Download "13. Вращение сферы"

Please wait. We're preparing links for easy ad-free video watching and downloading.

Next video: Религия для ученых / Савватеев, Панчин // Час Speak

Религия для ученых / Савватеев, Панчин // Час Speak

Религия для ученых / Савватеев, Панчин // Час Speak

Channel: RTVI Развлечения

Matlab

Mathworks

Simulink

матлаб

симулинк

алексей савватеев

леонард эйлер

математика

высшая математика

наука

алексей_савватеев

леонард_эйлер

высшая_математика

00:00:01

[music]

00:00:06

rotation of a sphere Euler’s next theorem in

00:00:10

general something called the

00:00:13

hole theorem is dozens if not hundreds of

00:00:15

statements of various branches of

00:00:18

mathematics Euler’s next theorem

00:00:21

sounds like this any motion of a sphere that

00:00:28

preserves orientation is a

00:00:31

rotation around a certain axis through a

00:00:34

certain angle

00:00:37

any now I will explain all the terms

00:00:39

any the movement of a sphere that maintains

00:00:49

its orientation is rotation around a

00:01:04

certain axis at a certain angle, which means

00:01:15

an explanation let’s start drawing our

00:01:17

sphere, in fact, each of you can

00:01:20

take just a soccer ball, forget

00:01:22

the firmware, well, well, or a soccer ball is

00:01:24

such a saint

00:01:25

if with this rubber and start twisting it, as it were.

