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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
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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
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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
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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?
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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.
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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.
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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.
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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.
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So it's a crazy notion.
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We're measuring distance
in terms of an angle.
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And if that's an angle, we can
take a trigonometric function
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of that angle, like
sine or cosine.
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And that blows the mind that
you can take trigonometric
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functions of a distance.
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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
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from A to B. All right.
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We've defined now
how we will draw
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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.
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I can pass a great circle
through A and C. I
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know how to do that.
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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
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as a spherical triangle.
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We know how to measure
the length of
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the spherical triangle.
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Let's call the arc opposite the
point of intersection A as
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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
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to C as little c.
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But there's something in between
these arcs that looks
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like an angle analogous to the
angle in a planar triangle.
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And how can we define that?
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Well, the arc AB is defined by
a plane, a great circle.
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The arc AC is similarly defined
by a plane that passes
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through a, c in the center
of the sphere.
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And what we will define as the
spherical angle BAC is the
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angle between the great circles
that define the two
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different arcs.
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So if there's one great circle
that defines the arc from A to
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B and another plane that defines
the arc from A to C,
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we'll define as the spherical
angle between those two arcs
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the angle between the planes
that define the great circles.
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So we're going to call this
angle in here between these
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two planes as angle BAC.
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Another construct that
is a useful one--
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suppose I look at the plane
that I've used to define a
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great circle and at the center
of the sphere construct a line
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that is perpendicular
to the great circle.
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And if I extend that line,
sooner or later it's going to
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poke out through the surface
of the sphere.
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And I will refer to this point
as the pole of arc AB, or the
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pole of the great circle that
we've used to define arc AB.
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So the North Pole is actually
the pole of the great circle
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that defines the equator.
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And clearly there
are two poles.
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There's one in either
direction.
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So there's a North Pole and a
South Pole to this arc AB and
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to the great circle
that defines it.
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AUDIENCE: I have a question.
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00:12:59,436 --> 00:13:02,922
Why couldn't you define the
angle BAC as the angle
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[INAUDIBLE]?
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00:13:09,894 --> 00:13:12,384
PROFESSOR: You want to
make a tangent here?
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00:13:12,384 --> 00:13:13,634
AUDIENCE: Yeah.
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PROFESSOR: I don't know if
that's really defined.
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In other words, if I'm saying I
want a line that is tangent
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to the sphere, it doesn't
fix its orientation.
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00:13:34,844 --> 00:13:36,460
AUDIENCE: In the plane
of the great circle.
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00:13:36,460 --> 00:13:36,760
PROFESSOR: OK.
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00:13:36,760 --> 00:13:38,010
In the plane of the
great circle.
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00:13:44,730 --> 00:13:46,350
Suppose you could if
you wanted to.
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00:13:46,350 --> 00:13:50,210
There are trigonometric
qualities to defining the
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00:13:50,210 --> 00:13:53,010
angle in the way that we have.
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And the construct really is not
something confined to the
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00:14:00,030 --> 00:14:02,390
surface of the sphere, and
everything else that
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00:14:02,390 --> 00:14:06,270
we are doing is.
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00:14:06,270 --> 00:14:06,780
OK?
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00:14:06,780 --> 00:14:12,940
So it's sort of the non sequitur
because we started
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00:14:12,940 --> 00:14:16,100
out by saying that everything
has to take place on the
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00:14:16,100 --> 00:14:17,730
surface of the sphere.
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There's things that we would
do in our three-dimensional
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world, like defining distances
between points as the shortest
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00:14:23,100 --> 00:14:27,980
straight line, that are ruled
out in spherical trigonometry.
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00:14:27,980 --> 00:14:32,160
And I think something
similar could be
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00:14:32,160 --> 00:14:35,340
levied at your proposal.
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00:14:35,340 --> 00:14:41,302
And the answer is we just
don't do it that way.
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00:14:41,302 --> 00:14:42,552
That's the real answer.
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00:14:46,040 --> 00:14:46,420
OK.
