WEBVTT
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PROFESSOR: OK.
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Before we get started, I'd
like to deal with a small
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matter of some unpleasantness.
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The class is sort of like
a football game.
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When there's two minutes to go,
you shoot off a pistol.
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But when there are two meetings
to go until we have
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the quiz, we shoot off a
pistol to wake you up.
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We're scheduled to have
a quiz on October 6.
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And even though October seems
far away when you're still in
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September, that is going
to be a week
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from this coming Thursday.
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So if you have problems that
you're working on, try to get
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them to me on Thursday, or just
come slide them under my
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office door, if I'm not in.
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And I'll have them back for you
on this coming Tuesday.
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The way the material has played
out is that we're
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really at a nice, convenient
juncture between one chunk of
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interconnected material
moving on to another.
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So I think the first quiz
we'll confine to
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two-dimensional symmetry.
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And beginning in about two
minutes flat, we'll begin to
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move into--
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take the first small steps,
at any rate, into
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three-dimensional symmetries,
which will be much more
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complicated in which we
will not deal with the
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exhaustiveness that we have been
able to afford the luxury
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of in two dimensions.
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Before I begin, let me-- does
everybody remember their
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spherical trigonometry?
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Has anybody had spherical
trigonometry?
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OK.
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What I will then do is give a
short primer on some of the
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definitions and concepts in
spherical trigonometry.
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And then we shall immediately
use this to combine rotation
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axes in space.
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Pass those back.
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AUDIENCE: I had a quick
question on this.
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PROFESSOR: Oh, sure, please.
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AUDIENCE: With the limiting
possible--
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PROFESSOR: Yep.
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AUDIENCE: --reflections and
conditions, that does say they
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have to be figured out?
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PROFESSOR: Actually, that was a
good question and something
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that we are not going to use at
all in the symmetry tables.
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Somebody asked what about this
notation in the far right-hand
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edge of all of the
plane groups--
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conditions limiting possible
reflections.
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One usually doesn't put a
two-dimensional crystal in an
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x-ray beam fairly often,
although I suppose a thin film
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actually is almost a
two-dimensional crystal.
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But you've probably all heard
one way or another about the
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magic conditions relating the
Miller indices of a plane that
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will require that the intensity
diffracted from that
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set of planes is identically
zero.
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And they're linear combinations
of H, K, and L.
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And one of the rules is that if
H plus K plus L is even or
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not, the intensity
may be zero.
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These are rules for systematic
absences.
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And the corresponding
information is given for you
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here for these not terribly
realistic real
00:03:29.930 --> 00:03:31.520
two-dimensional crystals.
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So, for example, if you turn to
number seven, P2MG, it says
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conditions limiting possible
reflections
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for the general position.
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For H and K, there's
no condition.
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For H zero, H has to be even,
if the reflection is to have
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non-zero intensity.
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And for the last two, for the
general reflections, H, K, if
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there's an atom occupying the
position either zero, zero or
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the position zero, one half,
then for the general planes
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with indices H, K, you will
see intensity only if H is
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even, as well-- same as
the condition above.
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This is something that is not
generally known that everybody
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knows-- that if the crystal is
face-centered cubic, there is
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a pattern of absences.
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But there are additional
absences if the atom is in a
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special position.
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And this can very often be used
to advantage because you
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can single out certain classes
of Miller indices for which
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one atom in the structure will
not diffract or which another
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atom in the structure
will not diffract.
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And that can be of
great utility in
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unraveling a structure.
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We'll see some examples of this
in useful form when we
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deal with three-dimensional
space groups.
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And the corresponding sheets
for the three-dimensional
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symmetries will be handed
out to you-- some of
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them, not all of them.
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Let me take a little bit of time
to remind you, if you've
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forgotten them, but to inform
you of certain definitions and
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trigonometry in spherical
geometry.
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Spherical trigonometry differs
from plane geometry in that
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all of the action takes place
on the surface of a sphere.
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And it'd be nice if I had
a spherical blackboard.
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Actually there is one in the
x-ray laboratory that I can
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draw things right on that
spherical surface.
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But this is a sphere.
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We'll see directly that
the radius of the
00:05:51.935 --> 00:05:53.840
sphere is not important.
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So we'll take that as a unity,
which is a nice, even number.
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And as my dichotomy of the
afternoon, if we pass a plane
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through that sphere, if the
plane hits the sphere, it will
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intersect it in a circle.
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If the plane passes through
the center of the sphere--
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and we've assigned the radius
of the sphere as unity, then
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this is a circle that's referred
to a great circle.
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Sounds like a value judgment,
but it's simply saying that's
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as large as the circle is going
to get is when it passes
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through the center of the
sphere., it would have unit
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radius, just as the
sphere does.
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So if you take any other plane
which intersects the sphere
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but doesn't pass through the
center, it's going to have a
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smaller radius.
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And this is something that's
called a small circle.
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OK, so if all of the action is
going to take place on the
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surface of the sphere, and we
have two points on the sphere,
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A and B, sitting on the surface
of the sphere, how do
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we measure the separation
of A and B?
