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00:00:08,870 --> 00:00:12,280
PROFESSOR: Any questions about
where we left off-- up to

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00:00:12,280 --> 00:00:13,530
where we left off?

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00:00:17,050 --> 00:00:21,700
OK, what I'll do then is give
you a few more examples of the

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00:00:21,700 --> 00:00:28,750
combinations in 22 to show which
ones we have to retain

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00:00:28,750 --> 00:00:32,590
as frameworks for
crystallographic point groups

6
00:00:32,590 --> 00:00:38,320
and which ones exist as groups
but which involve rotational

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00:00:38,320 --> 00:00:41,080
symmetries that are not
permitted to a lattice.

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00:00:41,080 --> 00:00:46,410
So we've seen a combination of
three orthogonal twofold axes

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00:00:46,410 --> 00:00:50,500
and then projection that
would look like this.

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00:00:50,500 --> 00:00:54,880
And the international symbol for
that point group is just a

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00:00:54,880 --> 00:01:03,190
running list of the different
axes that are present, 222.

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00:01:03,190 --> 00:01:08,940
The next group in the sequence
would be 3 2 2, where we took

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00:01:08,940 --> 00:01:11,940
a 120 degree rotation.

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00:01:11,940 --> 00:01:15,820
We combine that with a twofold
axis perpendicular to it and

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00:01:15,820 --> 00:01:20,120
the new twofold axis comes out
and reminds you again of

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00:01:20,120 --> 00:01:24,520
things that are quite clear but
which are easy to forget--

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00:01:24,520 --> 00:01:28,550
that this angle here is
1/2 of 2 pi over 3.

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00:01:28,550 --> 00:01:29,680
Don't forget that 1/2.

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So the neighboring twofold axis
is 60 degrees away and

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then if we allow these axes to
operate on each other, the net

21
00:01:43,750 --> 00:01:47,680
symmetry consistent set of
axes looks like this.

22
00:01:47,680 --> 00:01:51,630
But let us look at a solid
that has this symmetry.

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00:01:51,630 --> 00:01:55,350
And such a solid would
be a trigonal--

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00:01:55,350 --> 00:01:56,600
triangular prism.

25
00:01:59,260 --> 00:02:03,080
And again we can use the corners
of this polyhedron as

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00:02:03,080 --> 00:02:06,440
the reference locations
of our motifs.

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00:02:06,440 --> 00:02:13,170
So let us put a first motif
here, number 1.

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Let's rotate it by 120 degrees
to get a second motif here.

29
00:02:20,500 --> 00:02:25,770
And then let us rotate that
one down by a twofold axis

30
00:02:25,770 --> 00:02:27,950
coming out of one
of the edges.

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That will give us number 3 that
is down here or of the

32
00:02:31,930 --> 00:02:33,520
same chirality.

33
00:02:33,520 --> 00:02:37,630
And how do we get from 1 to
3 directly in one shot?

34
00:02:37,630 --> 00:02:41,240
And the answer is about a
twofold axis that comes out of

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00:02:41,240 --> 00:02:42,490
the face of the prism.

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00:02:45,490 --> 00:02:49,010
So again the prism that we've
used as our reference is a

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00:02:49,010 --> 00:02:52,800
prism that looks like this.

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We've got one twofold axis
coming out of the face.

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00:02:55,420 --> 00:03:00,960
The next twofold axis is the
one that is equivalent to a

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00:03:00,960 --> 00:03:03,320
threefold rotation followed
by a twofold rotation.

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00:03:06,460 --> 00:03:10,500
Now here we hit a situation in
deciding on the name for the

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combination that is analogous
to what we found in

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00:03:15,040 --> 00:03:18,550
two-dimensional plane
groups for 3mm.

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00:03:18,550 --> 00:03:22,170
We saw that only one mirror
plane was distinct.

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We look at this arrangement
of axes--

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00:03:25,440 --> 00:03:30,120
this was axis a, alpha,
this was axis b, beta.

47
00:03:30,120 --> 00:03:35,880
But if I repeat one of these
twofold axes by 120 degree

48
00:03:35,880 --> 00:03:40,130
rotations, this one is the
opposite end of this one, this

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00:03:40,130 --> 00:03:43,000
one is the opposite end of this
one, and this one is the

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opposite end of this one.

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00:03:44,280 --> 00:03:47,820
So there are only three kinds
of-- there are three, twofold

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00:03:47,820 --> 00:03:53,080
axes and they are related by
the 120 degree rotation.

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00:03:53,080 --> 00:03:56,680
Another way of saying that is
that if we look at a trigonal

54
00:03:56,680 --> 00:04:01,330
prism, each of the twofold axes
comes out of an edge and

55
00:04:01,330 --> 00:04:02,490
out of the opposite face.

56
00:04:02,490 --> 00:04:07,560
So there they all do the same
thing in this regular prism--

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twofold axes extend between
corners of the opposite face.

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So there are only--

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there's only one kind of twofold
axis present in terms

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of being symmetry independent.

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So just as we called 3mm, 3m,
and we call this one 3 2

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because all the twofold axes
are symmetry equivalent.

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00:04:30,030 --> 00:04:33,650
The next one that is
crystallographic would be a

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combination of a 90 degree
rotation with a pair of

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00:04:40,750 --> 00:04:47,710
twofold axes that are normal to
it and separated by 1/2 of

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pi over 2, the [? throw ?]
of a fourfold axis.

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00:04:55,490 --> 00:04:58,440
And the symbol for this one
would be a fourfold and now

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00:04:58,440 --> 00:05:01,380
there are two different
kinds of twofold axes.

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00:05:01,380 --> 00:05:07,280
And if we look at a regular
square prism, we can again

70
00:05:07,280 --> 00:05:10,070
show that what we've been
demonstrating for these other

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00:05:10,070 --> 00:05:11,750
prisms is true.

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One twofold axis would come
out of the face, the other

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00:05:15,490 --> 00:05:18,670
twofold axis would come out of
the midpoint of an edge, and

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the fourfold axis would
come up here.

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00:05:21,780 --> 00:05:28,030
And a 90 degree rotation, from
here to here, followed by a

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00:05:28,030 --> 00:05:32,320
twofold rotation about this
axis, gives us as a net

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00:05:32,320 --> 00:05:37,370
effect a 1, 2, 3.