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If I have not boggled your mind
so far, let me go a bit
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00:14:54,690 --> 00:14:56,885
further with another
useful construct.
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We can see how we can define a
pole of a great circle or a
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00:15:06,420 --> 00:15:11,050
pole on an arc that is a portion
of a great circle.
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00:15:11,050 --> 00:15:18,750
Let me take a spherical
triangle, A, B, and C. And
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00:15:18,750 --> 00:15:22,810
I've got three great circles
now, which have formed those
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00:15:22,810 --> 00:15:27,640
arcs that make up the sides
of my triangle.
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Let me now find the
pole of arc CB.
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00:15:33,620 --> 00:15:35,900
And that means we're going to
go out 90 degrees to that
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00:15:35,900 --> 00:15:37,720
plane through the center
of the sphere.
243
00:15:37,720 --> 00:15:40,390
And that's going to define some
point that I'll call A
244
00:15:40,390 --> 00:15:45,050
prime, that is the
pole of arc BC.
245
00:15:45,050 --> 00:15:48,260
And I'm going to do the same
thing for the other arcs that
246
00:15:48,260 --> 00:15:52,700
are sides of my great circle
of my spherical triangle.
247
00:15:52,700 --> 00:15:55,400
I'll find the pole of arc AC.
248
00:15:55,400 --> 00:16:00,070
And I'm going to label that
point as B prime.
249
00:16:00,070 --> 00:16:02,630
And, finally, there'll be
another pole that is
250
00:16:02,630 --> 00:16:04,480
the pole of arc AB.
251
00:16:04,480 --> 00:16:08,100
And that's going to define
a point C prime.
252
00:16:08,100 --> 00:16:11,440
Now I've got three points, I can
connect these together and
253
00:16:11,440 --> 00:16:13,998
make another spherical
triangle.
254
00:16:13,998 --> 00:16:17,670
AUDIENCE: How do you know to
determine where the pole is?
255
00:16:17,670 --> 00:16:19,700
PROFESSOR: If you think of it
in three dimensions, I got
256
00:16:19,700 --> 00:16:20,670
three different arcs.
257
00:16:20,670 --> 00:16:24,020
And for each one of them I am
drawing a perpendicular to the
258
00:16:24,020 --> 00:16:26,310
plane of that great circle
and looking at the
259
00:16:26,310 --> 00:16:28,460
point where it emerges.
260
00:16:28,460 --> 00:16:28,720
OK.
261
00:16:28,720 --> 00:16:32,060
So now if there's another arc,
there'll be another great
262
00:16:32,060 --> 00:16:33,940
circle coming around
like this.
263
00:16:33,940 --> 00:16:35,660
And I look for the pole of it.
264
00:16:35,660 --> 00:16:38,950
And that would be another
one of the corners.
265
00:16:38,950 --> 00:16:42,860
This thing that I've constructed
is bizarre.
266
00:16:42,860 --> 00:16:44,200
But it's given a special name.
267
00:16:44,200 --> 00:16:45,615
This is called the
polar triangle.
268
00:16:53,660 --> 00:16:56,185
And it has some useful
properties.
269
00:17:04,890 --> 00:17:09,530
A property of the polar triangle
is that the two
270
00:17:09,530 --> 00:17:15,569
triangles, A, B, and C, and A
prime, B prime, and C prime
271
00:17:15,569 --> 00:17:18,540
are mutually polar.
272
00:17:18,540 --> 00:17:21,800
That is, if I use the spherical
triangle ABC to
273
00:17:21,800 --> 00:17:24,095
define and locate the
three points A
274
00:17:24,095 --> 00:17:25,560
prime, B prime, C prime--
275
00:17:25,560 --> 00:17:30,985
now if I reverse the process
and find the whole of arc A
276
00:17:30,985 --> 00:17:36,960
prime C prime, that turns
out to be point B.
277
00:17:36,960 --> 00:17:41,120
And if I take the arc of A prime
B prime, that turns--
278
00:17:41,120 --> 00:17:45,070
I'm sorry-- take the arc of B
prime C prime, that turns out
279
00:17:45,070 --> 00:17:50,470
to be point A. So the two
triangles are mutually polar.
