WEBVTT
Kind: captions
Language: en
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So, last time we discussed about the representations
and I said that it is a good thing its quite
00:00:28.099 --> 00:00:36.400
handy thing for calculations to think of abstract
elements of a group as matrices or some other
00:00:36.400 --> 00:00:53.430
thing may permutation so that calculations
are easy. And we also saw certain matrix groups
00:00:53.430 --> 00:01:05.890
last time, and one of the matrix groups that
I promised I will discuss today is SO 3 R
00:01:05.890 --> 00:01:13.530
or simply SO 3.
And a couple of lectures ago we had seen that
00:01:13.530 --> 00:01:23.539
SO 3 can be understood through its double
cover which is unit quaternions, quaternions
00:01:23.539 --> 00:01:31.119
which have norm one you remember that,
right. Whenever I have a unit quaternion to
00:01:31.119 --> 00:01:43.729
that I can associate a map which is conjugation
by q inverse conjugation by q. So, that is
00:01:43.729 --> 00:01:52.740
q x q inverse that kind of map and in in the
understanding of rotations this map is
00:01:52.740 --> 00:01:58.229
quite useful.
So, what is the purpose for today's lecture
00:01:58.229 --> 00:02:16.180
today we want to understand finite sub groups
of SO 3 R. So, you have group of rotations
00:02:16.180 --> 00:02:23.060
group of rotations there are inverse so many
axis infinity many axis and there are infinitely
00:02:23.060 --> 00:02:29.650
many angles about which you can rotate. So,
thing is this this in finite group we want
00:02:29.650 --> 00:02:38.430
to find out number of finite sub groups.
So, before we do that it is very clear to
00:02:38.430 --> 00:02:47.612
us that SO 2 is a subgroup of SO 3. So, these
are rotations in 2 dimensions. So, I can always
00:02:47.612 --> 00:02:52.310
fix back see see I am fixing the z axis
and I am rotating. So, that rotation is just
00:02:52.310 --> 00:03:02.170
a rotation in 2 dimensions and therefore,
SO 2 can be thought of as a subgroup of SO
00:03:02.170 --> 00:03:11.329
3. Rotations in 2 dimensions can be thought
of as rotations in 3 dimensions as well
00:03:11.329 --> 00:03:17.640
yeah. And what are subgroups of this? What
are finite sub groups of SO 3 R? So, what
00:03:17.640 --> 00:03:25.859
is SO 3 R? It is not very hard to see that
SO 3 R is actually same as s one which is
00:03:25.859 --> 00:03:33.239
unique circular.
So, you want those complex numbers whose now
00:03:33.239 --> 00:03:45.470
whose mod whose norm is 1, no 1 complex numbers.
So, finite subgroups of S 1 finite subgroup
00:03:45.470 --> 00:03:57.209
of circle they are what they are cyclic groups
finite cyclic groups. So, therefore, it is
00:03:57.209 --> 00:04:05.749
clear that finite cyclic groups are also
subgroups of SO 3 R. But our these all we
00:04:05.749 --> 00:04:14.500
want to know all finite sub groups of SO 3
R and that is quite interesting problem which
00:04:14.500 --> 00:04:20.690
involves group action and heavenly uses
one side formula for counting orbits that
00:04:20.690 --> 00:04:31.900
we had discussed a couple of lectures ago.
Before we do that let me mention that lots
00:04:31.900 --> 00:04:48.650
of finite sub groups of SO 3, well I should
probably not be seen lots, so what essentially
00:04:48.650 --> 00:05:20.750
all. So, they occur as groups of rotational
symmetries of platonic solids I hope you remember
00:05:20.750 --> 00:05:25.910
what platonic solids were I had shown you
certain 3 dimensional objects. I will show
00:05:25.910 --> 00:05:35.629
you again and that is going to your first
job. First job is going to be understanding
00:05:35.629 --> 00:05:45.080
of groups of rotational symmetries of platonic
solids. In fact, in one case in case of tetrahedron
00:05:45.080 --> 00:05:54.880
we had seen this group. In fact, we had written
this group and we had also made any graph
00:05:54.880 --> 00:05:58.600
of this group in case of tetrahedron. Let
us recall platonic solids.
