WEBVTT
Kind: captions
Language: en
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So, we are back last time we did some platonic
solids we studied symmetry of the platonic
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solids. And we also saw certain graphs, Cayley
graphs, you remember I tried to draw the Cayley
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graph of free group with 2 generators. Here
is a nice picture I got it from the Wikimedia,
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and the key thing is that there are no circuits
here. There is no closed loops, like this.
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Just one minute sorry I I should have put
it should have cut it.
00:01:06.210 --> 00:01:16.900
So, there are no closed loops in this we shall
play with Cayley graphs after sometime. And
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as you can see this as an object as a geometric
object as a geometry it is a 2D object it
00:01:33.526 --> 00:01:48.180
has certain symmetry. What kind of symmetries?
For example, if you rotate it by 90 degree,
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there is a symmetry if you reflect it about
see one of the axis the y axis, there is a
00:01:59.470 --> 00:02:10.899
reflection symmetry and there they could be
more symmetries in this.
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I am going to shows again 2 dimensional
objects certain pictures which have beautiful
00:02:19.499 --> 00:02:35.109
symmetry and see that, right very nice pictures
quite fascinating some of them are very classic
00:02:35.109 --> 00:02:43.680
they are found all across the world in various
monuments. Our own Taj Mahal is an is a beautiful
00:02:43.680 --> 00:02:56.680
example of symmetry see this there are so
many, there are so many of them, very simple
00:02:56.680 --> 00:03:03.299
one.
So, how to understand symmetry in 2 dimension
00:03:03.299 --> 00:03:29.690
in in 2 dimension in R 2? R 2 is plane
real 2D plane then how to understand symmetry.
00:03:29.690 --> 00:03:35.879
How many symmetries are there all those are
questions or if I ask you to look at one of
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these pictures say this one and I asked you
how many symmetries does it have. Another
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interesting question I can ask here is wallpaper,
wallpaper symmetry. You have seen all the
00:03:50.780 --> 00:03:57.769
wallpapers which are there that is tiling
with those wallpapers you can also use them
00:03:57.769 --> 00:04:04.840
as a flooring for the flooring of the
surface of the of the floor using certain
00:04:04.840 --> 00:04:09.019
tiles maybe something like this.
So, what are the different types of tiles
00:04:09.019 --> 00:04:17.200
one can actually make so that they can
complete the white paper or the tiling pattern
00:04:17.200 --> 00:04:27.170
that is an interesting question. So, question
is
00:04:27.170 --> 00:04:41.030
how many different types of tilings. So, one
has to understand what is the meaning of different
00:04:41.030 --> 00:04:53.630
types of tilings are there interesting question
before I answer all these questions let me
00:04:53.630 --> 00:05:00.650
just talk of certain symmetries in R 2 at
the elimination all types of symmetries in
00:05:00.650 --> 00:05:12.940
R 2 and as you can guess they are infinitely
many symmetries, ok let me see.
00:05:12.940 --> 00:05:23.820
Let me first define certain types of symmetries
you are aware of rotations, see rotations
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by any degree theta and then reflection about
a line say y is equal to mx plus c. I hope
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you remember equation of line m is the slope.
What else? Translations, the translations
00:06:04.480 --> 00:06:10.650
I am going to talk about translations.
And then there is very interesting type of
00:06:10.650 --> 00:06:23.560
symmetry which is there quite loud in nature
which is called glide deflection. So, I am
00:06:23.560 --> 00:06:30.680
going to talk about all these 4 types of switches
rotations refraction, translations and glide
00:06:30.680 --> 00:06:36.270
reflections. And as you would recall in one
of the earlier lectures I had mentioned that
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composition composition of 2 symmetries is
again a symmetry, composition of symmetries
00:06:50.810 --> 00:07:05.490
is again a symmetry .
So, what should be all possibles which
00:07:05.490 --> 00:07:12.990
is a R 2? They should be compositions of what
I have written here, yeah I am going to explain
00:07:12.990 --> 00:07:18.780
one by one what all these switches are particularly
glide reflection which totally you might or
00:07:18.780 --> 00:07:29.530
not have heard off before I start I would
like to show you some interesting artwork.
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This one, it is a beautiful artwork on
the plate the the Russian style of painting
00:07:38.270 --> 00:07:48.160
and can you observe certain symmetry here.