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Well, if you think in terms of
a normal three-dimensional
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person, you say, zonk, connect
them by a line.
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And that's the distance between
A and B. Now, you
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can't do that because all the
action has to take place on
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the surface of the sphere.
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People who deal with spherical
trigonometry all the time are
00:07:28.750 --> 00:07:30.360
airplane pilots.
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And if your pilot is going to
take you from New York--
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where would you like to go?
00:07:34.860 --> 00:07:35.400
Paris?
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That sounds like a nice place.
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But if you're going to go from
New York to Paris, you don't
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plow your way through the
intervening earth.
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You follow something that is at
a constant radius out from
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the center the Earth, at a
height of 5,000 feet above the
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surface of the Earth, an
additional radius.
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So the way we'll define distance
is to pass a great
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circle through A, B, pass a
plane through A, B in the
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center of the sphere.
00:08:03.610 --> 00:08:08.040
And then we will define distance
between A and B as
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the smaller of the two angles
subtended at the
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center of the sphere.
00:08:15.340 --> 00:08:18.490
So this is a more reasonable
looking great circle.
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If this is point A and this is
point B, pass a plane through
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the center of the sphere, O, and
through A and through B.
00:08:25.410 --> 00:08:28.740
And then we'll measure the
length of the arc AB in terms
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of the angle alpha subtended at
the center of the sphere.
00:08:34.679 --> 00:08:35.799
So it's a crazy notion.
00:08:35.799 --> 00:08:39.630
We're measuring distance
in terms of an angle.
00:08:39.630 --> 00:08:43.740
And if that's an angle, we can
take a trigonometric function
00:08:43.740 --> 00:08:46.740
of that angle, like
sine or cosine.
00:08:46.740 --> 00:08:49.990
And that blows the mind that
you can take trigonometric
00:08:49.990 --> 00:08:51.130
functions of a distance.
00:08:51.130 --> 00:08:51.700
But we can.
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We'll see it's going to
be useful to us, too.
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And then I emphasize again,
we'll take this
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as the smaller distance.
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It'll be 360 degrees
minus alpha.
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That would be the
long way around
00:09:04.260 --> 00:09:10.432
from A to B. All right.
00:09:10.432 --> 00:09:14.270
We've defined now
how we will draw
00:09:14.270 --> 00:09:15.820
distances between two points.
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Suppose I have three points on
the surface of the sphere--
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A, B, and C. I can pass a great
circle through A and B.
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I know how to do that.
00:09:30.470 --> 00:09:35.180
I can pass a great circle
through A and C. I
00:09:35.180 --> 00:09:36.910
know how to do that.
00:09:36.910 --> 00:09:40.700
And I can pass a great circle
through B and C.
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So now I have defined something
that is referred to
00:09:44.130 --> 00:09:45.605
as a spherical triangle.
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We know how to measure
the length of
00:09:58.200 --> 00:10:00.010
the spherical triangle.
00:10:00.010 --> 00:10:06.780
Let's call the arc opposite the
point of intersection A as
00:10:06.780 --> 00:10:11.970
little a and the length of the
arc opposite B as a distance
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and angles little b, and
the distance from A
00:10:15.320 --> 00:10:18.400
to C as little c.
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But there's something in between
these arcs that looks
00:10:21.660 --> 00:10:25.810
like an angle analogous to the
angle in a planar triangle.
00:10:25.810 --> 00:10:29.520
And how can we define that?
00:10:29.520 --> 00:10:40.300
Well, the arc AB is defined by
a plane, a great circle.
00:10:40.300 --> 00:10:46.090
The arc AC is similarly defined
by a plane that passes
00:10:46.090 --> 00:10:49.090
through a, c in the center
of the sphere.
00:10:49.090 --> 00:10:56.200
And what we will define as the
spherical angle BAC is the
00:10:56.200 --> 00:11:02.120
angle between the great circles
that define the two
00:11:02.120 --> 00:11:03.920
different arcs.
00:11:03.920 --> 00:11:09.530
So if there's one great circle
that defines the arc from A to
00:11:09.530 --> 00:11:13.790
B and another plane that defines
the arc from A to C,
00:11:13.790 --> 00:11:18.140
we'll define as the spherical
angle between those two arcs
00:11:18.140 --> 00:11:23.390
the angle between the planes
that define the great circles.
00:11:23.390 --> 00:11:26.840
So we're going to call this
angle in here between these
00:11:26.840 --> 00:11:29.210
two planes as angle BAC.
00:11:41.960 --> 00:11:45.245
Another construct that
is a useful one--
00:11:49.720 --> 00:11:53.050
suppose I look at the plane
that I've used to define a
00:11:53.050 --> 00:11:59.260
great circle and at the center
of the sphere construct a line
00:11:59.260 --> 00:12:04.530
that is perpendicular
to the great circle.
