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So, during last lecture we saw Cayley graphs.
We saw directed Cayley graphs of various groups
00:00:20.450 --> 00:00:27.130
and we had some fun with that. And one question
that they asked last time was what is a purpose,
00:00:27.130 --> 00:00:30.740
what are the applications of trying candy
crabs like that.
00:00:30.740 --> 00:00:39.730
And before I come to applications let me show
few interesting objects I had shown you last
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time. This remember tetrahedron there are
few more objects that I am going to show and
00:00:49.240 --> 00:00:54.300
In later lectures we are going to talk
more about these objects that is tetrahedron,
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this of course you know is cube, that is a
dodecahedron it has 12 faces, dodecahedron,
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and this is a octahedron, it has 8 sides,
8 faces and then this is icosahedron it has
00:01:22.590 --> 00:01:29.960
20 faces. These are some quite nice objects
and you are going to have some fun with all
00:01:29.960 --> 00:01:38.390
these objects after sometime. Anyway let me
start with this, this object, tetrahedron.
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How do I write the group of rotations of this?
And as I am moving this actually what am I
00:01:51.420 --> 00:02:02.280
doing is I am doing a random walk on the Cayley
graph of the group associated to this. So,
00:02:02.280 --> 00:02:11.510
I can I can think of group of rotations
a group of rotational symmetries of tetrahedron.
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I can write the Cayley graph of that and I
can do random walk on those groups what does
00:02:18.470 --> 00:02:27.040
it mean. So, first let us see how can I write
this group of symmetries of this object.
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So, that is the purpose today
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rotational symmetries of of a tetrahedron.
How to put a notation to each vertex? So,
00:03:04.639 --> 00:03:16.040
here is the thing I put 1 to the top and 2
3 4 the bottom. Now, how can I get all the
00:03:16.040 --> 00:03:28.069
rotations of this? One ways I take one of
the vertices and then I take center of the
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opposite face. So, vortex and center of the
opposite face, I have like this maybe I will
00:03:36.559 --> 00:03:46.549
show you like this I am holding it let the
axis. This thumb and this finger they are
00:03:46.549 --> 00:03:55.650
forming an axis. And then there are 3 options
to rotate I can rotate by about this axis
00:03:55.650 --> 00:04:05.790
how much I can rotate by 120 degree and I
can rotate by further 120 degree which is
00:04:05.790 --> 00:04:08.810
240 degrees. So, this kind of rotations I
can do.
00:04:08.810 --> 00:04:15.919
So, how many choices for these vertices are
there? There are 4 choices, 1 2 3 4 4 choices
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vertices are there. So, those are certain
kind of rotations I have in this tetrahedron,
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these symmetries. So, first let me write
all those. So, I have a identity which is
00:04:32.470 --> 00:04:39.630
doing nothing and suppose this top vertex
is 1. So, when I am having this vertex 1 and
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when I am rotating this vertex is not moving
at all what is happening is 2 is going to
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3, 3 is going to 4, vertex 2 goes to 3, vertex
3 goes to 4. So, I will write it exactly in
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the same fashion as I take earlier in the
permutation group, group of permutations s
00:05:02.550 --> 00:05:11.700
n. Here vertex 1 is fixed and then I do it
once more. So, what I actually get? This 2
00:05:11.700 --> 00:05:27.280
4 3. In fact, the square of 2 3 4 is equal
to 2 4 3 that is easy that is easy to conflict,
00:05:27.280 --> 00:05:33.950
ok.
So, I have this and then I can fix say vertex
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number 2. So, I have 1 3 4 and 1 4 3 and then
I fix other vertices vertex number 3. So,
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I have 1 2 4 and 1 4 2 and when I fix vertex
number 4, I have 1 2 3 and 1 3. Is that all?
00:06:06.500 --> 00:06:18.710
Well, there are few more. You pick an edge
consider the midpoint of this and then there
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is a opposite edge which is perpendicular
to this. So, here is an edge, I take the midpoint
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and there is opposite edge which is actually
perpendicular to this. So, I pick the midpoint
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of that edge as well. So, I pick like this
and then I can rotate by 90 degree, I can
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rotate by 90 degree.