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00:05:37,370 --> 00:05:41,170
And that rotation from 1 to 3
about this twofold axis gives

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us the combined mappings.

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Notice that the order in which
we do them is unimportant.

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For example we could do the
same thing but do two 180

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degree rotations about
the twofold axes.

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00:06:08,260 --> 00:06:11,280
Let's say we start by doing
a rotation about

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this twofold axis.

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From here down to here.

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And then we do a twofold
axis about the--

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twofold axis that comes out the
face and that would take

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number 2, and rotate
it up to number 3.

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And the way we get from 1 to 3
directly is by a rotation of C

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00:06:32,830 --> 00:06:36,570
pi over 2 about the square
face of the prism.

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00:06:36,570 --> 00:06:38,760
So do them in any
order you like--

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two 180 degree rotations, or a
90 degree rotation, one of the

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two full rotations with 90
is the other twofold--

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the other type of
twofold axis.

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The 90 degree rotation plus the
second type of the twofold

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axis is the same as the first.

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So the result can be permuted
and turns out to be the same

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combination.

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The next one that we would hit
if we proceed systematically

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is non-crystallographic.

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And this would be
a fivefold axis.

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And I'm foolhardy to even start
trying to sketch this in

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three dimensions.

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Nevertheless, nothing ventured,
nothing gained.

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Start with a twofold axis out of
one of the edges and rotate

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from 1 down to 2.

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Follow that by a twofold
rotation about the twofold

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axis that comes out
of the face.

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And that gives us one number 3
up here, and lo and behold,

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the way you get from 1 to 3
directly is by a rotation

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through 1/5 of 2 pi.

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So this would be the
non-crystallographic point

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group, 5- well it's not
going to be 5 2 2.

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And that has a whole bunch
of twofold axes separated

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by 1/10 of 2 pi.

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And just as in 3 2, twofold axes
here all come out of the

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face and out of the
opposite edge.

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So this would be called 522.

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A nice symmetry but not
crystallographic, so we can

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promptly forget about.

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So what comes out of this is a
family of groups that are all

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of the form n 2 2.

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The crystallographic ones are
222, 32, 422 which we've

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looked at in detail, and one
that I won't draw because

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there's so much symmetry
it get's messy.

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This is 622.

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There is a Schoenflies
notation.

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You remember the language that
we encountered for our two

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dimensional point groups.

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We used m for mirror in the
international notation.

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We used C standing for cyclic
group, subscript s standing

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for Spiegel in the Schoenflies
notation.

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The Schoenflies notation for all
of this family of symmetry

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is Dn, and the D stands
for dihedral.

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And the reason for that name is
that the difference between

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all of these groups, besides the
n-fold axis, is this angle

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between adjacent twofolds and
this is a dihedral angle.

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I have a set of planes passing
through a common axis; the

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angle between those planes is
termed a dihedral angle.

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So this is called D for dihedral
and then a subscript

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that gives the rank
of the axis.

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So Dn generically.

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This is D2, this is D3, this
is D4, and this is D6.

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Comments or debate?

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00:10:21,270 --> 00:10:22,038
Yes, sir.

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00:10:22,038 --> 00:10:23,288
AUDIENCE: [INAUDIBLE]

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00:10:34,938 --> 00:10:37,270
PROFESSOR: [INAUDIBLE]

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00:10:37,270 --> 00:10:38,420
--out of this face.

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So I got from here down to the
diametrically opposed axis.

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And I rotate it about the
adjacent one which comes out

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of an edge and--

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00:10:46,790 --> 00:10:49,120
what did I do here.

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Here I did A pi over 2 from 1
to 2, and then I did B pi,

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where this is B pi, and they
turned out to be C pi, which

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00:11:02,400 --> 00:11:04,090
is this one here.

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00:11:06,650 --> 00:11:10,660
So going from here to here down
to number 3 is the same

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00:11:10,660 --> 00:11:14,010
as going from 1 to 3 in one
shot about a twofold axis

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00:11:14,010 --> 00:11:15,260
normal to the face.

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And actually I'm courageous
to try to do this in three

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00:11:21,180 --> 00:11:21,840
dimensions.

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We could do it in projection
and then things are used--

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this for a point that's up,
and use this for a point

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00:11:29,990 --> 00:11:31,600
that's down.

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And then what we've done is to
go from 1 that's up, to 2

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00:11:37,830 --> 00:11:48,200
that's up, and then we rotate
it about this twofold axis.

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00:11:48,200 --> 00:11:51,710
That was 3, that's down.

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So when we get into complicated
symmetries where

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00:11:54,990 --> 00:11:58,220
you just can't do a proper
job drawing them in an

169
00:11:58,220 --> 00:12:01,190
orthographic drawing, we'll do
them in projection and use a

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00:12:01,190 --> 00:12:04,140
solid dot for something that's
up and an open circle for

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00:12:04,140 --> 00:12:05,390
something that's down.

172
00:12:11,760 --> 00:12:15,410
Is there any other way we
can combine things?

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00:12:15,410 --> 00:12:19,820
Well what you would have to do
is use these three relations,

174
00:12:19,820 --> 00:12:22,470
and plug and chug your way
through all of the

175
00:12:22,470 --> 00:12:26,080
combinations which were
enumerated in the handout.

176
00:12:26,080 --> 00:12:29,020
And I'll save ourselves a lot
of work by saying that there

177
00:12:29,020 --> 00:12:32,410
are only two more
combinations.

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00:12:32,410 --> 00:12:37,020
And these are combinations of
axes at angles that have

179
00:12:37,020 --> 00:12:40,680
relevance to directions
in a cube.

180
00:12:40,680 --> 00:12:44,490
One of them is a twofold axis
with a threefold axis with a

181
00:12:44,490 --> 00:12:46,760
threefold rotation.

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00:12:46,760 --> 00:12:47,830
Again remember these are not

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00:12:47,830 --> 00:12:49,450
equations in symmetry elements.

184
00:12:49,450 --> 00:12:56,180
This is really A pi, combined
with B 2 pi over 3, combined

185
00:12:56,180 --> 00:12:59,810
with C 2 pi over 3.