00:01:27

twist this way and that if you

00:01:28

unscrew it like this decorated it so as soon as

00:01:31

you record some

00:01:32

movements of the football but of this

00:01:35

ball of ours you can actually

00:01:37

see two fixed points on

00:01:39

opposite sides of each other

00:01:41

around which there was actually just a

00:01:43

rotation quite amazing the result

00:01:45

actually you can’t avoid

00:01:47

this, you can

00:01:49

get away with it as you like, but as a result,

00:01:51

as a result of all these exercises,

00:01:55

it turns out that some two points

00:01:57

still remained in their places and everything

00:01:58

was spinning around the axis that passes through them,

00:02:00

they will be diametrically

00:02:01

opposite, this is also so visual

00:02:04

but very difficult an amazing fact,

00:02:06

absolutely a fact that belongs to me,

00:02:08

so let's

00:02:10

develop this science a little, what is movement,

00:02:12

movement is a transformation that

00:02:14

preserves the distance, simply that is,

00:02:15

formally speaking, and of from the sphere, the sphere

00:02:24

is such that for any x y

00:02:26

from our sphere, the distance from x to y

00:02:31

is equal to the distance from f from x

00:02:35

to y from y, well, here we need to explain what

00:02:40

distance means, how is it most convenient

00:02:42

to reconcile on a sphere, distances can of course be

00:02:43

measured, but usually the idea of an

00:02:46

angular angular measure of distance is accepted, that is,

00:02:48

distance is an angle if I took the center

00:02:51

of the sphere and headed to these two points along

00:02:54

radius, then this angle is, as it were, a

00:02:57

distance; if the angle is zero, then

00:02:59

the points coincide; if the angle is 180,

00:03:02

then they are opposite, and that is, the

00:03:04

maximum achievable

00:03:06

distance between points on a sphere; but if the angles

00:03:08

are equal, it is hidden that the distance here

00:03:10

will also be equal, that is correspondence

00:03:11

the triangles will be equal, which means

00:03:13

the distance is an angle of 1 century and I am not an

00:03:17

angle or that the same is the

00:03:20

shortest path traversed on a sphere

00:03:22

between these two points

00:03:23

and the shortest path is always an arc of a great

00:03:26

circle,

00:03:27

that is, if you want to get from

00:03:29

some point on the sphere at some point

00:03:31

the sphere is different, then you need to turn

00:03:34

to the center of the earth for help,

00:03:35

here and on the version of the earth we

00:03:37

are at the center of the sphere, draw the

00:03:39

plane of the only one that passes

00:03:41

through these three points and it will carve out

00:03:43

exactly that same king the shortest path

00:03:45

on the surface spheres, but there will be

00:03:48

two paths, one long and the other short, but

00:03:49

it should give a short one, if by chance these

00:03:51

two paths

00:03:52

coincided in length, this means that the starting

00:03:55

points were opposite and through the

00:03:57

center of the earth there

00:03:58

actually passes an infinite

00:04:00

number of two bones, which these

00:04:02

three connect together by extinguishing on

00:04:04

they lie on the same straight line,

00:04:05

but then all these paths are also

00:04:07

the same, it’s just the entire diameter,

00:04:08

well, well, not the diameter, as it were, the maximum

00:04:10

distance on a sphere, that’s how it’s the

00:04:11

floor of the equator, let’s say like that, that’s it

00:04:14

then iv is called

00:04:19

movement movement that is, it’s like

00:04:21

not only would it not glue different points

00:04:23

together into an obviously mutually significant

00:04:24

correspondence, in fact, it is strictly necessary to

00:04:27

prove that it

00:04:29

exists for any point on the sphere

00:04:32

since the one that goes into it

00:04:35

proves this to us, well, God bless him, let’s

00:04:37

leave it alone in general, from the fact that

00:04:39

it and the movement is the following, it is a one-

00:04:41

to-one correspondence between points,

00:04:42

as if it

00:04:44

one-to-one transforms the points of the

00:04:47

sphere, that is, different ones go into different ones,

00:04:49

and for any point on the sphere, what is it that

00:04:51

went into it, this is some kind of

00:04:53

strange somersault, well and now it is argued

00:04:56

that in fact, this

00:04:58

somersault property is that a

00:05:00

certain axis is taken and we simply make a rotation

00:05:03

around this axis at a certain

00:05:04

angle of rotation,

00:05:06

well, note not at all, as if before the

00:05:09

preparatory questions, I have not yet

00:05:11

explained what

00:05:12

orientation-preserving is look again, here it’s

00:05:18

easier for me to turn to the intuition of

00:05:21

orientation, this is a very complex thing

00:05:22

of mathematics, there are three different

00:05:24

definitions of how to introduce it, it’s

00:05:25

always some kind of linear algebra, these are the

00:05:27

legs of the determinants of their signs, that is, this is

00:05:29

something that can be simply explained, but

00:05:33

not strictly so Here’s a loose explanation:

00:05:37

we take our sphere and paint it, and

00:05:40

this side, for example, is blue,

00:05:43

and the inside of the sphere is in, but on the inside,

00:05:46

let’s say we got inside the

00:05:48

sphere and we do

00:05:50

n’t eat the inside, we decorate it with green, and we

00:05:54

need it so that when transforming f,

00:05:57

green does not become blue, that is, it is not

00:05:59

turned inside out, but how do

00:06:01

you imagine yourself turned inside out? Or, for

00:06:04

example, I take a circle here, I

00:06:08

fix it entirely, so it remains in

00:06:11

place and all the points on top go through.

00:06:15

from below and vice versa, that is, I turn

00:06:17

this hat inside out like this

00:06:18

completely, that is, this whole green one

00:06:20

will come up, it passes through the

00:06:22

surface opposite it from the inside, it does not

00:06:25

become externally, it also

00:06:27

becomes externally from the inside, a rose is such a fold, here are the

00:06:30

races and

00:06:31

received the sphere also again cerrado like this

00:06:34

turned inside out, we look at

00:06:36

the inside now as an outside, but we

00:06:38

need to preserve the orientation

00:06:40

from this, in particular, I need to have

00:06:42

only one property, and those that preserve

00:06:45

the orientation from mappings is that if

00:06:47

some kind of equator remains

00:06:51

in place, then all the spheres too must

00:06:52

remain in place, that's just one

00:06:54

property that is generally quite

00:06:55

obvious, yes, that is, if you roughly speaking,

00:06:57

if you took a soccer ball by the lower back, by the

00:07:01

waist, which we ask for the thickest thing in it,

00:07:04

just if you took it by

00:07:07

the waist, then you will move it from its place already

00:07:09

you can’t with the principle of preserving the distance, except for

00:07:12

this turning out,

00:07:14

you can’t move it; all the

00:07:16

other points are also fixed

00:07:18

forever, if it’s fixed, then it’s not

00:07:20

that it turns back and forth,

00:07:22

namely . at a point like this, all of it is

00:07:25

fixed and nothing else. also

00:07:27

cannot change her position

00:07:29

because she must remain at the same

00:07:31

distance from everyone else and

00:07:33

for her unity the option is

00:07:34

to jump here, well, as if at least one.