00:05:58.600 --> 00:06:30.979
I will show again this one this has 20 faces.
So, f is 20 number of vertices is 12 and number
00:06:30.979 --> 00:06:52.789
of edges is 30 and this is called sorry not
tetrahedron, icosahedrons this one, every
00:06:52.789 --> 00:07:06.039
face is a triangle. And then we have this
thing which is dual to it and this is called
00:07:06.039 --> 00:07:22.720
dodecahedron. And how many faces does it have?
It has 12 faces and how many verses it has
00:07:22.720 --> 00:07:43.580
it has let us count 5 up, 5 down, 10 and then
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20
00:07:43.580 --> 00:07:57.720
verses and number of edges is 30.
That is dodecahedron and this one is octahedron.
00:07:57.720 --> 00:08:21.370
For this number of faces is 8 octa number
of words is 6 and number of edges is 12
00:08:21.370 --> 00:08:38.500
I guess, so 1 2 3 4 4 up, 4 down and 3
in the middle 12. And then we have cube
00:08:38.500 --> 00:09:00.590
which is dual to it the cube has 6 faces
and 8 vertices and 12 ages for up for bottom
00:09:00.590 --> 00:09:19.690
for vertical and this one tetrahedron where
number of faces is 4 number of vertices
00:09:19.690 --> 00:09:36.220
is also 4 and number of edges is 6, 3 here,
3 vertical, 3 in the bottom 6 and this is
00:09:36.220 --> 00:09:40.870
dual to itself.
So, what is the duality? Duality it is between
00:09:40.870 --> 00:09:51.960
faces and vertices faces and vertices that
is duality, here again faces and vertices
00:09:51.960 --> 00:10:02.520
and number of edges is equal number of edges
is equal and this is dual to itself. So, for
00:10:02.520 --> 00:10:08.570
example, I can take this I can consider middle
of the face middle of this face and opposite
00:10:08.570 --> 00:10:16.180
face I hold like this and then I hold it like
this.
00:10:16.180 --> 00:10:24.160
So, here the vertex, vertex is you see if
I rotate about this about this axis this is
00:10:24.160 --> 00:10:29.160
order 4 rotation. So, I can rotate by 90 degree,
90 degree, 90 degree, 90 degree 4 times I
00:10:29.160 --> 00:10:37.410
am back and similarly here or therefore, rotation
90 degree each. So, the midpoint of the phase
00:10:37.410 --> 00:10:44.090
is acting as x midpoint of phases I am taking
it joining them opposite ones and those are
00:10:44.090 --> 00:10:50.520
acting as axis.
So, this is phase vertex duality and how the
00:10:50.520 --> 00:11:01.340
call them dual do any key. And now the question
is how they interesting in sub in
00:11:01.340 --> 00:11:18.820
determining subgroups of S 3.
So, platonic solids and sub groups of SO 3,
00:11:18.820 --> 00:11:27.920
SO 3 R. For that let us first determine what
are the groups of symmetries of that objectives,
00:11:27.920 --> 00:11:33.860
and it is not very hard everything is just
about observation and putting in the right
00:11:33.860 --> 00:11:38.320
mathematical language.
So, you do that and after that we are going
00:11:38.320 --> 00:11:50.680
to learn how can we determine all subgroups
of SO 3 in terms of platonic solids. So, let
00:11:50.680 --> 00:11:58.830
us take this one, this tetrahedron, and what
are all the symmetries. In fact, we wrote
00:11:58.830 --> 00:12:04.641
all the symmetries in terms of permutations
in terms of even permutations that we have
00:12:04.641 --> 00:12:10.750
already done couple of lectures ago where
I I said 1 3 and then 2 4 that kind of thing
00:12:10.750 --> 00:12:17.020
and then 2 3 4 those all are 3 and all
are 2 are elements in a 4.
00:12:17.020 --> 00:12:24.800
Nevertheless, let us have a look at it again.
So, first first job is to identify order 2
00:12:24.800 --> 00:12:31.240
elements in the symmetry group. So, how do
you find order 2 elements in this? So, here
00:12:31.240 --> 00:12:40.360
I take midpoint of edge and opposite side,
right. I am holding it from the midpoint of
00:12:40.360 --> 00:12:48.730
this edge and when I hold it naturally the
other side my thumb is also on the midpoint
00:12:48.730 --> 00:12:54.320
of opposite edge.