186
00:12:59,810 --> 00:13:03,320
And the angles that fall out of
this are all crazy things

187
00:13:03,320 --> 00:13:08,480
like 109 point something degrees
and they make no sense

188
00:13:08,480 --> 00:13:09,130
whatsoever.

189
00:13:09,130 --> 00:13:13,300
They're not nice things like
some multiples of 2 pi, 90

190
00:13:13,300 --> 00:13:15,030
degrees or 120 degrees.

191
00:13:15,030 --> 00:13:21,030
And they make no sense at all
until you refer them to

192
00:13:21,030 --> 00:13:22,530
directions in a cube.

193
00:13:26,080 --> 00:13:32,360
And this one, 2 3 3, consists
of a twofold axis coming out

194
00:13:32,360 --> 00:13:39,090
of the face of the cube, a 120
degree rotation, so this is B

195
00:13:39,090 --> 00:13:43,000
2 pi over 3, this is A pi.

196
00:13:43,000 --> 00:13:49,270
And the other one, C2 pi over 3,
corresponds to a threefold

197
00:13:49,270 --> 00:13:51,260
axis coming out of another
body diagonal.

198
00:14:04,030 --> 00:14:06,000
I don't know if I want to be
gutsy enough to try to

199
00:14:06,000 --> 00:14:08,140
illustrate that that
really works.

200
00:14:08,140 --> 00:14:12,540
But one thing that we should do
is to let these axes go to

201
00:14:12,540 --> 00:14:16,330
work on one another, and
see what comes out.

202
00:14:16,330 --> 00:14:19,650
First thing we can say is that
this threefold axis--

203
00:14:19,650 --> 00:14:22,270
if we extend it, it comes
out of the bottom

204
00:14:22,270 --> 00:14:24,360
diagonal of the cube.

205
00:14:24,360 --> 00:14:27,760
This one, if we extend it, will
come out of this diagonal

206
00:14:27,760 --> 00:14:28,550
of the cube.

207
00:14:28,550 --> 00:14:31,300
So there's a threefold here,
and a threefold here.

208
00:14:31,300 --> 00:14:35,700
The twofold axis gives us a
threefold axis that will come

209
00:14:35,700 --> 00:14:38,290
down this way.

210
00:14:38,290 --> 00:14:41,400
So there's a threefold axis
here and a threefold axis

211
00:14:41,400 --> 00:14:42,720
coming out here.

212
00:14:42,720 --> 00:14:46,000
And then the twofold axis will
rotate this threefold axis

213
00:14:46,000 --> 00:14:49,970
over to the remaining
pair of corners.

214
00:14:49,970 --> 00:14:54,940
So we have created this by
looking at a twofold rotation,

215
00:14:54,940 --> 00:14:57,490
combined with the threefold
axis, combined with the

216
00:14:57,490 --> 00:14:59,790
rotation of another
threefold axis.

217
00:14:59,790 --> 00:15:03,990
But in point of fact, if you let
the twofold axis operate

218
00:15:03,990 --> 00:15:11,620
on the threefold axis coming out
of one body diagonal, you

219
00:15:11,620 --> 00:15:13,980
get threefold axes
automatically

220
00:15:13,980 --> 00:15:16,340
out of all body diagonals.

221
00:15:16,340 --> 00:15:20,760
So this is given the
international symbol 2 3,

222
00:15:20,760 --> 00:15:23,900
because if you start with one
twofold axis out of a face

223
00:15:23,900 --> 00:15:27,420
normal and one threefold
axis out of--

224
00:15:27,420 --> 00:15:31,830
along a body diagonal, the
twofold axis, acting on that

225
00:15:31,830 --> 00:15:35,150
threefold axis gives you one
along every body diagonal.

226
00:15:35,150 --> 00:15:37,895
And the threefold axis--

227
00:15:37,895 --> 00:15:43,590
let me point out that a cube
standing up on its body

228
00:15:43,590 --> 00:15:48,150
diagonal, with a threefold axis
coming out here and these

229
00:15:48,150 --> 00:15:51,030
faces sloping down into
the blackboard.

230
00:15:51,030 --> 00:15:54,780
If I have a twofold axis coming
out of one face, the

231
00:15:54,780 --> 00:15:58,620
threefold axis puts a twofold
axis on this face and rotates

232
00:15:58,620 --> 00:16:02,510
again 120 degrees and puts a
twofold axis on this face.

233
00:16:02,510 --> 00:16:06,490
So the threefold axis relates
all twofold axes coming out

234
00:16:06,490 --> 00:16:09,790
normal to the cubed face.

235
00:16:09,790 --> 00:16:14,630
And they were really just two
independent axes in this

236
00:16:14,630 --> 00:16:16,680
combination.

237
00:16:16,680 --> 00:16:20,160
So 2 3 is the international
symbol-- one kind of twofold

238
00:16:20,160 --> 00:16:23,730
axis, one kind of
threefold axis--

239
00:16:23,730 --> 00:16:28,200
inclined at these crazy angles
that are in a cube.

240
00:16:28,200 --> 00:16:34,770
The Schoenflies notation for
this is T, and that stands for

241
00:16:34,770 --> 00:16:36,020
tetrahedron.

242
00:16:40,510 --> 00:16:47,640
And let me try to convince
you that if I look at a

243
00:16:47,640 --> 00:16:52,520
tetrahedron, that that is the
arrangement of pure rotation

244
00:16:52,520 --> 00:16:56,070
axes in a tetrahedron.

245
00:16:56,070 --> 00:17:00,110
And the way to show a
tetrahedron is to

246
00:17:00,110 --> 00:17:01,650
inscribe it in a cube.

247
00:17:13,119 --> 00:17:18,670
So if I connect these two faces
together and these two

248
00:17:18,670 --> 00:17:34,970
faces together, that will define
for me a solid that has

249
00:17:34,970 --> 00:17:37,910
four triangular faces.

250
00:17:37,910 --> 00:17:40,705
Easier to recognize it when
we put it up on one face.

251
00:17:45,860 --> 00:17:47,110
So this is a tetrahedron.