Well, if I just rotate by 180 degree there
00:07:48.160 --> 00:07:59.090
is a symmetry can you observe any other symmetry
in this well that is all that is all these
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symmetry it is having rotation by 180 degree
quite beautiful and making such artwork as
00:08:05.500 --> 00:08:17.070
I am sure quite difficult thing. What you
see again? This and I just rotate by 180 degree
00:08:17.070 --> 00:08:19.451
same.
You remember once I told it for the symmetry
00:08:19.451 --> 00:08:25.450
here I am showing you and then on my back
I am doing something and if I show you and
00:08:25.450 --> 00:08:32.740
it looks like same. So, that you cannot detect
what was done what was not done that is symmetry.
00:08:32.740 --> 00:08:38.690
So, doing something that you can detect is
symmetry.
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Rotation as you know, I can this R 2 this
is just for difference every point is same
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as any other point when we talk of R 2 as
a geometric object any 2 points are same there
00:09:00.269 --> 00:09:06.230
is no reason for us to distinguish between
distinguish 2 points. So, what is rotation
00:09:06.230 --> 00:09:21.819
I pick a point p and then I decide an angle
theta certain angle theta say and they rotate
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everything by an angle theta about p, about
p
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by angle theta. So, one can express all this
in terms of material oh in terms of matrix.
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So, if p is origin then I can express rotation
as a matrix.
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So, if p is origin then rotation matrix rotation
by theta is having very nice expression. It
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is equivalent it is multiplication by this
matrix cos theta minus sin theta sin theta
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cos theta. So, what is the meaning? Meaning
is if I have a point xy, so I am writing it
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as a column as a point in R 2 then we rotate
this point about origin by angle theta
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the answer is simply matches multiplication
of cos theta minus sin theta sin theta cos
00:10:55.559 --> 00:11:05.379
theta with this column matrix, that is why
as you recalled matrix multiplication I multiplied
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this row with this column. So, this is x cos
theta minus y sin theta and then I have x
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sin theta plus y cos theta. So, that is this
new coordinate. So, rotation matrices can
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be used to express rotations.
So, rotation is simply as I am saying you
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have this you fix a point just rotate as simple
as that, you are already aware of that. And
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then we have reflections, right.
So, for reflections what do we do we pick
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a line and we, so here is R 2 just for reference.
I pick any line say y is equal to mx plus
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c and then they reflect about this. If there
is a point p here I just and suppose this
00:12:23.990 --> 00:12:32.589
is ninety degree distances d, I further go
by distance of d that is all there is a point
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p dash.
So, I can reflect my points in R 2 and
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again you can use certain kind of matrix to
understand reflections if the point line passes
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through origin. So, if c is 0 yes that is
line of reflection line of reflection is just
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as mirror you can think of it as a mirror,
right. If c is 0 then that is the line of
00:13:27.089 --> 00:13:45.959
reflection passes through origin, then one
can actually write then the matrix actually
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represents reflection, it is not very difficult
to find out again in terms of cos theta and
00:14:00.509 --> 00:14:09.709
sin theta. So, for example, suppose this
is our mirror this is a mirror line through
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which we are supposed to find reflections
and suppose this is a point say p let me write
00:14:17.910 --> 00:14:32.179
this point point p as say cos um alpha sine,
if I just taking the point to be on a unit
00:14:32.179 --> 00:14:39.610
circle or maybe let me simply write it
R cos alpha R sine alpha.
00:14:39.610 --> 00:14:51.720
So, that this is alpha angle and the distance
of point p from origin is R and suppose
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this is like y is equal to mx, then this angle
is theta so m is actually just tan theta and
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then for reflection what we are supposed to
do this, this is angle theta minus alpha.
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So, we have to go further by angle theta minus
alpha and see what happens to this point because
00:15:16.879 --> 00:15:22.119
for reflection this angle theta minus alpha
will be same as this angle theta minus alpha
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and then see what happens, that is not very
difficult find this and this matrix is actually
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going to be in terms of cos 2 theta and sine
2 theta. So, matrix of reflection is also
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there.
And translation is easier thing I have say
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I am just I have something here I am just
translating it, a similar shape just assumed
00:15:54.660 --> 00:16:04.880
that the , is this ok. So, this is suppose
distance R, further distance R, further distance
00:16:04.880 --> 00:16:18.949
R like this, the same pattern is being
passed on to via distance R that is our translation.