00:12:04.530 --> 00:12:07.120
And if I extend that line,
sooner or later it's going to
00:12:07.120 --> 00:12:12.300
poke out through the surface
of the sphere.
00:12:12.300 --> 00:12:20.480
And I will refer to this point
as the pole of arc AB, or the
00:12:20.480 --> 00:12:24.190
pole of the great circle that
we've used to define arc AB.
00:12:27.680 --> 00:12:35.010
So the North Pole is actually
the pole of the great circle
00:12:35.010 --> 00:12:36.750
that defines the equator.
00:12:36.750 --> 00:12:38.180
And clearly there
are two poles.
00:12:38.180 --> 00:12:39.800
There's one in either
direction.
00:12:39.800 --> 00:12:44.505
So there's a North Pole and a
South Pole to this arc AB and
00:12:44.505 --> 00:12:46.460
to the great circle
that defines it.
00:12:57.942 --> 00:12:59.436
AUDIENCE: I have a question.
00:12:59.436 --> 00:13:02.922
Why couldn't you define the
angle BAC as the angle
00:13:02.922 --> 00:13:04.172
[INAUDIBLE]?
00:13:09.894 --> 00:13:12.384
PROFESSOR: You want to
make a tangent here?
00:13:12.384 --> 00:13:13.634
AUDIENCE: Yeah.
00:13:22.880 --> 00:13:26.796
PROFESSOR: I don't know if
that's really defined.
00:13:26.796 --> 00:13:31.800
In other words, if I'm saying I
want a line that is tangent
00:13:31.800 --> 00:13:34.844
to the sphere, it doesn't
fix its orientation.
00:13:34.844 --> 00:13:36.460
AUDIENCE: In the plane
of the great circle.
00:13:36.460 --> 00:13:36.760
PROFESSOR: OK.
00:13:36.760 --> 00:13:38.010
In the plane of the
great circle.
00:13:44.730 --> 00:13:46.350
Suppose you could if
you wanted to.
00:13:46.350 --> 00:13:50.210
There are trigonometric
qualities to defining the
00:13:50.210 --> 00:13:53.010
angle in the way that we have.
00:13:53.010 --> 00:14:00.030
And the construct really is not
something confined to the
00:14:00.030 --> 00:14:02.390
surface of the sphere, and
everything else that
00:14:02.390 --> 00:14:06.270
we are doing is.
00:14:06.270 --> 00:14:06.780
OK?
00:14:06.780 --> 00:14:12.940
So it's sort of the non sequitur
because we started
00:14:12.940 --> 00:14:16.100
out by saying that everything
has to take place on the
00:14:16.100 --> 00:14:17.730
surface of the sphere.
00:14:17.730 --> 00:14:19.760
There's things that we would
do in our three-dimensional
00:14:19.760 --> 00:14:23.100
world, like defining distances
between points as the shortest
00:14:23.100 --> 00:14:27.980
straight line, that are ruled
out in spherical trigonometry.
00:14:27.980 --> 00:14:32.160
And I think something
similar could be
00:14:32.160 --> 00:14:35.340
levied at your proposal.
00:14:35.340 --> 00:14:41.302
And the answer is we just
don't do it that way.
00:14:41.302 --> 00:14:42.552
That's the real answer.
00:14:46.040 --> 00:14:46.420
OK.
00:14:46.420 --> 00:14:54.690
If I have not boggled your mind
so far, let me go a bit
00:14:54.690 --> 00:14:56.885
further with another
useful construct.
00:15:01.930 --> 00:15:06.420
We can see how we can define a
pole of a great circle or a
00:15:06.420 --> 00:15:11.050
pole on an arc that is a portion
of a great circle.
00:15:11.050 --> 00:15:18.750
Let me take a spherical
triangle, A, B, and C. And
00:15:18.750 --> 00:15:22.810
I've got three great circles
now, which have formed those
00:15:22.810 --> 00:15:27.640
arcs that make up the sides
of my triangle.
00:15:27.640 --> 00:15:33.620
Let me now find the
pole of arc CB.
00:15:33.620 --> 00:15:35.900
And that means we're going to
go out 90 degrees to that
00:15:35.900 --> 00:15:37.720
plane through the center
of the sphere.
00:15:37.720 --> 00:15:40.390
And that's going to define some
point that I'll call A
00:15:40.390 --> 00:15:45.050
prime, that is the
pole of arc BC.
00:15:45.050 --> 00:15:48.260
And I'm going to do the same
thing for the other arcs that
00:15:48.260 --> 00:15:52.700
are sides of my great circle
of my spherical triangle.
00:15:52.700 --> 00:15:55.400
I'll find the pole of arc AC.
00:15:55.400 --> 00:16:00.070
And I'm going to label that
point as B prime.
00:16:00.070 --> 00:16:02.630
And, finally, there'll be
another pole that is
00:16:02.630 --> 00:16:04.480
the pole of arc AB.
00:16:04.480 --> 00:16:08.100
And that's going to define
a point C prime.
00:16:08.100 --> 00:16:11.440
Now I've got three points, I can
connect these together and
00:16:11.440 --> 00:16:13.998
make another spherical
triangle.