252
00:17:50,200 --> 00:17:55,540
Schoenflies symbol is T. And I
love to get to this part of

253
00:17:55,540 --> 00:18:00,280
the semester and be at this
point at the end of the hour,

254
00:18:00,280 --> 00:18:07,000
because if I draw a
stereographic projection of

255
00:18:07,000 --> 00:18:15,730
the twofold axes, in this
symmetry, I can take a couple

256
00:18:15,730 --> 00:18:19,700
of more twofold axes and add it
to this combination which

257
00:18:19,700 --> 00:18:21,660
destroys the group.

258
00:18:21,660 --> 00:18:25,670
But lets me wish everybody a
happy Halloween and exit to a

259
00:18:25,670 --> 00:18:28,020
stunned silence at the
end of the hour.

260
00:18:32,810 --> 00:18:35,340
So this is a nice point
group for October.

261
00:18:39,520 --> 00:18:43,780
There is one more and that is
the highest symmetry of all.

262
00:18:43,780 --> 00:18:47,420
And this is a combination
of a rotation--

263
00:18:47,420 --> 00:18:53,200
A pi over 2, a 90 degree
rotation, with a rotation B 2

264
00:18:53,200 --> 00:18:57,280
pi over 3, rotation through one
third of the circle, and

265
00:18:57,280 --> 00:18:59,630
the rotation C pi.

266
00:18:59,630 --> 00:19:04,650
And the directions between the
axes that come out there don't

267
00:19:04,650 --> 00:19:07,050
come out some multiples
of 2 pi.

268
00:19:07,050 --> 00:19:10,880
Again they are crazy angles
that make no sense at all

269
00:19:10,880 --> 00:19:13,700
unless you refer them
to directions

270
00:19:13,700 --> 00:19:16,160
that occur in a cube.

271
00:19:16,160 --> 00:19:19,310
The fourfold axis is in the
direction that corresponds to

272
00:19:19,310 --> 00:19:21,180
the normal to a face.

273
00:19:21,180 --> 00:19:24,910
The twofold axis corresponds
to a direction out

274
00:19:24,910 --> 00:19:26,300
of one of the edges.

275
00:19:26,300 --> 00:19:29,600
And the threefold axis
corresponds to a direction

276
00:19:29,600 --> 00:19:31,940
that is a body diagonal.

277
00:19:31,940 --> 00:19:33,240
So again this is a mess.

278
00:19:33,240 --> 00:19:37,010
If we let those axes work on one
another however, we will

279
00:19:37,010 --> 00:19:41,420
produce, more readily
appreciated in a stereographic

280
00:19:41,420 --> 00:19:47,310
projection, fourfold axes along
the directions that

281
00:19:47,310 --> 00:19:48,710
correspond to face normal.

282
00:19:48,710 --> 00:19:52,620
So the cube, threefold axes
coming out of the body

283
00:19:52,620 --> 00:20:00,060
diagonals and twofold axes in
between all of the fourfold

284
00:20:00,060 --> 00:20:06,690
axes in directions that
correspond to the lines from

285
00:20:06,690 --> 00:20:10,130
the center of the cube out
through the edges.

286
00:20:10,130 --> 00:20:14,650
So this is a group which
would be called 4 3 2.

287
00:20:14,650 --> 00:20:18,220
There's one kind of fourfold
axis related to all the others

288
00:20:18,220 --> 00:20:20,570
by the other rotation axes
that are present.

289
00:20:20,570 --> 00:20:23,840
One kind of threefold axis that
is related to all of the

290
00:20:23,840 --> 00:20:26,970
other threefold axes along the
body diagonal by other

291
00:20:26,970 --> 00:20:28,720
rotation axes that
are present.

292
00:20:28,720 --> 00:20:32,410
And twofold axes, one kind, all
coming out of the edges.

293
00:20:32,410 --> 00:20:35,070
So this is the group that, in
international tables, is

294
00:20:35,070 --> 00:20:36,760
called 4 3 2.

295
00:20:36,760 --> 00:20:41,650
And the Schoenflies notation,
this is called O. And that is

296
00:20:41,650 --> 00:20:44,590
the general reaction when one
sees this lovely combination.

297
00:20:44,590 --> 00:20:47,980
You go ooh and O is
what it's called.

298
00:20:47,980 --> 00:20:51,090
But the O doesn't stand for
a gasp, it stands for an

299
00:20:51,090 --> 00:20:52,340
octahedral.

300
00:20:56,080 --> 00:20:59,460
This is the symmetry
of an octahedron--

301
00:20:59,460 --> 00:21:00,910
rotational symmetry
of an octahedron.

302
00:21:04,680 --> 00:21:05,290
So that's it.

303
00:21:05,290 --> 00:21:09,280
That is the bestiary of ways
in which you can combine

304
00:21:09,280 --> 00:21:13,650
crystallographic rotation axes
in space about a fixed point

305
00:21:13,650 --> 00:21:15,220
of intersection.

306
00:21:15,220 --> 00:21:17,360
And there are eleven of them.

307
00:21:17,360 --> 00:21:18,760
There are the axes
by themselves--

308
00:21:18,760 --> 00:21:21,250
1, 2, 3, 4, and 6.

309
00:21:21,250 --> 00:21:29,540
There are the dihedral groups,
222, 32, 422, and 622..

310
00:21:29,540 --> 00:21:35,690
And then the two cubic groups,
T and O. In the international

311
00:21:35,690 --> 00:21:37,670
notation, I'm mixing
metaphors.

312
00:21:37,670 --> 00:21:45,970
These are 23 and 432.

313
00:21:45,970 --> 00:21:53,250
I call to your attention the
insidious similarity of the

314
00:21:53,250 --> 00:21:59,170
two combinations of
axes, 32 and 23.

315
00:21:59,170 --> 00:22:04,740
When the 3 comes first, this
is a group of the form n22.

316
00:22:04,740 --> 00:22:09,190
When the 2 comes first, that
is the tetrahedral group.

317
00:22:14,910 --> 00:22:19,580
So if you count them all up,
there are 4, 2 is 6, and 5.

318
00:22:19,580 --> 00:22:21,945
There are eleven axial
combinations.