So, what is happening? In the process suppose
00:06:47.270 --> 00:06:52.500
this is vertex number one this vertex number
2 and these are 3 and 4. So, position is the
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position of the vertex 1 is being changed
with position of vertex 2 and position of
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vertex 3 is being changed with the position
of vertex 4. So, what is happening? 1 goes
00:07:07.060 --> 00:07:15.600
to 2 and 3 goes to 4 and this may happen.
There are other choices of vertices I can
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pick the vertex 1 and 3 and go to the edge
which is joining those 2 vertices. So, I have
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1 in 3, 1 goes to 3 and the opposite 1 is
2, 4. And there is one more 1 4, I pick vertices
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1 and 4 and then I pick an another one opposite
side. So, there I am swapping 2 and 3. So,
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these are all the elements here in this
the symmetry in the group of rotational symmetries
00:07:51.550 --> 00:07:56.260
of tetrahedron
How many of these are? There are 3 of these,
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4 of these, 7 and 5 of these eleven 5 of
these twelve and this group has a name A 4.
00:08:06.740 --> 00:08:19.240
Why do we give it this name? Because these
are, so happens that these are precisely
00:08:19.240 --> 00:08:33.260
those elements of A 4 whose parity I use the
word parity couple of lectures ago or I also
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use the word signature is 0, 0 was for the
even. So, that is these are, these are product
00:08:51.589 --> 00:09:10.550
of, these elements are products of even number
of transpositions.
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And now what about the Cayley graph of this?
So, maybe that you should do in the assignment.
00:09:26.379 --> 00:09:45.970
So, exercise for you would be first find
generating set of this group is rotational
00:09:45.970 --> 00:09:57.959
symmetries of tetrahedron, right, rotational
symmetries of tetrahedron which is A4 and
00:09:57.959 --> 00:10:22.760
second exercise would be finding Cayley graph.
What is the directed Cayley graph of this
00:10:22.760 --> 00:10:42.009
group? So, well I am going to tell you what
exactly the Cayley graph is. How does it look
00:10:42.009 --> 00:10:48.420
like I am not going to label it, but just
for you for some choice of generators for
00:10:48.420 --> 00:10:51.959
some generating said the Cayley graph looks
like this.
00:10:51.959 --> 00:11:23.449
I am just drawing it, just as a picture .
So, Cayley graph say this is identity element
00:11:23.449 --> 00:11:37.611
this could be 1 2 3, this could be 1 3 2 likewise
this is 1 2 3 4. This is identity, this is
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1 2 3, 1 3 2 and so on other things I am not
labeling, so 1 2 3 4 5 6 7 8 9 10 11 12 and
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this this this. So, here in this graph let
me talk off random walk. So, what I am talking
00:12:24.509 --> 00:12:36.819
about is random walk on graphs of groups those
graphs which are those Cayley graphs which
00:12:36.819 --> 00:12:44.889
are coming out as graphs of groups and
groups are coming out of automorphisms of
00:12:44.889 --> 00:12:59.550
certain objects. So, that is quite interesting.
To start with what is there say a geometric
00:12:59.550 --> 00:13:14.089
object like this one, and then you have a
a group which is being cooked out of it and
00:13:14.089 --> 00:13:23.249
how does it come some kind of symmetry considerations
you have you consider certain symmetries of
00:13:23.249 --> 00:13:35.310
that object. And then you choose some set
of generators and make graph Cayley graph
00:13:35.310 --> 00:13:49.709
of this and on this Cayley graph you are considering
random walk. So, what is random walk and what
00:13:49.709 --> 00:13:55.749
is the advantage of considering certain things
the language of random walks?
00:13:55.749 --> 00:14:05.110
So, I am trying to I am trying to explain
it through an example. We take a point say
00:14:05.110 --> 00:14:13.170
here, that is the identity position say that
is the identity position of this and then
00:14:13.170 --> 00:14:22.800
at each node I have options to traverse
either to this or through this or through
00:14:22.800 --> 00:14:27.819
this. So, there are 3 options. At each node
I have 3 options I can go like this, I can
00:14:27.819 --> 00:14:34.399
go like this, I can go like this in 3 different
directions. So, what is assigned to me?