00:16:13.998 --> 00:16:17.670
AUDIENCE: How do you know to
determine where the pole is?
00:16:17.670 --> 00:16:19.700
PROFESSOR: If you think of it
in three dimensions, I got
00:16:19.700 --> 00:16:20.670
three different arcs.
00:16:20.670 --> 00:16:24.020
And for each one of them I am
drawing a perpendicular to the
00:16:24.020 --> 00:16:26.310
plane of that great circle
and looking at the
00:16:26.310 --> 00:16:28.460
point where it emerges.
00:16:28.460 --> 00:16:28.720
OK.
00:16:28.720 --> 00:16:32.060
So now if there's another arc,
there'll be another great
00:16:32.060 --> 00:16:33.940
circle coming around
like this.
00:16:33.940 --> 00:16:35.660
And I look for the pole of it.
00:16:35.660 --> 00:16:38.950
And that would be another
one of the corners.
00:16:38.950 --> 00:16:42.860
This thing that I've constructed
is bizarre.
00:16:42.860 --> 00:16:44.200
But it's given a special name.
00:16:44.200 --> 00:16:45.615
This is called the
polar triangle.
00:16:53.660 --> 00:16:56.185
And it has some useful
properties.
00:17:04.890 --> 00:17:09.530
A property of the polar triangle
is that the two
00:17:09.530 --> 00:17:15.569
triangles, A, B, and C, and A
prime, B prime, and C prime
00:17:15.569 --> 00:17:18.540
are mutually polar.
00:17:18.540 --> 00:17:21.800
That is, if I use the spherical
triangle ABC to
00:17:21.800 --> 00:17:24.095
define and locate the
three points A
00:17:24.095 --> 00:17:25.560
prime, B prime, C prime--
00:17:25.560 --> 00:17:30.985
now if I reverse the process
and find the whole of arc A
00:17:30.985 --> 00:17:36.960
prime C prime, that turns
out to be point B.
00:17:36.960 --> 00:17:41.120
And if I take the arc of A prime
B prime, that turns--
00:17:41.120 --> 00:17:45.070
I'm sorry-- take the arc of B
prime C prime, that turns out
00:17:45.070 --> 00:17:50.470
to be point A. So the two
triangles are mutually polar.
00:17:50.470 --> 00:17:53.190
The polar triangle of the polar
triangle is the triangle
00:17:53.190 --> 00:17:54.340
that we started with.
00:17:54.340 --> 00:17:54.810
Yeah?
00:17:54.810 --> 00:17:56.220
AUDIENCE: I guess
I missed that.
00:17:56.220 --> 00:17:58.570
So if you take B prime
through C prime, then
00:17:58.570 --> 00:17:59.510
all of that is going--
00:17:59.510 --> 00:18:05.002
PROFESSOR: Yeah, I'm saying that
the pole of this arc, A
00:18:05.002 --> 00:18:08.770
prime C prime, is
this point here.
00:18:08.770 --> 00:18:18.450
And the way I can show that
is to say that we got B by
00:18:18.450 --> 00:18:21.150
looking at the pole--
00:18:21.150 --> 00:18:22.000
I'm sorry--
00:18:22.000 --> 00:18:26.510
I got C by looking at the
pole of the arc AB.
00:18:26.510 --> 00:18:33.840
So B is 90 degrees away
from C prime.
00:18:33.840 --> 00:18:43.080
I found point A prime by finding
the pole of arc CB.
00:18:43.080 --> 00:18:47.440
So B is 90 degrees from
any point on that arc.
00:18:47.440 --> 00:18:51.140
So it's 90 degrees away
from A. So B is 90
00:18:51.140 --> 00:18:52.200
degrees from A prime.
00:18:52.200 --> 00:18:54.040
B is 90 degrees C prime.
00:18:54.040 --> 00:19:00.290
And, therefore, it has to
be the pole of that arc.
00:19:00.290 --> 00:19:01.510
Now that was a little
too quick.
00:19:01.510 --> 00:19:02.740
That's written down
in the notes.
00:19:02.740 --> 00:19:03.990
And that's why I
wrote them out.
00:19:09.030 --> 00:19:12.310
One final thing and then we can
put circle trigonometry to
00:19:12.310 --> 00:19:14.690
one side, and this is something
00:19:14.690 --> 00:19:18.490
that is not all obvious.
00:19:18.490 --> 00:19:26.500
If we look at a spherical
triangle and simultaneously
00:19:26.500 --> 00:19:27.540
the polar triangle--
00:19:27.540 --> 00:19:31.290
so let's say this is ABC.
00:19:31.290 --> 00:19:38.660
And here is the polar triangle
A prime, C prime, B prime.
00:19:45.527 --> 00:19:53.170
It turns out that the spherical
angle in one circle
00:19:53.170 --> 00:20:00.380
triangle and the length of the
arc opposite it, namely this
00:20:00.380 --> 00:20:04.665
arc B prime C prime,
are complementary--
00:20:10.040 --> 00:20:13.300
supplementary, not
complementary.