319
00:22:33,830 --> 00:22:38,450
Quite a few more than the
situation in two dimensions

320
00:22:38,450 --> 00:22:42,440
where we just had single
rotation axes 1, 2, 3, 4, 6.

321
00:22:42,440 --> 00:22:46,480
Now we have those as in two
dimensions but the dihedral

322
00:22:46,480 --> 00:22:51,630
groups and the two cubic
arrangements of axes as well.

323
00:22:51,630 --> 00:22:55,410
So we don't have to stretch our
vocabulary too much more

324
00:22:55,410 --> 00:22:57,260
to be all inclusive here.

325
00:23:09,460 --> 00:23:11,110
OK, comments?

326
00:23:11,110 --> 00:23:12,460
Takes your breath away,
doesn't it?

327
00:23:24,920 --> 00:23:31,160
All right, let me indicate
the next step in outline.

328
00:23:40,770 --> 00:23:45,470
What we will next do is
introduce our remaining two

329
00:23:45,470 --> 00:23:48,370
symmetry operations
into the picture.

330
00:23:48,370 --> 00:23:50,250
We have the eleven axial
combinations.

331
00:23:56,150 --> 00:23:59,920
As I said, we can regard these
as a framework that we can

332
00:23:59,920 --> 00:24:05,210
decorate with mirror planes and
or the inversion center,

333
00:24:05,210 --> 00:24:09,110
which has not appeared until
this point because inversion

334
00:24:09,110 --> 00:24:11,620
is inherently a three
dimensional transformation.

335
00:24:15,120 --> 00:24:18,300
So what we're going to do
is to take these axial

336
00:24:18,300 --> 00:24:21,450
combinations and add
another symmetry

337
00:24:21,450 --> 00:24:23,240
operation to the group.

338
00:24:23,240 --> 00:24:25,750
And this as--

339
00:24:25,750 --> 00:24:28,300
we used the term earlier,
this is an extender.

340
00:24:28,300 --> 00:24:31,380
We have something that
constitutes a group by itself

341
00:24:31,380 --> 00:24:34,650
then we muck things up by adding
another operation.

342
00:24:34,650 --> 00:24:37,580
Remember in all of this we are
combining operations, not

343
00:24:37,580 --> 00:24:38,880
symmetry elements.

344
00:24:38,880 --> 00:24:41,840
So we'll take an axial
combination and add the

345
00:24:41,840 --> 00:24:45,930
reflection sigma, a reflection
operation sigma.

346
00:24:45,930 --> 00:24:52,620
Or take a rotation and combine
it with the operation of

347
00:24:52,620 --> 00:24:53,930
inversion as an extender.

348
00:24:57,500 --> 00:25:02,710
So let's itemize the sorts of
extenders we should consider.

349
00:25:02,710 --> 00:25:08,090
And the ground rules are that
the extender should leave the

350
00:25:08,090 --> 00:25:12,620
arrangement of rotation
axes invariant.

351
00:25:12,620 --> 00:25:17,360
Because if it doesn't, we are
going to create a rotation

352
00:25:17,360 --> 00:25:23,110
operation that does not conform
to the constraints

353
00:25:23,110 --> 00:25:26,270
that we used in Euler's
construction.

354
00:25:26,270 --> 00:25:33,200
So for example, if we take
222, which contains the

355
00:25:33,200 --> 00:25:43,690
operations A pi, B pi,
C pi and identity,

356
00:25:43,690 --> 00:25:46,810
that's the group 222.

357
00:25:46,810 --> 00:25:52,600
If we would add to the
arrangement 222 a mirror plane

358
00:25:52,600 --> 00:25:56,850
that snaked through some
arbitrary fashion like this,

359
00:25:56,850 --> 00:26:00,240
that mirror plane is going to
reproduce the twofold axis

360
00:26:00,240 --> 00:26:01,970
over to here.

361
00:26:01,970 --> 00:26:07,120
And that is either going to not
constitute a group because

362
00:26:07,120 --> 00:26:10,670
this twofold, this twofold, and
this twofold don't conform

363
00:26:10,670 --> 00:26:12,480
to Euler's construction.

364
00:26:12,480 --> 00:26:16,150
Or alternatively, if we put it
in carefully at 45 degrees

365
00:26:16,150 --> 00:26:19,330
with this twofold axis, we're
going to get twofold axes that

366
00:26:19,330 --> 00:26:22,390
are 45 degrees apart and that's
going to change this

367
00:26:22,390 --> 00:26:24,460
into a fourfold axis.

368
00:26:24,460 --> 00:26:29,500
So if the addition of a mirror
plane does not leave the

369
00:26:29,500 --> 00:26:31,870
arrangement of rotation axes--

370
00:26:31,870 --> 00:26:33,020
rotation operations--

371
00:26:33,020 --> 00:26:36,050
invariant, we're either going
to get something that's

372
00:26:36,050 --> 00:26:39,160
impossible and does not
constitute a group.

373
00:26:39,160 --> 00:26:41,660
Or else we're going to get
a combination of rotation

374
00:26:41,660 --> 00:26:44,840
operations of higher symmetry
which we've already found

375
00:26:44,840 --> 00:26:48,270
because we went through that
process of combination using

376
00:26:48,270 --> 00:26:51,730
Euler's construction in
an exhaustive fashion.

377
00:26:51,730 --> 00:26:58,560
So the rule then is that if we
add the reflection operation

378
00:26:58,560 --> 00:27:11,410
sigma, then the arrangement of
rotation operations must be

379
00:27:11,410 --> 00:27:12,660
left invariant.

380
00:27:40,800 --> 00:27:44,320
OK let's look at the
single axes.

381
00:27:44,320 --> 00:27:50,380
If there's an n-fold axis, the
ways we can add a reflection

382
00:27:50,380 --> 00:27:57,080
operation to that axis is to
pass the reflection operation

383
00:27:57,080 --> 00:27:59,570
through the axis.

384
00:27:59,570 --> 00:28:03,750
And this is going to look very
much like the two dimensional

385
00:28:03,750 --> 00:28:09,630
point groups of the form nmm
except that rather than having

386
00:28:09,630 --> 00:28:12,940
a mirror line, imagine the
whole works as extending

387
00:28:12,940 --> 00:28:16,430
upwards along the rotation
axis and space.