And now I can rotate by 180 degree. So, if
00:12:54.320 --> 00:13:03.350
I rotate by 180 degree I am like this back,
to the similar kind of the symmetry motion,
00:13:03.350 --> 00:13:07.180
this is the symmetric rotation. So, there
was ordered 2 element in this. So, the order
00:13:07.180 --> 00:13:28.720
2 element is; let me call it s order 2 element.
And which is obtained by taking midpoints
00:13:28.720 --> 00:13:39.750
of edges as axis, midpoints of say opposite
edges.
00:13:39.750 --> 00:13:49.530
Now, what else is very natural order which
occurs here? In a group of this group of symmetry
00:13:49.530 --> 00:13:55.220
of this, right: triangle. So, there has to
be order 3 element naturally what which one,
00:13:55.220 --> 00:14:03.070
I take vertex and opposite phase midpoint
of the opposite phase I have and then I simply
00:14:03.070 --> 00:14:11.560
rotate, I rotate by R 2 pi by 3 120 degree
each time. And that is how I get order 3 element
00:14:11.560 --> 00:14:18.500
in this. I am just rotating with that as an
axis, right I am just rotating that as an
00:14:18.500 --> 00:14:30.430
axis.
So, here is t which is order 3 element. And
00:14:30.430 --> 00:14:46.200
how do I get that order 3 element? So, I take
a vertex and I join it with midpoint of opposite
00:14:46.200 --> 00:15:02.320
face. And that is how I get order 3 element.
What else? What if I compose two, what if
00:15:02.320 --> 00:15:08.100
I compose these 2 what would happen? And that
is going to be interesting.
00:15:08.100 --> 00:15:23.150
So, I take first I rotate like this and
then I take midpoints and then rotate again.
00:15:23.150 --> 00:15:33.280
was it What is it going to be? That is
interesting. And for this let us try to do
00:15:33.280 --> 00:15:43.710
a competition. Although, what you can do one
way to see it is. So, question is what is
00:15:43.710 --> 00:15:55.160
s t, right. So, one way is to actually
do the experiment and realize what is s
00:15:55.160 --> 00:16:09.620
t, what is the order of s t. Or the other
way as we did some lectures ago. So, other
00:16:09.620 --> 00:16:22.450
2 element would be something like 1 3 2 4
something like this. And what is t? T let
00:16:22.450 --> 00:16:40.260
us say t is so I am writing s here. So,
t say for example, 2 3 4. Let us compute what
00:16:40.260 --> 00:16:47.240
it is.
So, this is edge joining. So, this is the
00:16:47.240 --> 00:16:53.480
edge which is joining vertex 1 with 3 and
the other is vertex 2 and 4. And here the
00:16:53.480 --> 00:17:01.490
vertex 1 is having 1 end of the axis and midpoint
of the phase 2 3 4 is having midpoint of the
00:17:01.490 --> 00:17:08.140
he is having other point of that same
axis. So, what happens to 1? Here nothing
00:17:08.140 --> 00:17:15.400
happens, here nothing happens, and here 1
goes to 3; 1 goes to 3. What happens to 3?
00:17:15.400 --> 00:17:30.230
3 goes to 4 and 4 goes to 2 and that is all.
So, 3 eventually goes to 2. What happens to
00:17:30.230 --> 00:17:44.740
2? 2 goes to 3 and 3 goes to 1, this is over
and what about 4? 4 goes to 2 and 2 comes
00:17:44.740 --> 00:17:54.830
back to 4. So, this is it. So, order is 3;
the order is 3.
00:17:54.830 --> 00:17:59.990
So, when they are actually compose like this
order is 3. So, you should be realizing that,
00:17:59.990 --> 00:18:09.900
therefore s t is actually rotation and
this time this the the vertex is 4th vertex.
00:18:09.900 --> 00:18:15.740
So, this is a different vertex about which
you are having this rotation. So, the one
00:18:15.740 --> 00:18:30.510
end of this is vertex 4 and you are having
other end of the axis as midpoint of is
00:18:30.510 --> 00:18:39.009
2 3 4.