00:16:18.949 --> 00:16:28.870
So, I can do translation by any number, any
distance R, any R. So, there are infinitely
00:16:28.870 --> 00:16:32.350
many translations.
And then there is quite interesting one which
00:16:32.350 --> 00:16:52.879
is glide reflection. What is that? It is a
composition of reflection and translation
00:16:52.879 --> 00:17:13.620
, ok. So, here is some object let us take
this one some object I translate it and then
00:17:13.620 --> 00:17:29.350
I reflect it. Let me take some other color
I translate it and then I reflect it something.
00:17:29.350 --> 00:17:52.800
So, suppose this object is x x dash. So, x
dash is glide reflection of x. This kind of
00:17:52.800 --> 00:18:05.370
pattern is observed it leaves and then it
goes like this like this and so on, it
00:18:05.370 --> 00:18:12.540
is quite natural. Actually have you seen the
footprints when you walk suppose you are
00:18:12.540 --> 00:18:19.950
walking on a muddy road the way your footprints
are there see they they actually form a glide
00:18:19.950 --> 00:18:27.660
reflection pattern. So, left leg and, right
leg their imprint their footprints are actually
00:18:27.660 --> 00:18:40.240
glide reflections of each other. So, as you
are walking your one leg is here other leg
00:18:40.240 --> 00:18:49.550
is here, and then next leg is here, next leg
is here like this. So, that pattern this glide
00:18:49.550 --> 00:18:57.920
reflection, so very naturally.
So, when you walk then you have glide deflection.
00:18:57.920 --> 00:19:07.120
So, 4 types of symmetry is we have seen
glide reflection, translation, reflection
00:19:07.120 --> 00:19:15.610
and rotation and then you can compose them,
right and there are so many possibilities,
00:19:15.610 --> 00:19:28.110
compose them all.
So, which ones rotation, reflection, translation,
00:19:28.110 --> 00:19:47.110
and glide reflection, compose them all. So,
there could be some more complicating things.
00:19:47.110 --> 00:19:53.350
So, how many of these are there? Rotation
you can do by any angle reflection you can
00:19:53.350 --> 00:20:00.190
again reflect by any line translation, by
any amount you can do glide reflection again
00:20:00.190 --> 00:20:07.670
you can pick any axis of reflection and any
amount by which you will be translating.
00:20:07.670 --> 00:20:24.890
So, this is an infinite collection. So, what
you have is actually infinite group. And you
00:20:24.890 --> 00:20:33.360
may expect there are some complicated patterns
out there say for example, I can rotate by
00:20:33.360 --> 00:20:41.440
certain angle say theta then I can compose
by certain reflection by certain line and
00:20:41.440 --> 00:20:52.810
then I can do. So, I can glide reflection
then I can rotate again I can make life more
00:20:52.810 --> 00:20:58.730
complicated. So, life could be more complicated,
right I can have lots of compositions rotation
00:20:58.730 --> 00:21:08.761
reflection glide reflection and rotation and
then further maybe some translation. What
00:21:08.761 --> 00:21:16.280
is interesting is do not get anything new.
What is the output of this is either are single
00:21:16.280 --> 00:21:21.990
irradiation or a single reflection or single
translation or single glide reflection
00:21:21.990 --> 00:21:25.120
that is amazing
So, only 2 therefore, there are only 4 types
00:21:25.120 --> 00:21:30.160
of symmetries which I have already mentioned
and that is a theorem, a serious theorem.
00:21:30.160 --> 00:21:59.980
So, it is a theorem only 4 types of symmetries
are there in R 2. So, when I talk of symmetries
00:21:59.980 --> 00:22:17.390
here I mean isometries, right. So, here isometries
meaning, distance preserving maps. If I have
00:22:17.390 --> 00:22:23.430
2 points x and y after performing one of these
symmetric operations the distance should be
00:22:23.430 --> 00:22:29.270
preserved. So, if x goes to x dash, y goes
to y dash then the distance of x dash and
00:22:29.270 --> 00:22:36.080
y dash distance of x dash and y dash is same
as distance of x and y distance between
00:22:36.080 --> 00:22:43.160
x and y. So, there is a theorem there are
only 4 types of symmetries in R 2. And now
00:22:43.160 --> 00:22:52.680
let us have some fun I have shown you all
these cards , right. There are so many of
00:22:52.680 --> 00:23:00.940
them, what you can do is you can try to identify
what symmetries what symmetric patterns do
00:23:00.940 --> 00:23:10.540
they exhibit.