388
00:28:16,430 --> 00:28:19,950
So this extender is called
a vertical mirror plane.

389
00:28:24,080 --> 00:28:30,350
The other way we could add a
mirror plane to an n-fold

390
00:28:30,350 --> 00:28:37,010
rotation axis is to put the
mirror plane in an orientation

391
00:28:37,010 --> 00:28:40,090
that's perpendicular to
the rotation axis.

392
00:28:40,090 --> 00:28:43,450
That didn't exist in two
dimensions because that plane

393
00:28:43,450 --> 00:28:45,240
that's perpendicular
to the axis is the

394
00:28:45,240 --> 00:28:46,830
plane of our paper.

395
00:28:46,830 --> 00:28:50,060
And unless we wanted to have a
two-sided group that was on

396
00:28:50,060 --> 00:28:53,370
both the top of the paper and
the bottom of the paper, and

397
00:28:53,370 --> 00:28:55,890
we make up the rules since
it's our ball game.

398
00:28:55,890 --> 00:28:58,590
And that could be a group and
these would be the two-sided

399
00:28:58,590 --> 00:29:01,880
plane groups, a plane
point groups, but we

400
00:29:01,880 --> 00:29:03,000
didn't do that here.

401
00:29:03,000 --> 00:29:07,215
Adding the mirror plane, the
reflection operation sigma, in

402
00:29:07,215 --> 00:29:11,450
a fashion that is normal to the
rotation operation, A 2 pi

403
00:29:11,450 --> 00:29:14,450
over n, is another distinct
combination.

404
00:29:14,450 --> 00:29:16,865
And this is called, very
descriptively a horizontal

405
00:29:16,865 --> 00:29:18,115
mirror plane.

406
00:29:25,640 --> 00:29:31,710
That looks like about all you
can do except for the cases

407
00:29:31,710 --> 00:29:41,900
where we have more than one kind
of rotation axis present.

408
00:29:41,900 --> 00:29:46,600
So let me use 422
as an example.

409
00:29:46,600 --> 00:29:50,000
We could add a horizontal mirror
plane perpendicular to

410
00:29:50,000 --> 00:29:52,330
the principal axis of symmetry,
and that would be

411
00:29:52,330 --> 00:29:53,625
the horizontal sigma.

412
00:30:01,760 --> 00:30:05,430
We could add a vertical
mirror plane.

413
00:30:05,430 --> 00:30:08,480
And now there are two
ways we can do it.

414
00:30:08,480 --> 00:30:12,550
We could put the mirror plane,
the operation sigma, through

415
00:30:12,550 --> 00:30:16,620
the fourfold axis and in a
fashion that was perpendicular

416
00:30:16,620 --> 00:30:18,110
to the twofold axis.

417
00:30:18,110 --> 00:30:21,400
And we will retain the
term of vertical

418
00:30:21,400 --> 00:30:24,370
sigma for that addition.

419
00:30:24,370 --> 00:30:26,980
But the other thing that we
could do would be to put the

420
00:30:26,980 --> 00:30:31,140
reflection operation interleaved
between the

421
00:30:31,140 --> 00:30:33,900
twofold axes.

422
00:30:33,900 --> 00:30:36,390
That's going to take this one
and flip it into this one,

423
00:30:36,390 --> 00:30:38,550
this one flip it into this
one, flip these back and

424
00:30:38,550 --> 00:30:41,300
forth, and that doesn't create
any new reflection.

425
00:30:41,300 --> 00:30:50,100
And this is referred to as a
diagonal reflection plane.

426
00:30:50,100 --> 00:30:51,280
Diagonal to what?

427
00:30:51,280 --> 00:30:56,710
Diagonally interleaved between
the twofold axes.

428
00:30:56,710 --> 00:30:59,280
And these are distinct additions
and they will lead

429
00:30:59,280 --> 00:31:00,530
to different groups.

430
00:31:04,448 --> 00:31:09,120
In as far as addition of
reflection operations is

431
00:31:09,120 --> 00:31:13,620
concerned, that's about all
we can do that's distinct.

432
00:31:13,620 --> 00:31:20,810
And notice that this is
for the groups Dn, and

433
00:31:20,810 --> 00:31:23,570
tetrahedral, and octahedral
only.

434
00:31:23,570 --> 00:31:27,250
The distinction here
is not defined for

435
00:31:27,250 --> 00:31:28,500
just a single axis.

436
00:31:30,930 --> 00:31:35,010
So there are three possible
extenders here-- a vertical

437
00:31:35,010 --> 00:31:37,800
mirror plane, a horizontal
mirror plane, a diagonal

438
00:31:37,800 --> 00:31:41,920
mirror plane, added to each of
the eleven arrangements of

439
00:31:41,920 --> 00:31:44,640
rotation axes.

440
00:31:44,640 --> 00:31:46,860
And then the final extender
that we could

441
00:31:46,860 --> 00:31:49,025
add is to add inversion.

442
00:31:55,510 --> 00:31:59,390
And the symbol for the inversion
operation is 1 bar.

443
00:31:59,390 --> 00:32:04,200
And obviously if one point in
space is going to be left

444
00:32:04,200 --> 00:32:14,800
invariant, you either add this
on a single axis, and that is

445
00:32:14,800 --> 00:32:22,320
what you'd have to do for the
groups Cn, or at the point of

446
00:32:22,320 --> 00:32:23,570
intersection.

447
00:32:33,920 --> 00:32:45,620
And that would be the case if
more than one axis, and that's

448
00:32:45,620 --> 00:32:49,770
the groups of the form n22, T,

449
00:32:49,770 --> 00:32:54,912
and O. That's it.

450
00:32:54,912 --> 00:32:57,580
That's the job.

451
00:32:57,580 --> 00:33:02,070
So we should consider each of
these possible additions of an

452
00:33:02,070 --> 00:33:04,210
extender systematically.

453
00:33:04,210 --> 00:33:08,460
I don't propose to do every
single one independently.

454
00:33:08,460 --> 00:33:10,790
If we do a couple, you'll
get the general idea.

455
00:33:10,790 --> 00:33:13,160
And I think because I have an
honest face, and you've come

456
00:33:13,160 --> 00:33:16,010
to trust me, I can just describe
the remaining results

457
00:33:16,010 --> 00:33:18,900
to you and we won't grind
through every single one.