280
00:17:50,470 --> 00:17:53,190
The polar triangle of the polar
triangle is the triangle
281
00:17:53,190 --> 00:17:54,340
that we started with.
282
00:17:54,340 --> 00:17:54,810
Yeah?
283
00:17:54,810 --> 00:17:56,220
AUDIENCE: I guess
I missed that.
284
00:17:56,220 --> 00:17:58,570
So if you take B prime
through C prime, then
285
00:17:58,570 --> 00:17:59,510
all of that is going--
286
00:17:59,510 --> 00:18:05,002
PROFESSOR: Yeah, I'm saying that
the pole of this arc, A
287
00:18:05,002 --> 00:18:08,770
prime C prime, is
this point here.
288
00:18:08,770 --> 00:18:18,450
And the way I can show that
is to say that we got B by
289
00:18:18,450 --> 00:18:21,150
looking at the pole--
290
00:18:21,150 --> 00:18:22,000
I'm sorry--
291
00:18:22,000 --> 00:18:26,510
I got C by looking at the
pole of the arc AB.
292
00:18:26,510 --> 00:18:33,840
So B is 90 degrees away
from C prime.
293
00:18:33,840 --> 00:18:43,080
I found point A prime by finding
the pole of arc CB.
294
00:18:43,080 --> 00:18:47,440
So B is 90 degrees from
any point on that arc.
295
00:18:47,440 --> 00:18:51,140
So it's 90 degrees away
from A. So B is 90
296
00:18:51,140 --> 00:18:52,200
degrees from A prime.
297
00:18:52,200 --> 00:18:54,040
B is 90 degrees C prime.
298
00:18:54,040 --> 00:19:00,290
And, therefore, it has to
be the pole of that arc.
299
00:19:00,290 --> 00:19:01,510
Now that was a little
too quick.
300
00:19:01,510 --> 00:19:02,740
That's written down
in the notes.
301
00:19:02,740 --> 00:19:03,990
And that's why I
wrote them out.
302
00:19:09,030 --> 00:19:12,310
One final thing and then we can
put circle trigonometry to
303
00:19:12,310 --> 00:19:14,690
one side, and this is something
304
00:19:14,690 --> 00:19:18,490
that is not all obvious.
305
00:19:18,490 --> 00:19:26,500
If we look at a spherical
triangle and simultaneously
306
00:19:26,500 --> 00:19:27,540
the polar triangle--
307
00:19:27,540 --> 00:19:31,290
so let's say this is ABC.
308
00:19:31,290 --> 00:19:38,660
And here is the polar triangle
A prime, C prime, B prime.
309
00:19:45,527 --> 00:19:53,170
It turns out that the spherical
angle in one circle
310
00:19:53,170 --> 00:20:00,380
triangle and the length of the
arc opposite it, namely this
311
00:20:00,380 --> 00:20:04,665
arc B prime C prime,
are complementary--
312
00:20:10,040 --> 00:20:13,300
supplementary, not
complementary.
313
00:20:13,300 --> 00:20:21,900
And the way one would do that is
to say that the measure of
314
00:20:21,900 --> 00:20:25,160
alpha is the length
of this arc here.
315
00:20:27,710 --> 00:20:36,700
And this total side, B prime C
prime, is equal to this arc
316
00:20:36,700 --> 00:20:41,490
plus this arc minus
this length.
317
00:20:41,490 --> 00:20:43,910
And these two arcs
are 90 degrees.
318
00:20:43,910 --> 00:20:47,820
So let me do as I've
done in the notes.
319
00:20:47,820 --> 00:20:50,270
Let me call this
point P prime.
320
00:20:50,270 --> 00:20:53,860
And I'll call this
point Q prime.
321
00:20:53,860 --> 00:21:05,340
So my argument, it says that B
prime is the pole of arc AC.
322
00:21:05,340 --> 00:21:13,170
And, therefore, B prime
Q prime equals
323
00:21:13,170 --> 00:21:15,660
90 degrees in length.