Let me show you one, this one would you like
00:23:10.540 --> 00:23:20.680
this as one of the wallpapers their room.
Does it exhibit extraordinary symmetry? Well,
00:23:20.680 --> 00:23:33.920
not quite probably you would like something
like this as a wallpaper pattern as your
00:23:33.920 --> 00:23:40.370
wallpaper, this has more symmetry. So, what
do I mean by that? So in fact, on the basis
00:23:40.370 --> 00:23:46.980
of symmetry one can classify all these possibilities
whatever are the wallpaper patterns into certain
00:23:46.980 --> 00:23:55.270
numbers and that number I am going to tell
you later. Let us discuss what kind of symmetries
00:23:55.270 --> 00:24:15.730
are there in .
So, there are wallpaper symmetries or 2D crystal
00:24:15.730 --> 00:24:28.140
symmetries take this one. I mentioned 4 types
of symmetries which one you observe here only
00:24:28.140 --> 00:24:35.140
translational, right.
So, there are various names of these symmetries.
00:24:35.140 --> 00:24:53.510
So, this is called p 1, p 1 is the pattern
where only translational symmetries are there,
00:24:53.510 --> 00:25:10.470
not much interest. Here is another one let
me show you this one, this is called p 4.
00:25:10.470 --> 00:25:33.030
What does p 4 has? p 4 has 2 rotation axis
or maybe I should call center, this 90 degree.
00:25:33.030 --> 00:25:37.620
So, what are the rotation centers here? 90
degree. If I pick the center of this and then
00:25:37.620 --> 00:25:48.510
I rotate that is one or I can actually pick
the center of this one and then I can rotate
00:25:48.510 --> 00:25:54.870
now you imagine not just this square getting
rotated, but assume that this is a wallpaper.
00:25:54.870 --> 00:26:00.590
So, you have this or there is a tiling you
have this and then the similar tile, here
00:26:00.590 --> 00:26:05.641
similar tile everywhere and then what is being
rotated it is the whole pattern, ok symmetry
00:26:05.641 --> 00:26:08.780
operations happening over the whole system
the whole wallpaper.
00:26:08.780 --> 00:26:14.680
So, I can take this and I can rotate the whole
wallpaper system about this again you get
00:26:14.680 --> 00:26:24.850
a symmetry. So, there are 2 centers this one
and this one both are 2 rotation centers ninety
00:26:24.850 --> 00:26:39.720
degree and then there is one rotation center
which is a 180 degree. Can you identify that
00:26:39.720 --> 00:26:47.630
180 degree center? Well, that is somewhere
here on the edge you take this rotate the
00:26:47.630 --> 00:26:53.080
whole wallpaper cycle, rotate the whole a
wallpaper pattern by 180 degree.
00:26:53.080 --> 00:27:01.480
So, when wallpaper or single tile observes
this kind of property this kind of phenomena
00:27:01.480 --> 00:27:06.980
that with similar kind of tile you exhibit
you just complete the wallpaper pattern and
00:27:06.980 --> 00:27:12.480
then observe there are 2 rotation x centers
of 90 degree in one rotation centers of 180
00:27:12.480 --> 00:27:20.340
degree and there is no other symmetries then
we call this type p 4. The one with only translational
00:27:20.340 --> 00:27:32.220
symmetry is called p 1. Let me show you few
more, this one.
00:27:32.220 --> 00:27:46.890
So, this is example of what is called p 3,
p 3 wallpaper symmetry and what is going to
00:27:46.890 --> 00:27:55.440
keep it symmetric or keep the whole wallpaper
pattern symmetric. There are actually 3 centers
00:27:55.440 --> 00:28:14.790
of rotation, and they are by 120 degrees which
one are those can you identify. Well, you
00:28:14.790 --> 00:28:25.590
take the hexagon take the center of the hexagon
rotate or you take centre of the right triangle
00:28:25.590 --> 00:28:31.510
rotate or you take the center of the green
triangle and then rotate. So, there are 3
00:28:31.510 --> 00:28:39.490
types of centers of rotation in their by 120
degree. So, that is called p 3 symmetry.