458
00:33:27,300 --> 00:33:31,950
Now the enormity of what I've
proposed becomes apparent when

459
00:33:31,950 --> 00:33:36,640
I say that we now are going to
have need of a number of

460
00:33:36,640 --> 00:33:37,870
different--

461
00:33:37,870 --> 00:33:41,630
what I call combination
theorems, that let us complete

462
00:33:41,630 --> 00:33:43,850
the group multiplication
table.

463
00:33:43,850 --> 00:33:49,290
And deduce, as a consequence,
which symmetry operations must

464
00:33:49,290 --> 00:33:52,310
come into being because
of these additions.

465
00:33:52,310 --> 00:33:55,790
So we'll want to know what
happens when you add a

466
00:33:55,790 --> 00:34:01,370
vertical sigma to a rotation
operation, A 2 pi over n.

467
00:34:01,370 --> 00:34:03,330
We'll want to know what
happens when you add a

468
00:34:03,330 --> 00:34:08,650
horizontal sigma to a rotation
operation, A 2 pi over n.

469
00:34:08,650 --> 00:34:11,330
And we're going to want to know
what happens when you add

470
00:34:11,330 --> 00:34:16,719
a diagonal mirror plane, this
really is a special case of

471
00:34:16,719 --> 00:34:19,340
the vertical mirror plane.

472
00:34:19,340 --> 00:34:21,280
And we'll want to know what
happens when you add an

473
00:34:21,280 --> 00:34:24,089
inversion center to
a rotation axis.

474
00:34:27,670 --> 00:34:31,739
So let me do a few of these
and then next time we can

475
00:34:31,739 --> 00:34:34,969
start off and start driving
the three dimensional

476
00:34:34,969 --> 00:34:36,219
symmetries.

477
00:34:52,420 --> 00:34:58,920
So let's just repeat the ones
that we've already done.

478
00:34:58,920 --> 00:35:04,230
We said that if we have a
rotation operation A alpha,

479
00:35:04,230 --> 00:35:10,720
and we put a reflection plane
through it, that A alpha

480
00:35:10,720 --> 00:35:13,710
followed by a reflection plane
passing through it-- let me

481
00:35:13,710 --> 00:35:16,430
call this sigma V-- because
this is the so-called

482
00:35:16,430 --> 00:35:18,440
vertical, up orientation.

483
00:35:18,440 --> 00:35:20,580
We've already seen that
in two dimensions.

484
00:35:20,580 --> 00:35:24,650
This is a vertical mirror plane,
sigma prime, that is

485
00:35:24,650 --> 00:35:29,330
going to be alpha over 2
away from the first.

486
00:35:29,330 --> 00:35:33,200
So that is something that we've
already seen in it's

487
00:35:33,200 --> 00:35:36,290
entirety in the two dimensional
point groups.

488
00:35:36,290 --> 00:35:46,170
There was m sigma, there was
2mm, and that was C2V, 3m,

489
00:35:46,170 --> 00:35:57,020
that's C3V, 4mm, and that was
C6V, and 6mm, and that's C4V,

490
00:35:57,020 --> 00:35:59,510
and that's C6V.

491
00:35:59,510 --> 00:36:02,970
So that is the result that we
obtained for two dimensions.

492
00:36:02,970 --> 00:36:06,130
And you can see now the reason
for distinguishing the mirror

493
00:36:06,130 --> 00:36:09,810
plane by saying it is a vertical
mirror plane because

494
00:36:09,810 --> 00:36:12,530
this is in the three
dimensional sense.

495
00:36:12,530 --> 00:36:16,130
It's vertical parallel to and
passing through the rotation

496
00:36:16,130 --> 00:36:19,035
axis-- no longer a rotation
point but a rotation axis.

497
00:36:21,680 --> 00:36:22,930
So we've got those theorems.

498
00:36:27,140 --> 00:36:36,780
What happens if we take a
rotation operation, A pi, and

499
00:36:36,780 --> 00:36:42,590
put a horizontal reflection
operation through it

500
00:36:42,590 --> 00:36:44,230
normal to that axis?

501
00:36:44,230 --> 00:36:47,635
So I'll call this sigma h,
a horizontal operation.

502
00:36:52,640 --> 00:36:57,720
OK what we have to do is draw
it out once and for all.

503
00:36:57,720 --> 00:37:00,500
Here's the first one, let's
say it's right-handed.

504
00:37:00,500 --> 00:37:04,540
We'll rotate by A pi to get
a second one which stays

505
00:37:04,540 --> 00:37:05,620
right-handed.

506
00:37:05,620 --> 00:37:10,210
And then we'll reflect it down
in the horizontal mirror plane

507
00:37:10,210 --> 00:37:14,260
to get a third one which
is left-handed.

508
00:37:14,260 --> 00:37:19,550
And now the question is, how
is number one related to

509
00:37:19,550 --> 00:37:22,640
number three?

510
00:37:22,640 --> 00:37:23,970
Anybody want to hazard
a guess?

511
00:37:23,970 --> 00:37:25,730
We've got to go from a
right-handed one to a

512
00:37:25,730 --> 00:37:27,590
left-handed one.

513
00:37:27,590 --> 00:37:29,400
But these two guys are oriented

514
00:37:29,400 --> 00:37:31,480
anti-parallel to one another.

515
00:37:31,480 --> 00:37:34,830
So how do we relate the first
one to the third one?

516
00:37:38,410 --> 00:37:43,560
I heard somebody mumble softly
enough to remain anonymous.

517
00:37:43,560 --> 00:37:45,651
Inversion.

518
00:37:45,651 --> 00:37:47,560
Right.

519
00:37:47,560 --> 00:37:51,830
So as we go along making these
combinations, if we had not

520
00:37:51,830 --> 00:37:54,990
been bright enough to think of
the operation of inversion as

521
00:37:54,990 --> 00:37:58,120
a general transformation where
the sense of all three

522
00:37:58,120 --> 00:38:00,880
coordinates is changed, we
would have stumbled over

523
00:38:00,880 --> 00:38:02,470
headlong right here.