00:14:34.399 --> 00:14:40.480
What is prescribed to me is at each point
you see there are how many at each point there
00:14:40.480 --> 00:14:47.319
are 1 2 3, 1 2 3 there are 3 edges which
are coming out of it then that is how the
00:14:47.319 --> 00:14:53.519
graph looks like here.
So, I am just as an example I am mentioning.
00:14:53.519 --> 00:15:13.420
So, each point has 3 choices of edges. So,
from here you can either go here or you can
00:15:13.420 --> 00:15:22.350
go here or you can go here. What if we assign
probability? Let us say that there are probability
00:15:22.350 --> 00:15:31.490
of taking this direction taking that direction
taking that direction equal probability. So,
00:15:31.490 --> 00:15:51.630
3 choices say with uniform probability. So,
uniform probability is 1 by 3 in this case
00:15:51.630 --> 00:15:57.100
whether 3 choices and accordingly. So, suppose
with 1 by 3 probability it comes here again
00:15:57.100 --> 00:16:03.239
there are 3 possibilities. So, take your system
take the configuration which is initially
00:16:03.239 --> 00:16:08.689
say at the identity position, and through
a number of steps it can go anywhere it can
00:16:08.689 --> 00:16:13.709
come here you choose this with one type probability,
chooses this with one type probability, chooses
00:16:13.709 --> 00:16:19.279
this with one type probability.
So, when I am actually moving this all the
00:16:19.279 --> 00:16:26.790
motions that I am doing here rotational motions
I am actually doing random walk on this, and
00:16:26.790 --> 00:16:32.549
I have to be careful in deciding what are
the actions, however moving because those
00:16:32.549 --> 00:16:38.350
the the basic moves here should correspond
to an element in the generating side there.
00:16:38.350 --> 00:16:45.319
So, when I go from certain position to certain
other position I will be doing an action here
00:16:45.319 --> 00:16:52.930
which corresponds to a generating element
here. So, for example, when I go from here
00:16:52.930 --> 00:16:58.459
to here what am I doing? I am taking 1 and
2, suppose they are 1 and 2 vertices and they
00:16:58.459 --> 00:17:04.699
are 3 and 4 vertices and they are just moving
like this. So, this move which I take for
00:17:04.699 --> 00:17:12.310
one end to and vertex 3 and 4 which are below
that corresponds to going from 1 to this point.
00:17:12.310 --> 00:17:18.170
So, our actions are being converted into random
walks.
00:17:18.170 --> 00:17:24.560
Now, I can do this thing not just for these
kind of objects, but for more complicated
00:17:24.560 --> 00:17:44.260
objects as well. Did I mention some complicated
object in these lectures? Yes, rubiks cube,
00:17:44.260 --> 00:17:49.110
rubiks cube there are so many configurations
and in forthcoming lectures we are actually
00:17:49.110 --> 00:17:56.110
going to see how many configurations are there.
So, now, you imagine a huge graph, what are
00:17:56.110 --> 00:18:01.670
the nodes? What are the vertices of that graph?
Those are all possible configurations. So,
00:18:01.670 --> 00:18:09.272
all possible possible configurations of rubiks
cube. And what are the edges of that graph?
00:18:09.272 --> 00:18:13.710
Those edges are determined by the actions
that you do on the rubiks cube. What kind
00:18:13.710 --> 00:18:20.030
of actions you can do? Say for example, you
can rotate by left or the left face by say
00:18:20.030 --> 00:18:26.380
90 degree or the top face by 270 degree all
those options are there. So, those will form
00:18:26.380 --> 00:18:33.920
generators of rubiks cube and using that you
can create huge extremely large graph and
00:18:33.920 --> 00:18:38.830
then walking along that graph is essentially
making lots of moves on the rubiks cube.
00:18:38.830 --> 00:18:46.540
So, it is just a random play random random
scrambling of the rubiks cube, you can
00:18:46.540 --> 00:18:58.330
do using these Cayley graphs, right.
We are going to talk we are going to eventually
00:18:58.330 --> 00:19:04.650
talk of rubiks cube, but not through graphs,
but through relations and generators using
00:19:04.650 --> 00:19:09.490
the software called called gap that will come
later on.