00:07:37

soon here, it’s easy to understand that everyone

00:07:39

else will also jump there on the other

00:07:40

side, this is a turning

00:07:41

inside out,

00:07:43

but this is the only intuitive

00:07:46

fact that I will use as an

00:07:48

axiom, here you need to understand that I

00:07:50

buried the strict definition of orientation under the rug,

00:07:52

so you buried it under the rug and

00:07:55

now, based on this principle, I

00:07:58

begin to build a certain science,

00:08:01

our first statement is that if

00:08:06

I have 2 points that are diametrically

00:08:09

opposite and both remain in

00:08:11

place, then the entire circumference of the great circle

00:08:14

should also be

00:08:15

in place, well, what is called where and

00:08:18

kindergarten look at it this point

00:08:20

should be at the same distance

00:08:21

as it was from this, that is, on this

00:08:24

circle, and at the same distance

00:08:26

as it was from this, that is, to this

00:08:27

circle, but it cannot remain

00:08:29

at the same time on this and on this circle, well,

00:08:32

at least Having gone somewhere, she

00:08:34

can no longer stay; she can stay with at least one of

00:08:36

this circle; therefore,

00:08:38

the requirement that this point remain

00:08:39

at the same distance from here and here

00:08:40

means that it must remain in

00:08:42

place, and the same can be said about any other point

00:08:44

that is, and

00:08:45

if they are opposite, then these two

00:08:47

circles, imagine what will

00:08:50

happen to them if these points were about the

00:08:51

false type, in fact, these two

00:08:53

circles from the 1st floor are

00:08:55

viewed from different sides, but this one

00:08:57

and this one are very important, this is very important

00:09:01

building up geometric intuition is

00:09:03

terribly important, even there were some

00:09:06

studies somewhere around pedagogically which I do

00:09:08

n’t believe in a damn thing, but in this case,

00:09:10

that something similar to that the coolest

00:09:13

skill of a child is the skill of the space

00:09:15

of imagination, but in general a person is

00:09:17

very strong helps in life in general,

00:09:20

if these were opposite,

00:09:21

we wouldn’t say that she stayed on this

00:09:23

circle, but could go to any point of

00:09:24

this circle,

00:09:25

but if they are not opposite, these are these,

00:09:27

surround them, intersect exactly one point,

00:09:28

they touch me, and in this this. This is the

00:09:31

one that you first considered,

00:09:33

so there’s no escape for everyone

00:09:37

else. this diameter is also not suitable and therefore the

00:09:40

whole sphere itself also has nowhere to go, that

00:09:41

is, in fact, that we really

00:09:44

showed that

00:09:45

so our our postulate which we

00:09:49

put forward as a postulate because I didn’t

00:09:51

prove it very strictly

00:09:52

if the movement

00:09:56

f well, preserving orientation to

00:10:01

preserving

00:10:02

orientation leaves in place

00:10:10

2

00:10:12

non-opposite points a and btf is

00:10:22

identically equal to ai di ai di this

00:10:27

transformation is nothing to

00:10:28

do nothing to do nothing

00:10:33

identical transformation

00:10:34

identical transformations,

00:10:37

that is, this means that

00:10:39

no rotation really took place,

00:10:40

no movement was made with the sphere if two

00:10:43

points like this remained on places,

00:10:44

you would have the rest of the sphere and don’t excuse me, well,

00:10:48

we’ve talked about this postulate well, and now we

00:10:51

use it to prove

00:10:54

Euler’s theorem, the wire is now straight, here’s how to

00:10:58

do it, how can we get a let’s

00:11:02

start the situation, here we took some

00:11:09

one point one and

00:11:10

and let's see where she went, if we were

00:11:16

wildly lucky, then she went into herself,

00:11:20

and I also have a photo, this is the first interesting

00:11:23

case that we will consider that

00:11:26

such a thing exists. that is, there are two

00:11:29

different cases 22 different cases

00:11:38

1 exists and from our sphere such that

00:11:44

I photo is equal to and motionless and the second

00:11:49

case, well, as if it does not exist in

00:11:54

reality and we will prove that this cannot be this is the

00:11:55

second case does not exist such and

00:12:01

but this will come to a contradiction this will come

00:12:08

to a contradiction but for now we will consider the

00:12:10

first case, which is actually none.