So, whatever is that face 2 3 4. No here
00:18:39.009 --> 00:18:54.750
in this case 1 3 2; in this case its face
1 3 that face yeah. So, s t has order 3. So
00:18:54.750 --> 00:19:06.090
now, I have group consider the group G
which is generated by 2 elements s and t,
00:19:06.090 --> 00:19:15.460
such that square of s is 1; these are relations,
remember generators and relations t cube is
00:19:15.460 --> 00:19:30.149
1 and s t cube is 1. And by doing all this
what we have obtained is actually a homomorphism
00:19:30.149 --> 00:19:47.059
from this group to the group of rotations
of tetrahedral.
00:19:47.059 --> 00:20:03.660
So, here is this homomorphism and it is a
surjective homomorphism; surjective because,
00:20:03.660 --> 00:20:10.100
we have observed all possible symmetries in
this. So, that is how we are came here to
00:20:10.100 --> 00:20:22.900
be surjective.
And what is the order of this? So, observe
00:20:22.900 --> 00:20:33.700
by some work that order of G is 12 and the
symmetries of tetrahedron that order is also
00:20:33.700 --> 00:20:55.759
12. And I have this subjection. So, this has
to be isomorphism. So, what we have is that
00:20:55.759 --> 00:21:02.309
rotations of tetrahedron, the group is this
and it is not hard to see that this group
00:21:02.309 --> 00:21:11.240
is actually A 4. The group is A 4, you
can actually in longhand you can do some calculations
00:21:11.240 --> 00:21:18.590
and obtain that this group is A 4. You just
have to observe these relations and these
00:21:18.590 --> 00:21:24.830
generator the the this kind of relations on
these two generators.
00:21:24.830 --> 00:21:36.679
So, tetrahedral symmetry is group A 4. Let
us see few more, let me take my favorite
00:21:36.679 --> 00:21:51.419
one Icosahedron.
What to pick first, let us see take vertices.
00:21:51.419 --> 00:21:56.950
If I get vertices and rotate how much is the
angle which is admissible for symmetry? You
00:21:56.950 --> 00:22:06.350
can see 5. So here is an element of order
5, before that let us see element of order
00:22:06.350 --> 00:22:26.759
2. So, again the element of order 2 is obtained
by a midpoint of edges.. So, this is a what
00:22:26.759 --> 00:22:36.830
you say order 2 element. So, order 2 element
is present there what about order 3 elements,
00:22:36.830 --> 00:22:43.659
since triangles are there; since triangles
are there midpoints of triangles you rotate
00:22:43.659 --> 00:22:50.740
by 2 pi by 3 and that is what is going to
give me element of order 3.
00:22:50.740 --> 00:23:11.039
So, I take midpoints of faces these triangles
and this is order 3 anyway. So, how many order
00:23:11.039 --> 00:23:18.820
2 elements are here? So, there are total of
are 30 edges, there are 15 pairs of edges,
00:23:18.820 --> 00:23:29.919
right. So, since this is 15 pairs of edges
are there, I have 15 of these. So, one is
00:23:29.919 --> 00:23:35.710
this position other is that position, 15 of
them. And order 3 elements, how many order
00:23:35.710 --> 00:23:47.090
3 elements will be there? Again, there are
20 faces. So, I will have one identity position.
00:23:47.090 --> 00:23:57.749
So, there are total 10 possible choices
and each thing is giving you 2 nonidentity
00:23:57.749 --> 00:24:08.159
rotations, so what you have is 20 of them.
What about s t? Again the same question: what
00:24:08.159 --> 00:24:14.240
is the order of s t?. So, for that maybe you
have to do some certain experiment, you make
00:24:14.240 --> 00:24:22.450
your own icosahedron and try to do some
experiment. And what I can tell you is
00:24:22.450 --> 00:24:35.320
that s t to the power 5 is 1. So, s t is of
order 5 and that order 5 element is actually
00:24:35.320 --> 00:24:44.580
obtained like this by taking vertices. So,
this is obtained by taking midpoints of that,
00:24:44.580 --> 00:24:59.259
not midpoints but taking opposite vertices.