00:19:09.490 --> 00:19:16.710
So, you can have lots of random walks. So,
start to start with you have a geometric object
00:19:16.710 --> 00:19:21.540
and whenever actually play with the geometric
object you have random walk. And now this
00:19:21.540 --> 00:19:31.620
object need not be geometric all the time,
you have for example, cards playing cards.
00:19:31.620 --> 00:19:39.130
When you are shuffling when you are shuffling
playing cards very random fashion what is
00:19:39.130 --> 00:19:58.140
happening? Card shuffling is actually a random
walk. One can certainly ask various questions
00:19:58.140 --> 00:20:07.800
for example, one question could be you start
moving on a graph, on graph of the group say
00:20:07.800 --> 00:20:12.110
at this point and you are falling certain
probability distribution need not be reform
00:20:12.110 --> 00:20:16.580
probability distribution, but it would be
some other probability distribution.
00:20:16.580 --> 00:20:26.040
And after n number of steps where would you
be nothing is deterministic here. So, in terms
00:20:26.040 --> 00:20:32.710
of probability you can say, one interesting
statement would be that after how many steps
00:20:32.710 --> 00:20:38.400
you are likely to be they are equally likely
to be there on all the steps. So, after how
00:20:38.400 --> 00:20:43.030
many steps the probability of finding this
point after doing all this random walk would
00:20:43.030 --> 00:20:48.381
be a uniform probability. So, those are quite
interesting questions. And one name that I
00:20:48.381 --> 00:21:04.240
would like to mention is
this mathematician Persi Diaconis, he is not
00:21:04.240 --> 00:21:13.140
just a mathematician he is also a magician.
He has invented lots of card tricks, not
00:21:13.140 --> 00:21:20.670
just card tricks beyond that some really professional
magic and he has some theorems in shuffling
00:21:20.670 --> 00:21:27.910
of cards which are concerning for example,
in how many steps, how many steps are required
00:21:27.910 --> 00:21:36.980
to make a pack of card so to say random. So,
quite interesting things, right. Groups are
00:21:36.980 --> 00:21:42.360
there, symmetry is there, geometric objects
are there, graphs are there playing cards
00:21:42.360 --> 00:21:46.970
are there, rubiks cube so many interesting
things are there just in one book.
00:21:46.970 --> 00:21:56.080
So, the rest of the lecture let me talk about
siblings of this, what are those? I told you
00:21:56.080 --> 00:22:07.340
this is dodecahedron, this is cube, this is
icosahedrons, and this is octahedron. So,
00:22:07.340 --> 00:22:12.820
it would be fun for us to write symmetry groups
of all these things. Let me just tell you
00:22:12.820 --> 00:22:27.770
and that is also the exercise for you to find
what are symmetry groups for so called platonic
00:22:27.770 --> 00:22:41.710
solids and that is a key word, platonic solids.
So, what are platonic solids? So, these are
00:22:41.710 --> 00:22:47.260
some interesting objects which are quite symmetric.
So, there are some interesting groups associated
00:22:47.260 --> 00:22:53.320
to them and one can have lot of fun with those
groups. So, let me just explain what are all
00:22:53.320 --> 00:22:59.130
these platonic solids and later on in the
later part of the course we are going to
00:22:59.130 --> 00:23:07.400
understand the groups of symmetries of these
rotational symmetries of these as subgroups
00:23:07.400 --> 00:23:16.200
of what is called so 3, the group of rotations
of sphere. But nevertheless first let us understand
00:23:16.200 --> 00:23:28.760
what are platonic solids.
So, the order you understand them let me pick
00:23:28.760 --> 00:23:43.940
one of them. See, I am just say I am picking
cube there is some symmetry right, in what
00:23:43.940 --> 00:23:51.590
sense. Each face is a square and the square
of the same size, right. The square of the
00:23:51.590 --> 00:24:01.140
same size this is square each face is a square,
right and then each if you take an edge
00:24:01.140 --> 00:24:08.620
each edge is actually it intersection of 2
faces. So, 2 faces are meeting exactly on
00:24:08.620 --> 00:24:20.350
the edge. What else? We take this vertex 1
2 3, there are exactly 3 edges which are
00:24:20.350 --> 00:24:26.430
adjacent on to this vertex and this property
is same for all the vertices. So, if I pick
00:24:26.430 --> 00:24:33.050
some other vertex say this one again there
are 3 edges which are edges side to this.