00:12:12

there was found which drives all this with

00:12:15

this somersault, by

00:12:16

the way, science for astronauts, you

00:12:18

understand, yes, the astronauts are studying the quaternions of the

00:12:20

quattro and the aversion of the sphere is not a

00:12:23

run. I seriously heard from one of

00:12:25

his mathematical colleagues that the

00:12:27

astronauts have a three-day course in the theory of

00:12:29

quaternions because Potter does not

00:12:32

describe rotation. space, how

00:12:33

exactly I will tell you a little later, but

00:12:35

not in this video, so here it is, but understand that the

00:12:39

rotation of the sphere is what an

00:12:41

astronaut needs, that there is no such thing, it’s up and

00:12:45

down, yes, you’re hanging out like this, but

00:12:47

I won’t say how that’s what

00:12:50

you have here.

00:12:51

and so she remained in place, I claim

00:12:56

that then her antipode, I designated it like this: the

00:12:59

antipode is the opposite of

00:13:01

Iran's sphere.

00:13:02

being in the country of origin, the sphere is

00:13:04

here somewhere, and with a line I, as a child, I

00:13:07

studied where the antipode of Moscow turned out that

00:13:09

somewhere away from New Zealand, not

00:13:11

very far in the sea, the sea is somewhere around 1000

00:13:14

kondrat of New Zealand, which were accepted

00:13:17

then also on place, that is, it turns out

00:13:20

that this is also true and why and because

00:13:25

look at the distance from point a to the point

00:13:34

with the line is equal to 180 degrees and it should be

00:13:39

preserved during movement therefore the

00:13:45

distance from f from a to f from the aes with the line is

00:13:49

also equal to 180 degrees this is the maximum

00:13:53

possible distance

00:13:54

but also of a tight oao

00:13:57

therefore the distance from a to f from the AES

00:14:01

line should also be equal to 180 20 in

00:14:04

other words, the rush remained in place and with the

00:14:07

line it was the opposite. into

00:14:09

which the line went

00:14:11

should also be at a distance of 180

00:14:13

degrees from this point which is the gnome

00:14:15

because their images should also be but at a

00:14:17

distance of 180 degrees there is only

00:14:20

one.

00:14:21

namely, the line itself, which means it is also

00:14:23

motionless, and all the

00:14:25

other points closer to the point could not have changed, but we

00:14:27

would not have preserved the distance. of

00:14:37

any movement there is

00:14:39

only an even number of motionless ones. there

00:14:41

can’t be anything, but either there are two at once

00:14:43

or there are four at once, well, and

00:14:46

as if in this case, I’m just

00:14:47

proving specifically it turned out that

00:14:49

the points are like antipodes, they

00:14:51

move in pairs, that is, any movement of

00:14:52

the sphere is a movement at the same time until the

00:14:55

points can fall and in fact,

00:14:56

honestly, there is, as it were, for many pairs of

00:15:00

points of opposite Ferret masses,

00:15:01

there is a special name for the

00:15:03

projective plane, but

00:15:05

we won’t talk about this now, it’s complicated, that is,

00:15:08

any movement of the sphere

00:15:09

induces the movement of the projective

00:15:10

plane, but we return to this

00:15:15

situation, then then, in principle, it’s

00:15:18

clear because here

00:15:22

on this axis I have two points that are motionless and

00:15:25

opposite,

00:15:26

then it’s clear that each point moves

00:15:30

strictly along a circle that is around

00:15:33

the corresponding one, that is, this is 7, let’s say

00:15:36

it’s north-south to the very northern and southern

00:15:37

poles, then we’ll just call all the others

00:15:39

points they must remain at the same distance

00:15:41

100 from these two, which means they are

00:15:44

moving along these circles, how to

00:15:46

prove that this is really a rotation

00:15:47

at the same angle and not just that

00:15:49

. from this circle I went 5

00:15:52

degrees in and z and immediately 20 why this

00:15:54

can’t be so let’s understand this from

00:15:57

I’m like this because look what I’m

00:16:00

going to do now I’m going to an unknown movement

00:16:04

right now purely

00:16:06

algebraic logic to annoy their theory of

00:16:09

groups purity look how this

00:16:11

mapping is followed by a

00:16:15

rotation to the corners

00:16:16

where fe well, for example, this is this

00:16:19

film, that is, I take some point

00:16:21

b it is sent here to FSB and this

00:16:23

distance is equal to fi angular distance and

00:16:26

again and then I apply rotations it

00:16:28

back the angle is minus vi, that is, if

00:16:30

it went in a positive direction, it

00:16:32

positively resembles and

00:16:34

counterclockwise is always yes, but in this case it’s not

00:16:36

very clear what this means, that it’s a

00:16:38

sphere, but how will you look at it from above,

00:16:41

look at this sphere, look from above, and then

00:16:44

in the opposite direction we will twist it

00:16:46

and look, the rotation saves the point, but

00:16:51

of course yes, because it

00:16:54

was done around the same race and it saves.