So, when you take opposite vertices you get
00:24:59.259 --> 00:25:06.980
order 5 element. And again you can think
of this group which is generated by 2 elements
00:25:06.980 --> 00:25:19.009
s and t. And the relations are s square is
1, t cube is 1, and s t to the power 5 is
00:25:19.009 --> 00:25:25.950
1; this group. And this group it is not hard
to see some work is required that this is
00:25:25.950 --> 00:25:30.259
actually A 5. So, group generated by this
is A 5.
00:25:30.259 --> 00:25:43.100
So, from here to groups of rotations of
icosahedron you have again surjective map.
00:25:43.100 --> 00:25:51.340
And you know what this symbol as does it it
corresponds to the midpoint of edges rotation,
00:25:51.340 --> 00:25:55.460
and similarly t corresponds to midpoints of
faces rotation; that kind of rotation. And
00:25:55.460 --> 00:26:09.650
here it is again not very difficult to see
that order of this is 60.
00:26:09.650 --> 00:26:26.409
So, here actually if you look at order 3 elements
and they are 20, and order 5 elements they
00:26:26.409 --> 00:26:36.330
are 24; because there are 6 options there
are 12 vertices, so there are 6 options 6
00:26:36.330 --> 00:26:46.080
and each pair of vertex when you take as axis
and keep opposite ones then you get 4 nonidentity
00:26:46.080 --> 00:26:51.369
positions. So, 6 into 4 there are 24. And
then there is one order one element which
00:26:51.369 --> 00:26:59.369
is identity.
So, there is a total of 20 plus 15 - 35; 35
00:26:59.369 --> 00:27:09.580
plus 1 - 36 total of 60. So, group of rotations
of icosahedron has size 60 and if you look
00:27:09.580 --> 00:27:15.019
at these generators and the subjective map
these generators again have; again generate
00:27:15.019 --> 00:27:22.289
a group which is a for your 60. So, here is
an isomorphism. So, A 5 is group of symmetries
00:27:22.289 --> 00:27:32.200
of this object which is icosahedron.
Now, for this dodecahedron is the same thing,
00:27:32.200 --> 00:27:39.200
it is all about vertex and face duality. So,
rather than saying all this, I will write
00:27:39.200 --> 00:27:46.669
face, in case of face I will write vertex
here, in case of vertex which which I will
00:27:46.669 --> 00:27:59.590
write face here and that all and so for dodecahedron
as well I get same symmetry group which is
00:27:59.590 --> 00:28:07.919
A 5.
Now, let us look at cube or octahedron. Same
00:28:07.919 --> 00:28:22.019
thing, yes let us take this cube. So, what
would be symmetry is here? So, first of all
00:28:22.019 --> 00:28:32.840
order 2 element, and order 2 element is again
obtained as midpoints of edges. So, I take
00:28:32.840 --> 00:28:38.820
midpoint of edge opposite edge take the midpoint
and I get order 2 element; the same every
00:28:38.820 --> 00:28:51.840
time. And what if I take opposite vertices;
so t. So, order 2 element by taking midpoints
00:28:51.840 --> 00:29:13.700
of opposite edges, and t which is order 2
element sorry order 3 element I can obtain
00:29:13.700 --> 00:29:22.299
by considering opposite with or rather yeah;
so opposite vertices.
00:29:22.299 --> 00:29:32.390
Now, you can see from here I; so from top
if I look at there are 3 edges which are going
00:29:32.390 --> 00:29:37.519
so just rotate these edges among themselves,
just rotate these edges among themselves.
00:29:37.519 --> 00:29:43.700
And that is how you will get order 3 element
and now the question is what is s t.. So,
00:29:43.700 --> 00:29:49.580
again you can actually do experiment if you
want you can enable them so that you remember
00:29:49.580 --> 00:29:57.419
what is identity position. So if you do
order 2 element, if you do this and after
00:29:57.419 --> 00:30:05.640
that you do this what actually you obtain
is order 4 element. So, s to the s t to
00:30:05.640 --> 00:30:16.980
the power 4 is actually 1. And s t is obtained
by considering opposite faces rather midpoints
00:30:16.980 --> 00:30:24.429
of them; midpoints of opposite faces you take
and that is how you get s 2 to the power 4.
00:30:24.429 --> 00:30:31.179
And again you consider the group which is
on 2 generators s and t for which relations
00:30:31.179 --> 00:30:43.619
are s square is 1, t cube is 1, and s t to
the power 4 is 1. And from here to group of
00:30:43.619 --> 00:30:50.169
rotations of cube you give a map.