00:24:33.050 --> 00:24:47.790
So, I will just write this I would say that
each side or rather each face, each face is
00:24:47.790 --> 00:25:05.250
a regular n-gon, in this case n was 4 everything
was regular 4 gon which is a square. And then
00:25:05.250 --> 00:25:34.660
each edge is precisely the intersection of
2 faces, it is point 1, point 2 and then for
00:25:34.660 --> 00:25:59.730
each vertex the number of edges you see which
are incident on on this vertex the same and
00:25:59.730 --> 00:26:14.510
the same number we denoted by m. So, this
is what defines a platonic solid. Let me take
00:26:14.510 --> 00:26:22.510
some other see this v octahedron in octahedron
you have this triangle, so everywhere is the
00:26:22.510 --> 00:26:29.680
same triangle of the same size and if I pick
this thing one vertex. Now, how many edges
00:26:29.680 --> 00:26:43.080
are incident on this? 1 2 3 and 4, ok.
So, for octahedron n is 3 and m is 4. So,
00:26:43.080 --> 00:26:54.980
do each platonic sonic therefore, you can
associate what is called Schlafli symbol.
00:26:54.980 --> 00:27:07.250
Schlafli was mathematician from Germany. So,
what is the Schlafli symbol? n comma m n is
00:27:07.250 --> 00:27:16.550
that regular n-gon and m is number of edges
which are incident on a vertex. So, this is
00:27:16.550 --> 00:27:39.190
a number of sides of each face and this is
number of edges meeting at the vortex. And
00:27:39.190 --> 00:28:03.190
it is very interesting thing, how many platonic
solids are there? Only 5. That is quite interesting
00:28:03.190 --> 00:28:09.760
statement and how does one see because having
only 5 platonic solids has something to do
00:28:09.760 --> 00:28:19.100
with the subgroups of rotational symmetries
of the sphere.
00:28:19.100 --> 00:28:28.550
So, I am quickly going to connect it with
graph theory here is a very famous statement
00:28:28.550 --> 00:28:38.650
in graph theory which is attributed to Euler.
Call it Euler's formula which is you take
00:28:38.650 --> 00:28:45.180
any closed shape like this not necessarily
uniform not necessarily a platonic solid you
00:28:45.180 --> 00:28:57.030
take any closed shape then number of vertices
minus number of edges plus number of faces
00:28:57.030 --> 00:29:13.310
is a constant its actually 2. So, this is
number of vertices, this is number of edges
00:29:13.310 --> 00:29:24.480
and this is number of faces. They make this
computation always going to get the number
00:29:24.480 --> 00:29:35.090
2 that is quite curious. Let us see this one.
I have taken cube for cube the Schlafli symbol
00:29:35.090 --> 00:29:47.220
is n comma m n is 4 square every side is a
square 3 because each vertex is having 3 edges.
00:29:47.220 --> 00:30:04.070
So, this is Schlafli symbol.
And then cube how many vertices are there?
00:30:04.070 --> 00:30:12.530
Top 4 bottom 4 8 vertices 8 vertices are there.
How many edges are there? 4 up 4 down and
00:30:12.530 --> 00:30:27.340
4 vertical, so 12. And how many faces are
there? 6 faces are there f 6. So, 8 minus
00:30:27.340 --> 00:30:40.570
1, 2 plus 6 is indeed 2, we have verifying
this. Same is case with other platonic solids
00:30:40.570 --> 00:30:47.230
in fact, not just platonic solids, but as
I said any solid any solid graph any closed
00:30:47.230 --> 00:30:55.940
craft we have this property. So, how to see
that there are only 5 platonic solids?
00:30:55.940 --> 00:31:04.340
As I said in the previous slide proof is very
easy and very interesting it is just very
00:31:04.340 --> 00:31:31.040
nice interpretation of Euler's formula.