00:16:57

according to the survey condition, because around the

00:16:59

same race,

00:17:00

but attention, well, not a turn in the sense, but

00:17:04

this composition will also

00:17:06

accordingly save the point and this

00:17:09

saves the years, they save it, but attention, it

00:17:12

also saves point b because when

00:17:16

displaying f&b, she somehow went to the FSB

00:17:20

somehow the kulbi there and when

00:17:22

turning came back because I

00:17:24

specially picked it up at that angle

00:17:25

to compensate for this

00:17:27

thing, so it would be the same,

00:17:29

but a and b are no longer opposite in

00:17:33

construction, which

00:17:34

means this mapping is

00:17:39

identical, it doesn’t change anything

00:17:41

because she left two points in

00:17:43

place, there is some beautiful logic, that

00:17:46

is, I made another movement that was familiar to me,

00:17:49

familiar to me, and found out that

00:17:51

performing them one after another leads to an

00:17:54

identical

00:17:55

display, but if so, then it is clear that it

00:17:59

exists and rfi because just again let

00:18:02

's apply, for example, from that side now

00:18:04

also shipyards

00:18:06

and this is the same as candy, apply

00:18:09

rfi well, and this is harp and this is f

00:18:15

because these two reduced

00:18:16

transformations can be applied

00:18:18

associatively, this is separately so beautiful,

00:18:21

beautiful, you can also get away with any

00:18:22

transformation in general, any mappings

00:18:24

have by the property that

00:18:26

a f.f. asterisk asterisk right there

00:18:30

the composition, no matter how you place the brackets, the

00:18:32

result will be the same because it’s

00:18:34

easy to follow how each

00:18:36

of the points to which we apply this

00:18:39

mapping, how it runs after us, here it is, just

00:18:42

in the beginning with the help of yours, then that’s why it’s

00:18:43

always here in Hebrew from right to left

00:18:45

because we need to substitute

00:18:50

this starting point at the beginning here

00:18:52

then here so we get

00:18:55

yes p minus feast of from and well, it just

00:18:58

turns out that if we write

00:19:00

off the second

00:19:03

transformation performed on the left, then this

00:19:05

will happen the first is carried out, this is the

00:19:06

second, this is why the transformation is always

00:19:09

performed from right to left, you just

00:19:10

need to get used to it, well, that is, in

00:19:14

fact, the case when we have a fixed one

00:19:15

. we have sorted it out,

00:19:17

it remains to deal with the case when there are

00:19:20

supposedly no fixed points and

00:19:23

actually bring this case to a contradiction, in

00:19:24

fact,

00:19:25

well, let’s draw sulfur, let there be no

00:19:33

such htf from and equals and look what I’m

00:19:41

doing, I now do the following, which means I

00:19:48

take some kind of a

00:19:50

and I'm looking at where I'm taking the photo,

00:19:53

let's choose to choose any

00:19:58

point a from c 2

00:20:03

let f from a which as as as we already

00:20:08

understand the damage on piston 1. don’t

00:20:10

go overboard, let me, here they are exactly, and

00:20:13

her image and we’ll draw it here before

00:20:18

we’ll draw the image, now I do the following,

00:20:22

I draw a diameter perpendicular to

00:20:27

this segment, well, a curved segment on

00:20:30

this one, that is, I’m now taking it to call the

00:20:38

perpendicular bisector plane

00:20:39

plane of the median perpendicular,

00:20:41

that is, I connect them, draw

00:20:43

a perpendicular to it, it cuts out for me a

00:20:46

plane passing through the center of the earth,

00:20:48

naturally through the center of the sphere, for

00:20:50

obvious reasons, I designate the

00:20:53

plane through as much and make

00:20:58

transformations, look, I’m proving

00:21:01

something about rotation, but I make

00:21:03

inversions from

00:21:04

on ngu I need to go beyond the limit, in order to

00:21:07

prove Euler's theorem, you need

00:21:12

to add turning inside out to your own culbis, that is, a transformation that does not