324
00:21:15,660 --> 00:21:23,810
And then I would say that C
prime is the pole of arc AB.
325
00:21:23,810 --> 00:21:30,710
And, therefore, the distance C
prime P prime is also exactly
326
00:21:30,710 --> 00:21:32,670
90 degrees.
327
00:21:32,670 --> 00:21:39,940
And that says that B prime Q
prime plus C prime P prime--
328
00:21:39,940 --> 00:21:44,530
if I add those two together,
it has to be 180 degrees.
329
00:21:44,530 --> 00:21:54,880
But I can write B prime C prime
as B prime P prime plus
330
00:21:54,880 --> 00:22:03,020
P prime Q prime plus Q prime
C prime plus the
331
00:22:03,020 --> 00:22:04,870
side P prime Q prime.
332
00:22:08,620 --> 00:22:10,080
And that's 180 degrees.
333
00:22:14,560 --> 00:22:17,390
But these three things that I've
lumped together here are
334
00:22:17,390 --> 00:22:20,780
exactly the same as the length
of the spherical polar
335
00:22:20,780 --> 00:22:22,280
triangle A prime.
336
00:22:22,280 --> 00:22:26,170
So what we've shown then is
that A prime plus alpha is
337
00:22:26,170 --> 00:22:30,580
equal to 180 degrees--
338
00:22:30,580 --> 00:22:31,780
QED.
339
00:22:31,780 --> 00:22:38,100
So this angle plus the side of
the polar triangle add up to
340
00:22:38,100 --> 00:22:39,350
180 degrees.
341
00:22:43,400 --> 00:22:45,230
And that is not obvious
at all.
342
00:22:50,350 --> 00:22:51,430
One final relation--
343
00:22:51,430 --> 00:22:55,635
and this I will simply hand
to you on a platter.
344
00:22:55,635 --> 00:22:57,570
I'm not about to derive it.
345
00:23:00,500 --> 00:23:05,140
Sides and angles in planar
geometry are related.
346
00:23:05,140 --> 00:23:09,130
And there's a particularly
useful relation in plane
347
00:23:09,130 --> 00:23:11,340
geometry that's called
the Law of Cosines.
348
00:23:15,440 --> 00:23:16,800
So this is in plane geometry.
349
00:23:22,040 --> 00:23:27,280
And if you have a triangle
that has sides a, b, c--
350
00:23:27,280 --> 00:23:29,300
a general oblique triangle--
351
00:23:29,300 --> 00:23:35,030
and it has angles A, B, and C,
the Law of Cosines says that
352
00:23:35,030 --> 00:23:40,410
the side A is determined
by c and b and the
353
00:23:40,410 --> 00:23:41,900
angle between them.
354
00:23:41,900 --> 00:23:42,630
And that's clear.
355
00:23:42,630 --> 00:23:46,440
If I specify this length,
specify this length, specify
356
00:23:46,440 --> 00:23:49,960
that angle, things set up like
a bowl of supercooled jello.
357
00:23:49,960 --> 00:23:52,830
And the triangle's completely
specified.
358
00:23:52,830 --> 00:23:56,565
So a squared in the Law of
Cosines is b squared plus c
359
00:23:56,565 --> 00:24:03,880
squared minus 2bc times
the cosine of angle A.
360
00:24:03,880 --> 00:24:08,670
In a spherical triangle there
is a similar sort of
361
00:24:08,670 --> 00:24:10,030
constraint.
362
00:24:10,030 --> 00:24:14,580
If we have a spherical triangle
with sides a, b, and
363
00:24:14,580 --> 00:24:19,750
c, and spherical angles capital
A, capital B, capital
364
00:24:19,750 --> 00:24:23,640
C, in the same way as specifying
the spherical angle
365
00:24:23,640 --> 00:24:28,650
A and the lengths of the two
sides c and b, specifies and
366
00:24:28,650 --> 00:24:33,080
fixes the spherical
triangle entirely.