So, let me just do this. So, why only 5 platonic
00:31:31.040 --> 00:31:47.720
solids? If you know this book by Euclid which
is called elements, why story book which is
00:31:47.720 --> 00:32:04.900
2000 years, may be even more maybe 2300 years
story book which was the first we can document
00:32:04.900 --> 00:32:14.250
available for geometry for the formal geometry
that we have Euclid actually mentions catalytic
00:32:14.250 --> 00:32:23.870
solids quite interesting, ok. So, why only
5 cationic solids? So, what do I do? I keep
00:32:23.870 --> 00:32:36.980
this at the back of my mind and then I count
in this number of edges in 2 different face
00:32:36.980 --> 00:32:45.790
number of edges. How can I count? You remember
Schlafli symbol n comma m what does it denote,
00:32:45.790 --> 00:33:04.390
n was number of sides of the face and m was
number of edges per vertex.
00:33:04.390 --> 00:33:13.260
So, if I am going to count number of edges
then maybe I should not say the number of
00:33:13.260 --> 00:33:26.190
edges per vertex, but number of edges meeting
at a vertex, yeah. So, this is a Schlafli
00:33:26.190 --> 00:33:31.380
symbol and his number of sides of the face
and n is number of edges which are meeting
00:33:31.380 --> 00:33:44.170
at a vertex. So, if I consider edges per face,
edges per face how many? m times, so
00:33:44.170 --> 00:33:55.610
the n times number of faces end times number
of faces, but then this edge is being counted
00:33:55.610 --> 00:34:01.480
twice it is counted for this as well as this
one its being counted twice. So, I have to
00:34:01.480 --> 00:34:11.069
divide by 2 that is e number of edges. But
similarly I could also count this is actually
00:34:11.069 --> 00:34:20.230
total number of edges yeah. So, that is e
number of edges e is n f by 2. But then I
00:34:20.230 --> 00:34:27.260
could have also counted it in different way
because I have a vertex and at that vertex
00:34:27.260 --> 00:34:38.190
I have these these many edges which are
coming out of it. So, maybe m v and each edge
00:34:38.190 --> 00:34:42.960
is shared by 2 vertices.
So, each edge is having this vertex and that
00:34:42.960 --> 00:34:57.420
vertex. So, m u by 2. So, from this what do
I have? I have that f is 2 e by m and v is
00:34:57.420 --> 00:35:16.690
sorry f is 2 e by m f is 2 e by and v is 2
e by m, now all this information I put here
00:35:16.690 --> 00:35:34.440
and then what do I get 2 e by m minus e plus
2, e by n is 2. So, from all this you can
00:35:34.440 --> 00:35:54.119
conclude that at 1 by m plus 1 by n is actually
strictly greater than half.
00:35:54.119 --> 00:36:00.090
And certainly number of sides each phase has
to be greater than equal to 3. So, this quantity
00:36:00.090 --> 00:36:07.800
is greater than equal to 3 and in order to
maintain this inequality one has to have that
00:36:07.800 --> 00:36:15.260
n is less than equal to 5. So, this is what
puts a restriction and when you have all the
00:36:15.260 --> 00:36:24.860
possibilities. So, when you have say 3 comma
5 that is a possibility which is icosahedron
00:36:24.860 --> 00:36:32.720
each side is a triangle and each vertex is
having 5 edges which are coming out of it.
00:36:32.720 --> 00:36:41.540
5 comma 3 could be possibility this is n this
is m, 5 comma 3 would be a dodecahedron, and
00:36:41.540 --> 00:36:54.460
then icosahedrons, dodecahedron, and then
this would be 4 coma 3 which is a cube, and
00:36:54.460 --> 00:37:12.910
then 3 comma 4 would be octahedron, this is
3 4 and then and have this 3 comma 3 and
00:37:12.910 --> 00:37:18.750
that would be tetrahedral and these are
the only possibilities.
00:37:18.750 --> 00:37:25.800
So, there are only 5 platonic solids. In fact,
everything can be understood in terms of the
00:37:25.800 --> 00:37:33.230
Schlafli symbol that I had mentioned. So,
you can actually have fun with all this you
00:37:33.230 --> 00:37:37.850
can try to find generators of all these groups,
you can write Cayley graphs of all these
00:37:37.850 --> 00:37:43.160
groups, and in coming lectures we are going
to understand the groups of symmetries of
00:37:43.160 --> 00:37:52.721
these objects in quite interesting fashion
and some relations of these with some
00:37:52.721 --> 00:37:57.790
interesting statements in group theory are
also going to be made. So, keep watching.
00:37:57.790 --> 00:37:58.589
Thank you.