00:21:14

preserve the orientation,

00:21:15

without this it will be much

00:21:17

more difficult to prove and so I transform, I have

00:21:21

one with aj, this is turning

00:21:24

inside out relative to a given plane,

00:21:26

well, that's what I already told you

00:21:29

about growing inside out relative to the

00:21:34

plane, so that’s what I’m doing,

00:21:42

accordingly, its image then

00:21:45

f on c aj already saves the point and the fns

00:21:52

already saves the point and correctly it

00:21:55

saves.

00:21:56

so great, now I’m working with this

00:22:00

transformation that is already

00:22:03

saved.

00:22:04

and prime the next move I

00:22:09

used to reverse it and now I’ll also do

00:22:12

a somersault but I’ll also do some twisting

00:22:15

but it will be a different twisting

00:22:17

absolutely means what I now want to

00:22:20

do and I’m considering some

00:22:26

other point b I some point b and

00:22:31

such that it was not opposite to

00:22:34

so let it be. sphere is not equal, well, in short,

00:22:42

unequal as much as a line, that is, not the antipode of

00:22:45

the point, but let’s choose a point that is

00:22:48

not a like then it remains at a

00:22:52

distance from b to a equal to the distance from the

00:23:01

application this is from as much as f

00:23:04

this is double

00:23:07

from bks as much as f from

00:23:11

a but this is a because we built it that way,

00:23:15

we specially turned it inside out

00:23:16

so that a returns to its place, so it

00:23:19

should be the same as from this one,

00:23:24

yes and this is simple, then I’ll replace it, that is,

00:23:27

the result of applying this

00:23:29

mapping kb

00:23:31

leaves b on the same circle on

00:23:33

which it was in relation to the point

00:23:36

ike and and infinite which passes

00:23:39

through it towards the antipode

00:23:41

and with the line like this, now the

00:23:46

fatal number is executed, let it be

00:23:50

c this one. let it be the purpose, I draw the

00:23:54

perpendicular bisector, here is

00:24:02

the order of what happened, I took and drew

00:24:11

a plane that passes through a and with

00:24:16

a line, the center of the earth and the midpoint between b and c, the middle of the

00:24:21

arc, here is the right angle,

00:24:24

here it is, and now I return c,

00:24:31

turning it inside out with this plane

00:24:34

welle back and so f after him with aj

00:24:40

after him sl2 turning inside out

00:24:48

returns point b to point b because

00:24:51

that’s all I drove it to cs easier, turn it inside out and

00:24:54

return it back to b

00:24:55

but these two turnings after f.

00:25:01

but they didn’t change it because

00:25:05

they didn’t change this according to our construction, but this

00:25:08

passed through and the

00:25:10

eversion plane passes through this one here

00:25:12

.