So, it is not very difficult to realize here
00:30:50.169 --> 00:30:57.450
that groups of rotations of cube has how many
elements let us count it. So, here how many
00:30:57.450 --> 00:31:03.950
pairs of edges are there? There are 12 edges.
So, 6 elements you will get this way, right.
00:31:03.950 --> 00:31:09.929
And here order 3 elements you will get 2 times
pairs of opposite vertices. So, there are
00:31:09.929 --> 00:31:18.859
8 vertices. So, you you will get 4 pairs
and each pair is going to be of 2. So,
00:31:18.859 --> 00:31:27.659
you get 4 into 2 6 of them.
And what about midpoints of faces, so how
00:31:27.659 --> 00:31:38.389
many faces are there? So, you have 3
pairs. So, 3 pairs of faces and each phase
00:31:38.389 --> 00:31:56.249
rotation is game going to give you 3; 1
2 3 4 its 9, yes wait a minute sorry some
00:31:56.249 --> 00:31:58.950
editing will be required I have written something
wrong. Mohan, some editing will be required.
00:31:58.950 --> 00:32:10.179
yeah
Then else one we start from the .
00:32:10.179 --> 00:32:15.419
And this.; so here in this cube let us count
how many order 2 elements are there, how many
00:32:15.419 --> 00:32:34.132
order 3 elements are there, and how many order
4 elements are there.. So, for order 2 elements
00:32:34.132 --> 00:32:42.210
I will see how many different edges are there.
So, there are total of 12 edges. So, they
00:32:42.210 --> 00:32:52.840
are total 12 by 2 6 pairs, so there are 6
elements of order 2. What about order 3 elements?
00:32:52.840 --> 00:33:03.659
So, when I do order 3 elements I am considering
opposite vertices. And in each rotation
00:33:03.659 --> 00:33:15.009
I am getting 2 non trivial 2 nonidentity positions.
So, I will multiply with not with 3, but with
00:33:15.009 --> 00:33:22.200
2. And how many pairs of opposite of vertices
are there? So, there are total of 8 vertices:
00:33:22.200 --> 00:33:32.789
1 2 3 4 4 up 4 down, so I will get total of
4 4 into 2 is 8.
00:33:32.789 --> 00:33:41.820
And what about opposite phases; so how many
opposite phases are there. There are 3 pairs
00:33:41.820 --> 00:33:47.389
and each of them is going to give me three
different three nonidentity positions,
00:33:47.389 --> 00:34:01.330
so 3 into 3 9. And then you have one identity
element. So, how much is this? This is 24.
00:34:01.330 --> 00:34:12.210
So, size of rotation group of cube is therefore
24, it is not hard to see that this is also
00:34:12.210 --> 00:34:20.450
of order 24 and in fact this is as for permutation
group on 4 letters. And then again this surjective
00:34:20.450 --> 00:34:28.620
map becomes isomorphism as in earlier cases.
So, for cube the group of rotational symmetries
00:34:28.620 --> 00:34:44.940
is S 4. And again by duality, for octahedron
as well I have a group of rotational symmetries
00:34:44.940 --> 00:34:51.470
which is same as S 4.
So, what we did in this lecture today? We
00:34:51.470 --> 00:34:59.550
just saw all possibilities of groups of rotations,
rotational symmetries of all these platonic
00:34:59.550 --> 00:35:06.150
solids. And as you recall I had already proved
that there are only 5 platonic solids, and
00:35:06.150 --> 00:35:20.570
these are right in front.
And observation is: that each of these objects
00:35:20.570 --> 00:35:40.670
can be put inside a sphere, each platonic
solid can be put inside sphere. So, a unit
00:35:40.670 --> 00:35:50.650
sphere or whatever is the radius of these
objects inside a sphere. And that is going
00:35:50.650 --> 00:35:58.010
to help us in determining what are all subgroups
of; what are all finite sub groups of SO 3.
00:35:58.010 --> 00:36:04.460
So, in next lecture we are going to see all
that, it is quite a fun, is very interesting
00:36:04.460 --> 00:36:08.140
application of group actions. See you next
time.