524
00:38:02,470 --> 00:38:08,280
Combine a rotation operation, A
pi, with a mirror reflection

525
00:38:08,280 --> 00:38:10,210
that is perpendicular
to the axis.

526
00:38:10,210 --> 00:38:13,680
The way you get from one to
three in one shot is by

527
00:38:13,680 --> 00:38:19,510
inversion through a point that
is at the intersection between

528
00:38:19,510 --> 00:38:22,610
the rotation axis and
the mirror plane.

529
00:38:22,610 --> 00:38:30,500
So rotation followed by rotation
in a vertical mirror

530
00:38:30,500 --> 00:38:34,640
plane that's perpendicular to
the axis is the operation of

531
00:38:34,640 --> 00:38:36,545
inversion at the point
of intersection.

532
00:38:43,260 --> 00:38:47,070
So again, if we had not been
clever enough to invent it or

533
00:38:47,070 --> 00:38:49,320
tell you about in advance,
there it is.

534
00:38:49,320 --> 00:38:52,750
When we start forming the group
multiplication table, we

535
00:38:52,750 --> 00:38:55,800
would have had to have defined
this operation to describe

536
00:38:55,800 --> 00:38:57,514
this relation.

537
00:38:57,514 --> 00:39:00,280
AUDIENCE: Is that sigma sub h?

538
00:39:00,280 --> 00:39:01,672
PROFESSOR: Ah yeah.

539
00:39:01,672 --> 00:39:04,010
Let me write that, sigma h.

540
00:39:04,010 --> 00:39:06,795
I thought that V didn't look
like a V, so I changed it so

541
00:39:06,795 --> 00:39:08,500
it looked like a V but it
shouldn't be a V. That's a

542
00:39:08,500 --> 00:39:09,750
horizontal mirror plane.

543
00:39:21,970 --> 00:39:31,990
OK this is a new combination and
if we see what operations

544
00:39:31,990 --> 00:39:35,080
are going to be present in the
group, we've got the two

545
00:39:35,080 --> 00:39:39,930
operations of the twofold
axis, 1 and A pi.

546
00:39:39,930 --> 00:39:46,230
And what we have added is
sigma h as an extender.

547
00:39:46,230 --> 00:39:53,910
So 1 and A pi, this little box
here is the subgroup that we

548
00:39:53,910 --> 00:39:56,470
know and love as the
twofold axis.

549
00:39:56,470 --> 00:39:59,680
And then we'll write sigma h
here and now let's fill in the

550
00:39:59,680 --> 00:40:00,970
group multiplication table.

551
00:40:00,970 --> 00:40:04,840
Doing the identity operation
twice is identity.

552
00:40:04,840 --> 00:40:08,020
Doing the identity operation
followed by A pi is A pi.

553
00:40:08,020 --> 00:40:12,830
Identity followed by
sigma h is sigma h.

554
00:40:12,830 --> 00:40:17,380
Identity followed by A
pi, sigma h, lets me

555
00:40:17,380 --> 00:40:18,830
fill in those boxes.

556
00:40:18,830 --> 00:40:22,470
Do a rotation twice, that's
the identity operation.

557
00:40:22,470 --> 00:40:26,670
Do a rotation of A pi and follow
that by a reflection, A

558
00:40:26,670 --> 00:40:29,260
pi followed by reflection.

559
00:40:29,260 --> 00:40:33,580
This is the inversion
operation.

560
00:40:33,580 --> 00:40:37,630
And so I should add inversion to
my list of operations since

561
00:40:37,630 --> 00:40:38,880
it's come up.

562
00:40:41,420 --> 00:40:44,950
So 1 followed by inversion
is inversion.

563
00:40:44,950 --> 00:40:50,135
A pi followed by inversion is
the horizontal reflection.

564
00:40:53,890 --> 00:40:58,060
Horizontal reflection
followed by A pi

565
00:40:58,060 --> 00:40:59,310
is the same as inversion.

566
00:41:01,810 --> 00:41:07,530
Horizontal reflection followed
by horizontal reflection is

567
00:41:07,530 --> 00:41:08,720
the identity operation--

568
00:41:08,720 --> 00:41:10,220
brings me back to where
I started from.

569
00:41:10,220 --> 00:41:13,770
And a horizontal reflection
followed by inversion is the

570
00:41:13,770 --> 00:41:16,310
same as A pi.

571
00:41:16,310 --> 00:41:21,330
So I'll have another object
down here to complete the

572
00:41:21,330 --> 00:41:23,730
three dimensional arrangement.

573
00:41:23,730 --> 00:41:26,980
And this is the group
for operations--

574
00:41:26,980 --> 00:41:29,920
A pi, inversion, horizontal
reflection, and

575
00:41:29,920 --> 00:41:31,170
the identity operation.

576
00:41:36,340 --> 00:41:40,180
So what do we call this one?

577
00:41:40,180 --> 00:41:45,920
The operation A pi plus a
horizontal reflection

578
00:41:45,920 --> 00:41:51,410
operation gives rise to a group
that is a twofold axis

579
00:41:51,410 --> 00:41:53,510
of its operations,

580
00:41:53,510 --> 00:41:55,710
perpendicular to a mirror plane.

581
00:41:55,710 --> 00:41:59,770
And this written as a fraction
means that the 2 is

582
00:41:59,770 --> 00:42:02,620
perpendicular to the mirror
plane rather than being

583
00:42:02,620 --> 00:42:05,760
parallel and in the plane
of the mirror plane.

584
00:42:05,760 --> 00:42:07,880
So two codes for writing
symbols.

585
00:42:07,880 --> 00:42:12,090
A 2 followed on the same line
by the m means that the m is

586
00:42:12,090 --> 00:42:13,070
parallel to 2.

587
00:42:13,070 --> 00:42:15,810
We know that when we write them
as a fraction, that will

588
00:42:15,810 --> 00:42:18,740
be our way of designating that
this mirror plane is

589
00:42:18,740 --> 00:42:20,660
perpendicular to
a twofold axis.

590
00:42:27,350 --> 00:42:32,230
OK it's the witching hour,
4 o'clock exactly.

591
00:42:32,230 --> 00:42:35,440
That's when we ought to quit,
so let's stop there.