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PROFESSOR: My watch says five
after 2:00, so why don't we

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get started?

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This is going to be
a strange lecture.

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I'm expecting some wise guy in
the back to say, all your

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lectures are rather peculiar!

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But I have a phone call from
Europe coming in sometime

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around 2:15, 2:30.

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Somebody's coming to visit us
next week, and we're going to

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set up a seminar.

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So I don't know exactly when
it's going to come in, but I

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have to be in my office.

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So what I will do is
adjourn at quarter

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after, 20 past the hour.

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And then so I don't keep you
here sitting restively

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wondering when I'm going
to come back, we'll

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adjourn until 3 o'clock.

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So you seem to have been taken
longer and longer breaks

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during intermission.

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So I'll give you a full 3/4 of
an hour so you can get it out

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of your systems.

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And then, we'll take shorter
breaks from here on in.

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OK this is going to
be pretty much our

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last lecture on symmetry.

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And we will begin, at least in
half of the next class on

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Tuesday, to begin talking about
physical properties, and

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in particular tensor
properties.

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But I'd like to say a little
bit about the derivation of

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space groups and take a look
at how this information is

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tabulated for you by the kind
folks who prepare the

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international tables.

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And there are really very strict
parallels between what

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we did in two dimensions.

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And that's the reason why we
did it so thoroughly and

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systematically, except that
there are three dimensions.

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And because we are taking 32
point groups and putting them

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into 14 lattices, there are a
lot more combinations to be

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considered.

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And there are a lot more of
them that are unique.

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Also, they're a little bit more
difficult to visualize

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and depict.

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So there are some new
conventions in preparing a

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representation of three
dimensional symmetries on a

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sheet of paper or page of a book
which, of necessity, has

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to be two dimensional.

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So we'll go over some of
those conventions.

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But the main reason for going
a little bit further is that

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there is another surprise
lurking in there for us as

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soon as we begin to combine
translation in three

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dimensions with a symmetry
element.

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But let me go through
in some detail the

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monoclinic space groups.

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So we have at our disposal
to decorate with symmetry

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elements two lattices.

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And they are a primitive
monoclinic lattice with two

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translations, a and b, that
define an oblique net, and a

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third translation, c, at right
angles to that net.

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And then, either two choices
for a double cell--

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and I'll draw these in
projection because I can do

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both of them with one diagram.

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Either this is a and b and the
extra lattice point is in the

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center of the cell halfway
up from the base.

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This would be body centered.

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And that's represented by
I for innenzentriert.

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In German, that's German
for body centered.

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Or the alternative would be to
pick a cell like this, in

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which case the extra lattice
point gets caught in the

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center of one of the faces.

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So this depiction here would
be a side centered lattice.

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And it could have the extra
lattice point in the middle of

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the face out of which b comes.

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And in that case, the symbol
for the lattice is B. Or

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alternatively, without changing
any of the nature of

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the specialness of the lattice,
the extra lattice

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point could be in the
middle of the face

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out of which a comes.

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And that would be
an A lattice.

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With the first, setting with the
c axis unique, there is no

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c lattice because the oblique
base of the cell would not

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give us anything new if
it were centered.

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All right, so there are
those two lattices.

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And then, there are three
point groups.

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And they were 2, m, and then a
combination of a twofold axis

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perpendicular to
a mirror plane.

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So it looks as though in
principle there are going to

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be 6 combinations.

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In point of fact, there are
a lot more because of the

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surprise that we have
in store for us.

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Before proceeding further,
though, let me ask

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rhetorically the question.

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Which lattice type would you
pick as the standard one?

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And here, there's a major
decision that has to be made.

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Should the labels on an
axes give a unique

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symbol for the lattice?

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In other words, if you have
the double cell for a

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monoclinic crystal, should you
define the cell to make it

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body centered always, or to
make it A centered or B

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centered always?

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Or should the labels be defined
by the relative

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lengths of the axes?

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So those are the two
possibilities.

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And there's no right
or wrong way.

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You pays your money and
you makes your choice.

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And on pragmatic grounds,
this is the

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convention that's followed.

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And it's purely a pragmatic
decision because lattice

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constants have been determined
for literally tens of

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thousands of materials.

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Sometimes, the lattice constants
are known and the

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space group is not.

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But it would be terrible if the
label that you applied to

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the axis could be C attached
to the longest one, the

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shortest one, or the
intermediate one.

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It would be impossible to
calculate any database for

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lattice constants displayed
by particular materials.

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So there's all this-- this is
what the decision was made.