00:25:13

therefore, she also did not change it is very

00:25:15

difficult to imagine, but this is

00:25:17

geometric intuition, space

00:25:19

that is absolutely necessary for life for everyone

00:25:22

who lives in this world, take swords,

00:25:25

draw on it, you see what is

00:25:26

happening, but then from these two

00:25:29

statements it

00:25:30

follows that s el is made after

00:25:33

hh and in the light zaev is equal to

00:25:36

identical, the transformation does not

00:25:38

change anything because these points were chosen

00:25:40

not opposite to each other, they

00:25:42

turned their state around, which means that here

00:25:46

we can again, as if multiplying, yes, I will

00:25:49

give a knife, I’m on the left, it follows that f

00:25:53

is sh on I’m just a goal, that is, if

00:25:57

we click on dreams, he will die with el

00:25:59

because double turning means they

00:26:01

did nothing and then again on sh and

00:26:04

he will die and sage and here, accordingly,

00:26:06

a goal will arise and then massage, that is, the

00:26:10

only transformation that is

00:26:11

back to f Well, it’s like she’s the only one,

00:26:15

but this one, but the opposite of this, in fact, this is the

00:26:20

turning inside out again turning

00:26:21

back turning inside out will

00:26:24

do nothing therefore iv this is a double

00:26:26

turning

00:26:27

and now I’m claiming that a double

00:26:29

turning is always a turning and that

00:26:32

in fact it was wrong that

00:26:33

there is no such point, but because

00:26:35

he has two more double inversions, I

00:26:37

have inverted planes relative to me

00:26:39

and some other but some other one

00:26:47

necessarily intersects with this one because there

00:26:54

are no parallel lines on a sphere, a straight

00:27:00

line on a sphere is only an arc of a large

00:27:03

circle, no other of a small circle

00:27:05

is not a mother not to shout distance

00:27:06

that is, these here eversion

00:27:08

occurs only in relation to large

00:27:10

circles, otherwise you, as if for us,

00:27:12

move the entire sphere somewhere and not as a single whole,

00:27:14

which is forbidden, but any two arcs of a great

00:27:19

circle are natural, like any two

00:27:20

planes in space up to containing

00:27:22

intersect in a straight line means these

00:27:23

intersect along this axis and that means

00:27:25

these two points actually

00:27:27

exist, they are motionless because

00:27:31

with one

00:27:32

inversion and with the other both points

00:27:34

remained in place, that is, in

00:27:37

fact, no such a case is impossible and any

00:27:40

transformation of the sphere this is a turn

00:27:42

because if the point is stationary and exists

00:27:44

then this is a turn it can be proven

00:27:46

this theorem was proven in the evening in 1700

00:27:49

shaggy year

00:27:50

I don’t remember which one but about whom he was on tour

00:27:54

and before that we need to talk separately this is a

00:27:55

very serious big topic well you can

00:27:58

as a thread and here Let's talk about them in another video,

00:28:00

thanks for your attention

Description:

Российская платформа математических вычислений и динамического моделирования Engee: сайт: https://start.engee.com/ Телеграм канал: https://t.me/engee_com ############### Теорема вращения Эйлера. Что такое движение. Вращение сферы. "Если движение f оставляет на месте две непротивоположные точки a и b, то f тождественно равно id". Мы в соцсетях: VK ‣ https://vk.com/mathworks Telegram ‣ https://t.me/exponenta_ru

Preparing download options

Popular

HD video

Only sound

All

* — If the video is playing in a new tab, go to it, then right-click on the video and select "Save video as..."

** — Link intended for online playback in specialized players

Questions about downloading video

How can I download "13. Вращение сферы" video?

http://unidownloader.com/ website is the best way to download a video or a separate audio track if you want to do without installing programs and extensions.
The UDL Helper extension is a convenient button that is seamlessly integrated into YouTube, Instagram and OK.ru sites for fast content download.
UDL Client program (for Windows) is the most powerful solution that supports more than 900 websites, social networks and video hosting sites, as well as any video quality that is available in the source.
UDL Lite is a really convenient way to access a website from your mobile device. With its help, you can easily download videos directly to your smartphone.

Which format of "13. Вращение сферы" video should I choose?

The best quality formats are FullHD (1080p), 2K (1440p), 4K (2160p) and 8K (4320p). The higher the resolution of your screen, the higher the video quality should be. However, there are other factors to consider: download speed, amount of free space, and device performance during playback.

Why does my computer freeze when loading a "13. Вращение сферы" video?

The browser/computer should not freeze completely! If this happens, please report it with a link to the video. Sometimes videos cannot be downloaded directly in a suitable format, so we have added the ability to convert the file to the desired format. In some cases, this process may actively use computer resources.

How can I download "13. Вращение сферы" video to my phone?

You can download a video to your smartphone using the website or the PWA application UDL Lite. It is also possible to send a download link via QR code using the UDL Helper extension.

How can I download an audio track (music) to MP3 "13. Вращение сферы"?

The most convenient way is to use the UDL Client program, which supports converting video to MP3 format. In some cases, MP3 can also be downloaded through the UDL Helper extension.

How can I save a frame from a video "13. Вращение сферы"?

This feature is available in the UDL Helper extension. Make sure that "Show the video snapshot button" is checked in the settings. A camera icon should appear in the lower right corner of the player to the left of the "Settings" icon. When you click on it, the current frame from the video will be saved to your computer in JPEG format.

What's the price of all this stuff?

It costs nothing. Our services are absolutely free for all users. There are no PRO subscriptions, no restrictions on the number or maximum length of downloaded videos.