00:20:13.300 --> 00:20:21.900
And the way one would do that is
to say that the measure of
00:20:21.900 --> 00:20:25.160
alpha is the length
of this arc here.
00:20:27.710 --> 00:20:36.700
And this total side, B prime C
prime, is equal to this arc
00:20:36.700 --> 00:20:41.490
plus this arc minus
this length.
00:20:41.490 --> 00:20:43.910
And these two arcs
are 90 degrees.
00:20:43.910 --> 00:20:47.820
So let me do as I've
done in the notes.
00:20:47.820 --> 00:20:50.270
Let me call this
point P prime.
00:20:50.270 --> 00:20:53.860
And I'll call this
point Q prime.
00:20:53.860 --> 00:21:05.340
So my argument, it says that B
prime is the pole of arc AC.
00:21:05.340 --> 00:21:13.170
And, therefore, B prime
Q prime equals
00:21:13.170 --> 00:21:15.660
90 degrees in length.
00:21:15.660 --> 00:21:23.810
And then I would say that C
prime is the pole of arc AB.
00:21:23.810 --> 00:21:30.710
And, therefore, the distance C
prime P prime is also exactly
00:21:30.710 --> 00:21:32.670
90 degrees.
00:21:32.670 --> 00:21:39.940
And that says that B prime Q
prime plus C prime P prime--
00:21:39.940 --> 00:21:44.530
if I add those two together,
it has to be 180 degrees.
00:21:44.530 --> 00:21:54.880
But I can write B prime C prime
as B prime P prime plus
00:21:54.880 --> 00:22:03.020
P prime Q prime plus Q prime
C prime plus the
00:22:03.020 --> 00:22:04.870
side P prime Q prime.
00:22:08.620 --> 00:22:10.080
And that's 180 degrees.
00:22:14.560 --> 00:22:17.390
But these three things that I've
lumped together here are
00:22:17.390 --> 00:22:20.780
exactly the same as the length
of the spherical polar
00:22:20.780 --> 00:22:22.280
triangle A prime.
00:22:22.280 --> 00:22:26.170
So what we've shown then is
that A prime plus alpha is
00:22:26.170 --> 00:22:30.580
equal to 180 degrees--
00:22:30.580 --> 00:22:31.780
QED.
00:22:31.780 --> 00:22:38.100
So this angle plus the side of
the polar triangle add up to
00:22:38.100 --> 00:22:39.350
180 degrees.
00:22:43.400 --> 00:22:45.230
And that is not obvious
at all.
00:22:50.350 --> 00:22:51.430
One final relation--
00:22:51.430 --> 00:22:55.635
and this I will simply hand
to you on a platter.
00:22:55.635 --> 00:22:57.570
I'm not about to derive it.
00:23:00.500 --> 00:23:05.140
Sides and angles in planar
geometry are related.
00:23:05.140 --> 00:23:09.130
And there's a particularly
useful relation in plane
00:23:09.130 --> 00:23:11.340
geometry that's called
the Law of Cosines.
00:23:15.440 --> 00:23:16.800
So this is in plane geometry.
00:23:22.040 --> 00:23:27.280
And if you have a triangle
that has sides a, b, c--
00:23:27.280 --> 00:23:29.300
a general oblique triangle--
00:23:29.300 --> 00:23:35.030
and it has angles A, B, and C,
the Law of Cosines says that
00:23:35.030 --> 00:23:40.410
the side A is determined
by c and b and the
00:23:40.410 --> 00:23:41.900
angle between them.
00:23:41.900 --> 00:23:42.630
And that's clear.
00:23:42.630 --> 00:23:46.440
If I specify this length,
specify this length, specify
00:23:46.440 --> 00:23:49.960
that angle, things set up like
a bowl of supercooled jello.
00:23:49.960 --> 00:23:52.830
And the triangle's completely
specified.
00:23:52.830 --> 00:23:56.565
So a squared in the Law of
Cosines is b squared plus c
00:23:56.565 --> 00:24:03.880
squared minus 2bc times
the cosine of angle A.
00:24:03.880 --> 00:24:08.670
In a spherical triangle there
is a similar sort of
00:24:08.670 --> 00:24:10.030
constraint.
00:24:10.030 --> 00:24:14.580
If we have a spherical triangle
with sides a, b, and
00:24:14.580 --> 00:24:19.750
c, and spherical angles capital
A, capital B, capital
00:24:19.750 --> 00:24:23.640
C, in the same way as specifying
the spherical angle
00:24:23.640 --> 00:24:28.650
A and the lengths of the two
sides c and b, specifies and
00:24:28.650 --> 00:24:33.080
fixes the spherical
triangle entirely.
00:24:33.080 --> 00:24:36.810
This side must be determined by
the length of side c, the
00:24:36.810 --> 00:24:40.460
length of side b, and the
angle between them.