367
00:24:33,080 --> 00:24:36,810
This side must be determined by
the length of side c, the
368
00:24:36,810 --> 00:24:40,460
length of side b, and the
angle between them.
369
00:24:40,460 --> 00:24:43,960
And that, since everything
is in terms of angles, is
370
00:24:43,960 --> 00:24:46,160
something that doesn't
involve squares.
371
00:24:46,160 --> 00:24:49,510
It involves totally
trigonometric expressions.
372
00:24:49,510 --> 00:24:53,490
And it turns out the cosine of
this missing side a is given
373
00:24:53,490 --> 00:24:57,475
by the product of the cosines
of the two other sides.
374
00:24:57,475 --> 00:25:00,680
So as I said, you can take a
trigonometric function of a
375
00:25:00,680 --> 00:25:05,260
length, which sounds
like an oxymoron.
376
00:25:05,260 --> 00:25:09,820
And it's the product of the
cosines of the two known sides
377
00:25:09,820 --> 00:25:15,380
and times the sine of b sine of
c times the cosine of the
378
00:25:15,380 --> 00:25:18,090
spherical angle A.
And that is also
379
00:25:18,090 --> 00:25:19,340
called the Law of Cosines.
380
00:25:24,400 --> 00:25:28,490
And this is the corresponding
case in spherical geometry.
381
00:25:34,130 --> 00:25:34,430
OK.
382
00:25:34,430 --> 00:25:35,740
So there's some machinery--
383
00:25:35,740 --> 00:25:36,674
yes, sir?
384
00:25:36,674 --> 00:25:39,009
AUDIENCE: What's the difference
between the sines
385
00:25:39,009 --> 00:25:41,811
of lowercase a and--
386
00:25:41,811 --> 00:25:44,850
PROFESSOR: OK, the angles
are the capital letters.
387
00:25:44,850 --> 00:25:47,230
This would be the angle between
the great circles that
388
00:25:47,230 --> 00:25:47,740
defines the--
389
00:25:47,740 --> 00:25:51,065
AUDIENCE: Since the radius is
one, there's no difference
390
00:25:51,065 --> 00:25:54,446
between the angles and the--
391
00:25:54,446 --> 00:25:56,378
[INAUDIBLE]?
392
00:25:56,378 --> 00:25:59,200
PROFESSOR: No.
393
00:25:59,200 --> 00:26:02,910
This angle is something, for
example, we can choose.
394
00:26:02,910 --> 00:26:06,270
And depending on how long we
want this arc to be, we can
395
00:26:06,270 --> 00:26:09,270
put the arc BC anywhere
we like.
396
00:26:09,270 --> 00:26:10,722
AUDIENCE: Does that mean your
radius [INAUDIBLE]?
397
00:26:14,600 --> 00:26:15,190
PROFESSOR: No.
398
00:26:15,190 --> 00:26:19,000
This would be, say, two points
of the spherical triangle.
399
00:26:19,000 --> 00:26:22,120
Now we can pick any third point
on the surface of the
400
00:26:22,120 --> 00:26:25,540
sphere, connect that with great
circles, and here is a
401
00:26:25,540 --> 00:26:26,790
spherical triangle.
402
00:26:29,264 --> 00:26:29,710
OK?
403
00:26:29,710 --> 00:26:31,130
So I see.
404
00:26:31,130 --> 00:26:35,290
I think I see what
your problem is.
405
00:26:35,290 --> 00:26:37,610
Here are the two planes.
406
00:26:37,610 --> 00:26:41,870
We define the angle of the
spherical triangle as the
407
00:26:41,870 --> 00:26:44,940
angle subtended buy
the great circle.
408
00:26:44,940 --> 00:26:48,590
So this is the definition
of A.
409
00:26:48,590 --> 00:26:51,940
But now the other two points on
the spherical triangle can
410
00:26:51,940 --> 00:26:54,060
be any point on these
great circles.
411
00:26:54,060 --> 00:26:57,750
So this can be point B, and this
can be point C. And my
412
00:26:57,750 --> 00:27:01,340
spherical triangle can be
something like this.