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This is the decision that
was made many years ago.

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And for monoclinic crystals,
the labeling of the axes is

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determined by the c axis
being the unique axis.

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That is, the axis that is along
the twofold axis or

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perpendicular to the
mirror plane.

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And then, the labels b and c are
defined by magnitude of b

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greater than the
magnitude of a.

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So I inherently drew this is
in the proper fashion.

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So the labels are applied
to monoclinic

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crystals in that fashion.

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The price you pay for that is
that the symbol for the

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lattice for a monoclinic crystal
that has the double

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cell could either be A, B, or
I depending on the relative

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lengths of the axes.

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If these were the two shortest,
it would be a body

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centered lattice.

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On the other hand, if this and
this were the shortest pair,

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then it would be a side
centered lattice.

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So the symbol for the lattice
type will bounce around

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depending on the dimensions
of the translations.

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So that's a complication
in the space groups.

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In the two dimensional
plane group, that

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issue never came up.

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We just took B greater than A
for the rectangular and the

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oblique nets.

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OK, so let me do quickly
a couple of the

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monoclinic space groups.

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Let's take a twofold axis and
put it into a primitive

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monoclinic net.

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And this is exactly the same as
our plane group, P2, with

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the twofold axes extended
parallel to the c translation.

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So we would have twofold axes in
all of these orientations.

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And that's simply P2, little p
in two dimensions, with the

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twofold axes extended
normal to the plane

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of the plane group.

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So this is p2.

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This is capital P2.

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And that is way we
distinguish plane

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groups from space groups.

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So the symbol for the lattice
is a capital symbol for the

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space groups.

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So I didn't have to use any
new combinations theorems.

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I simply say there's
a third translation

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perpendicular to the net.

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So everything protrudes
out of the plane group

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into a third dimension.

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Let me now combine a twofold
axis this a and this b and

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this c with a lattice that is
not the primitive lattice but

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one of the double cells.

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And I'll take it, for
convenience, to be the body

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centered flavor because
it's easier to draw.

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And the operation that we've
added to the lattice, if this

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is a combination of a twofold
axis, is the symbol A pi.

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And I know what happens if
I combine A pi with a

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translation that is
perpendicular to

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the rotation axis.

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I get a new rotation axis, B pi,
that's halfway along the

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perpendicular part of
the translation.

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And that's where all
of these additional

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twofold axes came in.

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But if a lattice is a body
centered lattice, there is now

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a translation that goes up to
the centered lattice point.

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So what's that going to be?

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What is T followed by A pi
followed by T, where T is 1/2

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A plus 1/2 of B?

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And that is a part that is
perpendicular to the axis, and

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then a third component, c, which
is parallel to the axis.

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As with all these theorems, what
you do is you draw it out

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once and for all and see what
it turns out to be.

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So let's do that.

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Let's say that this is the
translation that goes up to

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the point 1/2, 1/2, 1/2.

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Here is the parallel part--

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the perpendicular
part, rather.

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This is 1/2 of A
plus 1/2 of B.

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And then, we have a part
that's parallel to the

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rotation operation, A pi.

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And that is 1/2 of c.

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So let's just do it.

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Here's my first object
number one.

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It's right handed.

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00:13:04,260 --> 00:13:09,520
I'll rotate 180 degrees
to get a second one

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00:13:09,520 --> 00:13:10,720
that's right handed.

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00:13:10,720 --> 00:13:15,220
And then, I will translate
it up to here.

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And here sits the third one.

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It's also right handed.

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00:13:22,120 --> 00:13:24,580
How do I get from
one to three?

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00:13:29,530 --> 00:13:32,180
It's not really clear
how I do that.

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00:13:32,180 --> 00:13:34,640
And again, I'll draw it in
projection because this is

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looking pretty messy in
three dimensions.

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00:13:37,000 --> 00:13:40,150
So here's my first one.

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00:13:40,150 --> 00:13:43,615
I rotate 180 degrees to
get a second one.

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00:13:43,615 --> 00:13:45,460
The chirality is all the same.

209
00:13:45,460 --> 00:13:48,190
And then, I translate
up to this point.

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00:13:48,190 --> 00:13:51,610
So if this one is at z and this
one is at z, this one

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00:13:51,610 --> 00:13:59,800
here will sit at z plus 1/2.

212
00:13:59,800 --> 00:14:02,860
And this is the third one.

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00:14:02,860 --> 00:14:04,110
Anybody got any idea?

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00:14:07,760 --> 00:14:08,800
Can't be reflection.