00:24:40.460 --> 00:24:43.960
And that, since everything
is in terms of angles, is
00:24:43.960 --> 00:24:46.160
something that doesn't
involve squares.
00:24:46.160 --> 00:24:49.510
It involves totally
trigonometric expressions.
00:24:49.510 --> 00:24:53.490
And it turns out the cosine of
this missing side a is given
00:24:53.490 --> 00:24:57.475
by the product of the cosines
of the two other sides.
00:24:57.475 --> 00:25:00.680
So as I said, you can take a
trigonometric function of a
00:25:00.680 --> 00:25:05.260
length, which sounds
like an oxymoron.
00:25:05.260 --> 00:25:09.820
And it's the product of the
cosines of the two known sides
00:25:09.820 --> 00:25:15.380
and times the sine of b sine of
c times the cosine of the
00:25:15.380 --> 00:25:18.090
spherical angle A.
And that is also
00:25:18.090 --> 00:25:19.340
called the Law of Cosines.
00:25:24.400 --> 00:25:28.490
And this is the corresponding
case in spherical geometry.
00:25:34.130 --> 00:25:34.430
OK.
00:25:34.430 --> 00:25:35.740
So there's some machinery--
00:25:35.740 --> 00:25:36.674
yes, sir?
00:25:36.674 --> 00:25:39.009
AUDIENCE: What's the difference
between the sines
00:25:39.009 --> 00:25:41.811
of lowercase a and--
00:25:41.811 --> 00:25:44.850
PROFESSOR: OK, the angles
are the capital letters.
00:25:44.850 --> 00:25:47.230
This would be the angle between
the great circles that
00:25:47.230 --> 00:25:47.740
defines the--
00:25:47.740 --> 00:25:51.065
AUDIENCE: Since the radius is
one, there's no difference
00:25:51.065 --> 00:25:54.446
between the angles and the--
00:25:54.446 --> 00:25:56.378
[INAUDIBLE]?
00:25:56.378 --> 00:25:59.200
PROFESSOR: No.
00:25:59.200 --> 00:26:02.910
This angle is something, for
example, we can choose.
00:26:02.910 --> 00:26:06.270
And depending on how long we
want this arc to be, we can
00:26:06.270 --> 00:26:09.270
put the arc BC anywhere
we like.
00:26:09.270 --> 00:26:10.722
AUDIENCE: Does that mean your
radius [INAUDIBLE]?
00:26:14.600 --> 00:26:15.190
PROFESSOR: No.
00:26:15.190 --> 00:26:19.000
This would be, say, two points
of the spherical triangle.
00:26:19.000 --> 00:26:22.120
Now we can pick any third point
on the surface of the
00:26:22.120 --> 00:26:25.540
sphere, connect that with great
circles, and here is a
00:26:25.540 --> 00:26:26.790
spherical triangle.
00:26:29.264 --> 00:26:29.710
OK?
00:26:29.710 --> 00:26:31.130
So I see.
00:26:31.130 --> 00:26:35.290
I think I see what
your problem is.
00:26:35.290 --> 00:26:37.610
Here are the two planes.
00:26:37.610 --> 00:26:41.870
We define the angle of the
spherical triangle as the
00:26:41.870 --> 00:26:44.940
angle subtended buy
the great circle.
00:26:44.940 --> 00:26:48.590
So this is the definition
of A.
00:26:48.590 --> 00:26:51.940
But now the other two points on
the spherical triangle can
00:26:51.940 --> 00:26:54.060
be any point on these
great circles.
00:26:54.060 --> 00:26:57.750
So this can be point B, and this
can be point C. And my
00:26:57.750 --> 00:27:01.340
spherical triangle can be
something like this.
00:27:01.340 --> 00:27:07.630
So the arc that defines the
spherical angle A is a value
00:27:07.630 --> 00:27:12.240
that is independent from the
length of the arc AC.
00:27:12.240 --> 00:27:15.910
That would be what is
subtended at the
00:27:15.910 --> 00:27:17.160
center of the sphere.
00:27:25.610 --> 00:27:30.050
I'm going to have time just to
set the stage for how we're
00:27:30.050 --> 00:27:32.336
going to use these relations.
00:27:35.200 --> 00:27:38.900
And the problem that I would
like to raise and then apply
00:27:38.900 --> 00:27:46.720
spherical trigonometry to is
the question if I go into
00:27:46.720 --> 00:27:54.990
three-dimensional space, there
is no longer any requirement
00:27:54.990 --> 00:27:58.340
that rotation axes be all
00:27:58.340 --> 00:28:00.170
parallel to the same direction.
00:28:00.170 --> 00:28:03.120
In two dimensions the rotation
points were really--
00:28:03.120 --> 00:28:06.420
could be viewed as axes that
were always perpendicular to
00:28:06.420 --> 00:28:10.140
the plane of the blackboard, the
plane of the plane group.
00:28:10.140 --> 00:28:16.250
But now when I'm dealing with
three-dimensional spaces, this
00:28:16.250 --> 00:28:20.420
could be the operation
A alpha.