413
00:27:01,340 --> 00:27:07,630
So the arc that defines the
spherical angle A is a value
414
00:27:07,630 --> 00:27:12,240
that is independent from the
length of the arc AC.
415
00:27:12,240 --> 00:27:15,910
That would be what is
subtended at the
416
00:27:15,910 --> 00:27:17,160
center of the sphere.
417
00:27:25,610 --> 00:27:30,050
I'm going to have time just to
set the stage for how we're
418
00:27:30,050 --> 00:27:32,336
going to use these relations.
419
00:27:35,200 --> 00:27:38,900
And the problem that I would
like to raise and then apply
420
00:27:38,900 --> 00:27:46,720
spherical trigonometry to is
the question if I go into
421
00:27:46,720 --> 00:27:54,990
three-dimensional space, there
is no longer any requirement
422
00:27:54,990 --> 00:27:58,340
that rotation axes be all
423
00:27:58,340 --> 00:28:00,170
parallel to the same direction.
424
00:28:00,170 --> 00:28:03,120
In two dimensions the rotation
points were really--
425
00:28:03,120 --> 00:28:06,420
could be viewed as axes that
were always perpendicular to
426
00:28:06,420 --> 00:28:10,140
the plane of the blackboard, the
plane of the plane group.
427
00:28:10,140 --> 00:28:16,250
But now when I'm dealing with
three-dimensional spaces, this
428
00:28:16,250 --> 00:28:20,420
could be the operation
A alpha.
429
00:28:20,420 --> 00:28:23,220
And there is no reason
whatsoever why we should not
430
00:28:23,220 --> 00:28:26,940
try to combine with this
first operation a
431
00:28:26,940 --> 00:28:31,790
second rotation B beta--
432
00:28:31,790 --> 00:28:35,090
a rotation through an angle beta
about this axis B and a
433
00:28:35,090 --> 00:28:40,060
rotation of angle alpha through
this axis A. If we're
434
00:28:40,060 --> 00:28:43,690
going to come up with a
crystallographic combination,
435
00:28:43,690 --> 00:28:46,970
the angles alpha and beta have
to be restricted to the
436
00:28:46,970 --> 00:28:51,410
angular rotations of either
a onefold, a twofold, a
437
00:28:51,410 --> 00:28:54,420
threefold, a fourfold, or
a sixfold axis, if the
438
00:28:54,420 --> 00:28:56,550
combination is going to
be crystallographic.
439
00:28:56,550 --> 00:28:57,012
Yes, sir?
440
00:28:57,012 --> 00:28:59,322
AUDIENCE: You're just rotating
around those lines?
441
00:28:59,322 --> 00:29:00,750
PROFESSOR: I'm rotating around
those lines, yeah.
442
00:29:00,750 --> 00:29:05,200
So I'm saying that we're going
to rotate an angle alpha about
443
00:29:05,200 --> 00:29:06,930
axis A.
444
00:29:06,930 --> 00:29:09,480
And now what I'm going
to raise as
445
00:29:09,480 --> 00:29:11,100
a rhetorical question--
446
00:29:11,100 --> 00:29:16,550
what is the rotation A alpha
followed by B beta?
447
00:29:16,550 --> 00:29:23,440
So I rotate through the angle
alpha about A. So here's my
448
00:29:23,440 --> 00:29:27,920
first motif, right-handed.
449
00:29:27,920 --> 00:29:31,560
Then I'll rotate
alpha degrees.
450
00:29:31,560 --> 00:29:33,290
And this here's number two.
451
00:29:33,290 --> 00:29:36,380
And that will stay
right-handed.
452
00:29:36,380 --> 00:29:40,840
Now, I will place on axis
B the two constraints.
453
00:29:40,840 --> 00:29:44,010
These have to be
crystallographic rotation
454
00:29:44,010 --> 00:29:52,380
angles, namely 360, 180,
120, 90, or 60.