00:28:39.490 --> 00:28:59.760
And then there is this one and name is pg
just named. What does it have? Can you observe
00:28:59.760 --> 00:29:19.820
carefully? So, this has show you guide reflections
only. Just carefully look at it glide reflection
00:29:19.820 --> 00:29:26.610
as the record is a combination its composition
of reflection and a translation it has only
00:29:26.610 --> 00:29:33.460
drive reflections and pg. Now, I have so many
of these right, I have shown you I have so
00:29:33.460 --> 00:29:47.320
many of these all these and we have seen that
there are infinitely many symmetries for R
00:29:47.320 --> 00:29:57.470
2, how many symmetries will there be for wallpaper
patterns, that sounds interesting.
00:29:57.470 --> 00:30:12.730
So, how many different
00:30:12.730 --> 00:30:35.300
wallpapers matrix, different wallpapers within
types of? The answer is quite interesting
00:30:35.300 --> 00:30:49.320
and it is not very easy, 17 very curious.
So, there are 17 different wallpaper pattern,
00:30:49.320 --> 00:30:59.510
there are 17 different tiles types of tiles
that you can make up to symmetry, up to
00:30:59.510 --> 00:31:05.860
symmetry considerations.
So, who who found all this and how did define
00:31:05.860 --> 00:31:20.380
it. So, this is Fedorov in 1891 he had observed
all this and then much famous person many
00:31:20.380 --> 00:31:32.940
of you must have heard of him Polya, George
Polya who 1924 wrote article on wallpaper
00:31:32.940 --> 00:31:52.680
symmetries. And very nice reference for this
is an article by R.L.E Schwarzenberger and
00:31:52.680 --> 00:32:16.590
this article appeared in mathematical Gazette,
mathematical Gazette volume 58. In fact,
00:32:16.590 --> 00:32:32.960
there are some higher dimension versions of
this
00:32:32.960 --> 00:32:38.320
and in higher dimensions higher than 4 the
question is still open and how many different
00:32:38.320 --> 00:32:54.480
wallpaper patterns are there, and high dimension
versions were kind of part of Hilbert's 18th
00:32:54.480 --> 00:33:03.320
problem.
Those who know what Hilbert's problems are
00:33:03.320 --> 00:33:15.420
in in the year 1900 in Paris, Hilbert proposed
23 problems and that was kind of his expectation
00:33:15.420 --> 00:33:21.390
from mathematicians for last for next 100
years what he would like people into work
00:33:21.390 --> 00:33:30.100
upon. So, he proposed 23 problems and problem
18th has a portion where he is asking about
00:33:30.100 --> 00:33:36.270
higher dimension versions of this theorem
of Fedorov which says that there are only
00:33:36.270 --> 00:33:40.360
17 wallpaper patterns. And there are certain
names, I can just mention some of those
00:33:40.360 --> 00:33:49.690
names here you are seen p 1, p 3, p 4, p g
and then there are many in fact, I just write
00:33:49.690 --> 00:34:18.581
all those names, p 1, p 1, p 2, p 3, p 4,
p 6, p m, p g, c m, p m m, p m g, p g g, c
00:34:18.581 --> 00:34:39.740
m m, p 4 m, p 4 g, p 3 m 1, p 3 1 m, p 6 m;
I hope here 16, 1 2 3 4 5 6 7 8 9 10 11 12
00:34:39.740 --> 00:34:44.840
13 14 15 16 17 here and this nomenclature
is according to what kind of symmetry is do
00:34:44.840 --> 00:34:52.450
they exhibit. In fact, one can divides algorithms
on how to determine what kind of wallpaper
00:34:52.450 --> 00:34:58.240
symmetry is there is one of the wallpapers
and one of the tiles, it is not very difficulty
00:34:58.240 --> 00:35:02.440
device once you understand what all these
symmetries are.
00:35:02.440 --> 00:35:10.290
So, this time we do not talked about symmetry
next time we are going to talk about one
00:35:10.290 --> 00:35:17.560
interesting software that is used to understand
symmetry that is useful understand groups
00:35:17.560 --> 00:35:23.990
and we would make that software solve Rubik's
cube for us. So, keep watching.
00:35:23.990 --> 00:35:25.420
Thank you.