objective necessity.

The level of cultural development determines the boundaries of human freedom, therefore human freedom is historical. However, it is not enough to have the right to freedom, one must be able to use it. Throughout modern civilization, man has striven to realize the ideal of freedom as fully as possible, but always remains mismatch between “want” and “can”.

The issue of personal freedom in the history of philosophy was considered from different positions, for example:

and) deterministic position (causation), whose representatives believe that in any situation choice there is only alone a real opportunity, and the rest of the possibilities are only formal. An extreme manifestation of determinism is fatalism (from Lat. fatalis - fatal, fatum - rock, fate) is a worldview that denies human free will and does not recognize anything accidental. According to this view, all events and actions of people predetermined by any external forces with a common name fate, rock (or by an inexorable coincidence of circumstances);

b) indeterministic position - it is the complete opposite of determinism, its alternative. Representatives of indeterminism deny the causality of the phenomena of the world and believe that a person does not have any destiny, but there are many possibilities and he is completely free to choose one of them, without any coercion from outside. An extreme form of indeterminism is voluntarism (from Lat.voluntas - will) - a direction in philosophy that considers will as the highest principle of being.Its representatives believe that the world is "ruled by will", ie. the viability of an individual, a community depends solely on willpower, and not from other qualities;

at) position of alternativeism (from Lat. alter - another), recognizing that in each choice there are necessarily at least two real possibilities. A person is not free in choosing the objective conditions of his activity - this choice is also limited, but he is free to set himself the goal of action, more or less free in choosing the means for realizing these goals. Freedom is always relative, because it is limited. It is limited by life itself, its spatio-temporal framework, the possibilities of choice, the laws of the state, the freedom of other individuals. And finally, freedom is limited to responsibility for one's choice of actions and their consequences to other people. Responsibility is a historically specific type of relationship between an individual, family, team, society, which is a conscious fulfillment of mutual requirements by them. If necessity ("must") is mainly an external restriction of freedom, then responsibility is an internal characteristic. how rights person related to his responsibilitiesand liberty a person corresponds to his responsibility... The hero of the story "The Little Prince" by Antoine de Saint Exupery says: "We are responsible for those we have tamed", "To be human means to feel that you are responsible for everything."

Whom they say about responsibility, they mean responsibility not only to other people, society, but also the internal responsibility of a person to himself, before the "project" of his life, which the person considers necessary to implement. Many life paths open up for young people, and the difficult problem of choosing one option arises. No matter how a person is subject to the influence of circumstances, friends, relatives, the media, in the end he himself determines his own path in life and carries full responsibility for your choice.And the more libertypossessed by a person, the more a responsibility for the specific choice he makes: “To whom much has been given, from that it will be required” (from a Russian proverb).

Modern man quite often has a separate understanding of freedom. without her connection with responsibility... And freedom of the individual is replaced by freedom of instinct, when the cult of the naked body is promoted, the legalization of swearing, pornography, drugs, prostitution, etc. But history shows that the more place in an individual's life takes instinct, the less time and space is left for reasonable activity... A person becomes a toy of his own instincts, forgetting about his main difference from an animal - to be Homo sapiens.

Unwillingness to be responsible for the consequences of their activities, to fulfill their debt in front of previous and subsequent generations sometimes clothed in such a fashionable phrase: "I owe nothing to anyone." This person is like disclaims responsibility for everything that happens, but thereby isolates himself from society, falling out of the social whole that provided him freedom... therefore a responsibility assumes that a person has a sense of duty and conscience, the ability to exercise self-control and self-government. Conscience acts as a controller of all human actions. A choice made by a person, a decision made means that a person ready to take full responsibility and even for what he could not foresee. The inevitability of the risk of doing something "wrong" and "wrong" assumes that a person has couragenecessary at all stages of its activity: both when making a decision, and in the process of its implementation, and, especially, in case of failure.

Currently, the pinnacle of ethical thought is ethics of responsibility, many schools of thought are working on it. This is due to the fact that modern society has many risks that threaten life on the planet. Immediate benefits are incompatible with the principle of responsibility.