00:28:20.420 --> 00:28:23.220
And there is no reason
whatsoever why we should not
00:28:23.220 --> 00:28:26.940
try to combine with this
first operation a
00:28:26.940 --> 00:28:31.790
second rotation B beta--
00:28:31.790 --> 00:28:35.090
a rotation through an angle beta
about this axis B and a
00:28:35.090 --> 00:28:40.060
rotation of angle alpha through
this axis A. If we're
00:28:40.060 --> 00:28:43.690
going to come up with a
crystallographic combination,
00:28:43.690 --> 00:28:46.970
the angles alpha and beta have
to be restricted to the
00:28:46.970 --> 00:28:51.410
angular rotations of either
a onefold, a twofold, a
00:28:51.410 --> 00:28:54.420
threefold, a fourfold, or
a sixfold axis, if the
00:28:54.420 --> 00:28:56.550
combination is going to
be crystallographic.
00:28:56.550 --> 00:28:57.012
Yes, sir?
00:28:57.012 --> 00:28:59.322
AUDIENCE: You're just rotating
around those lines?
00:28:59.322 --> 00:29:00.750
PROFESSOR: I'm rotating around
those lines, yeah.
00:29:00.750 --> 00:29:05.200
So I'm saying that we're going
to rotate an angle alpha about
00:29:05.200 --> 00:29:06.930
axis A.
00:29:06.930 --> 00:29:09.480
And now what I'm going
to raise as
00:29:09.480 --> 00:29:11.100
a rhetorical question--
00:29:11.100 --> 00:29:16.550
what is the rotation A alpha
followed by B beta?
00:29:16.550 --> 00:29:23.440
So I rotate through the angle
alpha about A. So here's my
00:29:23.440 --> 00:29:27.920
first motif, right-handed.
00:29:27.920 --> 00:29:31.560
Then I'll rotate
alpha degrees.
00:29:31.560 --> 00:29:33.290
And this here's number two.
00:29:33.290 --> 00:29:36.380
And that will stay
right-handed.
00:29:36.380 --> 00:29:40.840
Now, I will place on axis
B the two constraints.
00:29:40.840 --> 00:29:44.010
These have to be
crystallographic rotation
00:29:44.010 --> 00:29:52.380
angles, namely 360, 180,
120, 90, or 60.
00:29:52.380 --> 00:29:55.330
And I'll also, since I would
like to obtain point group
00:29:55.330 --> 00:30:00.450
symmetries initially, I will
require that axis A and axis B
00:30:00.450 --> 00:30:03.620
intersect at some point.
00:30:03.620 --> 00:30:07.090
And one of the variables in the
combination will be this
00:30:07.090 --> 00:30:10.000
angle between the two
rotation axes.
00:30:10.000 --> 00:30:14.660
So let's complete our
combination of operations.
00:30:14.660 --> 00:30:17.490
I'll rotate from one
to two by A alpha.
00:30:17.490 --> 00:30:19.560
If the first one is
right-handed, the second motif
00:30:19.560 --> 00:30:21.310
is right-handed, as well.
00:30:21.310 --> 00:30:27.840
And then I will rotate beta
degrees about B. And here will
00:30:27.840 --> 00:30:31.680
sit number three.
00:30:31.680 --> 00:30:32.930
And it will stay right-handed.
00:30:35.490 --> 00:30:41.190
So, again, the $64 question that
we raise periodically--
00:30:41.190 --> 00:30:46.020
what operation is the net
effect of two successive
00:30:46.020 --> 00:30:48.755
rotations about a point
of intersection?
00:30:51.984 --> 00:30:53.234
[INTERPOSING VOICES]
00:30:55.337 --> 00:30:56.290
PROFESSOR: Lots of opinions.
00:30:56.290 --> 00:30:57.230
Let's sort them out.
00:30:57.230 --> 00:30:58.980
I heard translation.
00:30:58.980 --> 00:31:02.720
Well, let's put down
what it could be.
00:31:02.720 --> 00:31:07.720
We know that only translation
and rotation leaves the
00:31:07.720 --> 00:31:09.550
chirality of the motif
unchanged.
00:31:09.550 --> 00:31:12.263
So it's got to be one
or the other.
00:31:12.263 --> 00:31:14.187
AUDIENCE: Since there's no
reason for the general
00:31:14.187 --> 00:31:16.592
orientation of the two to say
it has to be rotation around
00:31:16.592 --> 00:31:18.035
the third axis, so
the question is
00:31:18.035 --> 00:31:19.385
what angle and what--
00:31:19.385 --> 00:31:19.760
PROFESSOR: OK.
00:31:19.760 --> 00:31:21.790
That is exactly the problem.
00:31:21.790 --> 00:31:25.080
Now that we know the problem,
we can go home early because
00:31:25.080 --> 00:31:28.380
we know what we're going
to do next time.
00:31:28.380 --> 00:31:32.030
Well, let me expand
a little bit.