455
00:29:52,380 --> 00:29:55,330
And I'll also, since I would
like to obtain point group
456
00:29:55,330 --> 00:30:00,450
symmetries initially, I will
require that axis A and axis B
457
00:30:00,450 --> 00:30:03,620
intersect at some point.
458
00:30:03,620 --> 00:30:07,090
And one of the variables in the
combination will be this
459
00:30:07,090 --> 00:30:10,000
angle between the two
rotation axes.
460
00:30:10,000 --> 00:30:14,660
So let's complete our
combination of operations.
461
00:30:14,660 --> 00:30:17,490
I'll rotate from one
to two by A alpha.
462
00:30:17,490 --> 00:30:19,560
If the first one is
right-handed, the second motif
463
00:30:19,560 --> 00:30:21,310
is right-handed, as well.
464
00:30:21,310 --> 00:30:27,840
And then I will rotate beta
degrees about B. And here will
465
00:30:27,840 --> 00:30:31,680
sit number three.
466
00:30:31,680 --> 00:30:32,930
And it will stay right-handed.
467
00:30:35,490 --> 00:30:41,190
So, again, the $64 question that
we raise periodically--
468
00:30:41,190 --> 00:30:46,020
what operation is the net
effect of two successive
469
00:30:46,020 --> 00:30:48,755
rotations about a point
of intersection?
470
00:30:51,984 --> 00:30:53,234
[INTERPOSING VOICES]
471
00:30:55,337 --> 00:30:56,290
PROFESSOR: Lots of opinions.
472
00:30:56,290 --> 00:30:57,230
Let's sort them out.
473
00:30:57,230 --> 00:30:58,980
I heard translation.
474
00:30:58,980 --> 00:31:02,720
Well, let's put down
what it could be.
475
00:31:02,720 --> 00:31:07,720
We know that only translation
and rotation leaves the
476
00:31:07,720 --> 00:31:09,550
chirality of the motif
unchanged.
477
00:31:09,550 --> 00:31:12,263
So it's got to be one
or the other.
478
00:31:12,263 --> 00:31:14,187
AUDIENCE: Since there's no
reason for the general
479
00:31:14,187 --> 00:31:16,592
orientation of the two to say
it has to be rotation around
480
00:31:16,592 --> 00:31:18,035
the third axis, so
the question is
481
00:31:18,035 --> 00:31:19,385
what angle and what--
482
00:31:19,385 --> 00:31:19,760
PROFESSOR: OK.
483
00:31:19,760 --> 00:31:21,790
That is exactly the problem.
484
00:31:21,790 --> 00:31:25,080
Now that we know the problem,
we can go home early because
485
00:31:25,080 --> 00:31:28,380
we know what we're going
to do next time.
486
00:31:28,380 --> 00:31:32,030
Well, let me expand
a little bit.
487
00:31:32,030 --> 00:31:35,080
It can't be translation because
clearly the separation
488
00:31:35,080 --> 00:31:38,030
of number one and number three
depend on exactly where I
489
00:31:38,030 --> 00:31:39,230
place the first one.
490
00:31:39,230 --> 00:31:41,350
If I place it a little further
out from A, then it's going to
491
00:31:41,350 --> 00:31:42,620
rotate to here.
492
00:31:42,620 --> 00:31:44,230
And then B is going
to swing it off to
493
00:31:44,230 --> 00:31:45,960
some different location.
494
00:31:45,960 --> 00:31:49,160
So it can't be translation.
495
00:31:49,160 --> 00:31:52,220
And I don't think these guys, if
I rotate this way and then
496
00:31:52,220 --> 00:31:54,600
I rotate this way, are going to
be parallel to one another.
497
00:31:54,600 --> 00:31:56,200
I doubt that very much.
498
00:31:56,200 --> 00:31:57,450
So it's got to be a rotation.