00:31:32.030 --> 00:31:35.080
It can't be translation because
clearly the separation
00:31:35.080 --> 00:31:38.030
of number one and number three
depend on exactly where I
00:31:38.030 --> 00:31:39.230
place the first one.
00:31:39.230 --> 00:31:41.350
If I place it a little further
out from A, then it's going to
00:31:41.350 --> 00:31:42.620
rotate to here.
00:31:42.620 --> 00:31:44.230
And then B is going
to swing it off to
00:31:44.230 --> 00:31:45.960
some different location.
00:31:45.960 --> 00:31:49.160
So it can't be translation.
00:31:49.160 --> 00:31:52.220
And I don't think these guys, if
I rotate this way and then
00:31:52.220 --> 00:31:54.600
I rotate this way, are going to
be parallel to one another.
00:31:54.600 --> 00:31:56.200
I doubt that very much.
00:31:56.200 --> 00:31:57.450
So it's got to be a rotation.
00:32:03.250 --> 00:32:08.500
So without knowing how to find
it, let's say that we can get
00:32:08.500 --> 00:32:12.520
from number one to number three
in one shot through
00:32:12.520 --> 00:32:20.340
rotation of an angle gamma about
some third axis, C. So
00:32:20.340 --> 00:32:23.575
the answer, in general, without
being specific, is A
00:32:23.575 --> 00:32:26.650
alpha followed B beta has got
to be equal to a third
00:32:26.650 --> 00:32:30.980
rotation, C, about a direction
that has the same point of
00:32:30.980 --> 00:32:32.870
intersection with the
first two axes.
00:32:36.880 --> 00:32:40.163
Now we've got some really,
really tough constraints.
00:32:43.140 --> 00:32:46.860
Alpha is restricted to
one of five values.
00:32:46.860 --> 00:32:49.785
Beta is restricted to
one of five values.
00:32:53.470 --> 00:32:57.290
The third rotation, gamma,
jolly well better be a
00:32:57.290 --> 00:33:03.360
crystallographic rotation and
not something that is not a
00:33:03.360 --> 00:33:04.720
sub-multiple of 2 pi.
00:33:04.720 --> 00:33:07.730
And even if it is a sub-multiple
of 2 pi, it has
00:33:07.730 --> 00:33:13.830
to be either 0 degrees,
120, so on.
00:33:13.830 --> 00:33:14.580
It has to be one of the
00:33:14.580 --> 00:33:16.530
crystallographic rotation angles.
00:33:16.530 --> 00:33:20.690
So what sort of relation can
we get that would give us,
00:33:20.690 --> 00:33:26.180
first, the value of gamma in
terms of alpha, beta, and the
00:33:26.180 --> 00:33:29.560
angle at which we
combine them?
00:33:29.560 --> 00:33:34.890
So taking A and B as the five
crystallographic rotation axes
00:33:34.890 --> 00:33:40.460
two at a time, we want to put
them together, if we can, such
00:33:40.460 --> 00:33:45.630
that the angle makes the third
rotation axis also be
00:33:45.630 --> 00:33:46.820
crystallographic.
00:33:46.820 --> 00:33:49.580
And then we would have
to find its location.
00:33:49.580 --> 00:33:52.810
It looks like an impossible
constraint .
00:33:52.810 --> 00:33:56.010
It looks absolutely
impossible to do.
00:33:56.010 --> 00:34:00.280
We've got to put this first in
quantitative form and then
00:34:00.280 --> 00:34:04.400
simply put in the values for
alpha, beta, and gamma and
00:34:04.400 --> 00:34:07.360
find the angle that they have
to be combined on to make
00:34:07.360 --> 00:34:11.010
this, if possible, be one
of the crystallographics
00:34:11.010 --> 00:34:12.260
sub-multiples.
00:34:13.929 --> 00:34:16.830
That is not an easy problem
to formulate.
00:34:16.830 --> 00:34:20.449
And, as I said a couple of times
ago, the geometry that
00:34:20.449 --> 00:34:25.219
is the basis of this derivation
was originally
00:34:25.219 --> 00:34:27.889
proposed by Euler.
00:34:27.889 --> 00:34:30.659
And it's known as Euler's
Construction.
00:34:30.659 --> 00:34:35.090
I will have for you next time
my own set of notes on this.
00:34:35.090 --> 00:34:37.010
We have finished with
two-dimensional
00:34:37.010 --> 00:34:37.780
crystallography.
00:34:37.780 --> 00:34:41.850
So we are back to Buerger's
book again.
00:34:41.850 --> 00:34:44.260
We had that little interlude.
00:34:44.260 --> 00:34:46.659
Buerger deals with Euler's
Construction.
00:34:46.659 --> 00:34:49.850
But I don't think he's
at his best in
00:34:49.850 --> 00:34:51.380
this particular section.
00:34:51.380 --> 00:34:53.300
So we'll take it a little
more slowly.
00:34:53.300 --> 00:34:55.820
And next time we'll get around
to deriving Euler's
00:34:55.820 --> 00:34:57.070
Construction.