499
00:32:03,250 --> 00:32:08,500
So without knowing how to find
it, let's say that we can get
500
00:32:08,500 --> 00:32:12,520
from number one to number three
in one shot through
501
00:32:12,520 --> 00:32:20,340
rotation of an angle gamma about
some third axis, C. So
502
00:32:20,340 --> 00:32:23,575
the answer, in general, without
being specific, is A
503
00:32:23,575 --> 00:32:26,650
alpha followed B beta has got
to be equal to a third
504
00:32:26,650 --> 00:32:30,980
rotation, C, about a direction
that has the same point of
505
00:32:30,980 --> 00:32:32,870
intersection with the
first two axes.
506
00:32:36,880 --> 00:32:40,163
Now we've got some really,
really tough constraints.
507
00:32:43,140 --> 00:32:46,860
Alpha is restricted to
one of five values.
508
00:32:46,860 --> 00:32:49,785
Beta is restricted to
one of five values.
509
00:32:53,470 --> 00:32:57,290
The third rotation, gamma,
jolly well better be a
510
00:32:57,290 --> 00:33:03,360
crystallographic rotation and
not something that is not a
511
00:33:03,360 --> 00:33:04,720
sub-multiple of 2 pi.
512
00:33:04,720 --> 00:33:07,730
And even if it is a sub-multiple
of 2 pi, it has
513
00:33:07,730 --> 00:33:13,830
to be either 0 degrees,
120, so on.
514
00:33:13,830 --> 00:33:14,580
It has to be one of the
515
00:33:14,580 --> 00:33:16,530
crystallographic rotation angles.
516
00:33:16,530 --> 00:33:20,690
So what sort of relation can
we get that would give us,
517
00:33:20,690 --> 00:33:26,180
first, the value of gamma in
terms of alpha, beta, and the
518
00:33:26,180 --> 00:33:29,560
angle at which we
combine them?
519
00:33:29,560 --> 00:33:34,890
So taking A and B as the five
crystallographic rotation axes
520
00:33:34,890 --> 00:33:40,460
two at a time, we want to put
them together, if we can, such
521
00:33:40,460 --> 00:33:45,630
that the angle makes the third
rotation axis also be
522
00:33:45,630 --> 00:33:46,820
crystallographic.
523
00:33:46,820 --> 00:33:49,580
And then we would have
to find its location.
524
00:33:49,580 --> 00:33:52,810
It looks like an impossible
constraint .
525
00:33:52,810 --> 00:33:56,010
It looks absolutely
impossible to do.
526
00:33:56,010 --> 00:34:00,280
We've got to put this first in
quantitative form and then
527
00:34:00,280 --> 00:34:04,400
simply put in the values for
alpha, beta, and gamma and
528
00:34:04,400 --> 00:34:07,360
find the angle that they have
to be combined on to make
529
00:34:07,360 --> 00:34:11,010
this, if possible, be one
of the crystallographics
530
00:34:11,010 --> 00:34:12,260
sub-multiples.
531
00:34:13,929 --> 00:34:16,830
That is not an easy problem
to formulate.
532
00:34:16,830 --> 00:34:20,449
And, as I said a couple of times
ago, the geometry that
533
00:34:20,449 --> 00:34:25,219
is the basis of this derivation
was originally
534
00:34:25,219 --> 00:34:27,889
proposed by Euler.
535
00:34:27,889 --> 00:34:30,659
And it's known as Euler's
Construction.
536
00:34:30,659 --> 00:34:35,090
I will have for you next time
my own set of notes on this.
537
00:34:35,090 --> 00:34:37,010
We have finished with
two-dimensional
538
00:34:37,010 --> 00:34:37,780
crystallography.
539
00:34:37,780 --> 00:34:41,850
So we are back to Buerger's
book again.
540
00:34:41,850 --> 00:34:44,260
We had that little interlude.
541
00:34:44,260 --> 00:34:46,659
Buerger deals with Euler's
Construction.
542
00:34:46,659 --> 00:34:49,850
But I don't think he's
at his best in
543
00:34:49,850 --> 00:34:51,380
this particular section.
544
00:34:51,380 --> 00:34:53,300
So we'll take it a little
more slowly.
545
00:34:53,300 --> 00:34:55,820
And next time we'll get around
to deriving Euler's
546
00:34:55,820 --> 00:34:57,070
Construction.