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00:14:08,800 --> 00:14:10,410
They're of the same chirality.

216
00:14:10,410 --> 00:14:12,640
It can't be translation
because the

217
00:14:12,640 --> 00:14:16,325
orientation is different.

218
00:14:16,325 --> 00:14:16,814
Yes, sir?

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00:14:16,814 --> 00:14:19,750
AUDIENCE: Then it has to be
some form of rotation.

220
00:14:19,750 --> 00:14:21,680
PROFESSOR: Right, has to be.

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00:14:21,680 --> 00:14:28,040
But how do we get the object up
to a different elevation?

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00:14:28,040 --> 00:14:35,220
Well, we've stumbled headlong
over a new type of operation,

223
00:14:35,220 --> 00:14:37,780
just as we discovered the glide
plane when we started

224
00:14:37,780 --> 00:14:40,330
putting mirror planes into a
lattice in combination with

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00:14:40,330 --> 00:14:41,670
translation.

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00:14:41,670 --> 00:14:45,950
The only way I can get from the
first to the third is to

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00:14:45,950 --> 00:14:50,990
rotate 180 degrees about exactly
the same point that I

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00:14:50,990 --> 00:14:54,050
would have before, namely at
1/2 of the part of the

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00:14:54,050 --> 00:14:55,820
translation that is

230
00:14:55,820 --> 00:14:59,200
perpendicular to the operation.

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00:14:59,200 --> 00:15:01,680
But then before putting it
down, I've got to take a

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00:15:01,680 --> 00:15:02,910
second step.

233
00:15:02,910 --> 00:15:08,470
I've got to slide it up by a
translation component that is

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00:15:08,470 --> 00:15:11,550
parallel to the axis about
which I've rotated.

235
00:15:11,550 --> 00:15:16,320
So to complete my theorem,
A alpha followed by a

236
00:15:16,320 --> 00:15:22,070
translation that has a
perpendicular part plus a

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00:15:22,070 --> 00:15:26,210
parallel part is a new type
of operation that involves

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00:15:26,210 --> 00:15:30,020
rotating through 180 degrees.

239
00:15:30,020 --> 00:15:36,450
And this is located at 1/2
of T perpendicular.

240
00:15:36,450 --> 00:15:39,360
But it has a translation
component which I'll call

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00:15:39,360 --> 00:15:40,970
generically tau.

242
00:15:40,970 --> 00:15:45,540
And tau is equal to the
parallel part of the

243
00:15:45,540 --> 00:15:48,120
translation.

244
00:15:48,120 --> 00:15:53,060
So if I separate out this new
sort of operation in all of

245
00:15:53,060 --> 00:15:59,180
its grandeur, what it does is
it takes a first object,

246
00:15:59,180 --> 00:16:02,920
rotates us 180 degrees, does
not yet put it down, first

247
00:16:02,920 --> 00:16:06,160
slides up by the amount tau.

248
00:16:06,160 --> 00:16:09,200
Doing the operation, again
rotates it 180 degrees,

249
00:16:09,200 --> 00:16:13,320
doesn't yet put it down,
slides it up by tau.

250
00:16:13,320 --> 00:16:18,700
Doing it again moves us to
another one up here.

251
00:16:18,700 --> 00:16:24,950
So what we've generated is a
helical spiral of objects

252
00:16:24,950 --> 00:16:29,450
about the central axis about
which we're rotating.

253
00:16:29,450 --> 00:16:32,140
And this is a new two step
operation that's called a

254
00:16:32,140 --> 00:16:34,310
screw axis.

255
00:16:34,310 --> 00:16:38,730
There is a persistent rumor that
it derives its name from

256
00:16:38,730 --> 00:16:41,860
the effect that it has
on 360 quizzes.

257
00:16:41,860 --> 00:16:43,460
But that is just
an ugly rumor.

258
00:16:43,460 --> 00:16:45,310
There's nothing to
that at all.

259
00:16:45,310 --> 00:16:48,930
So this is a new two step
operation that we'll call, in

260
00:16:48,930 --> 00:16:53,910
general, A alpha tau.

261
00:16:53,910 --> 00:16:57,900
And what it will do is to
generate a screw like

262
00:16:57,900 --> 00:16:59,680
distribution of objects.

263
00:16:59,680 --> 00:17:04,700
For those of you who have come
in late, since you've been

264
00:17:04,700 --> 00:17:07,290
demonstrating a predilection
towards longer and longer

265
00:17:07,290 --> 00:17:11,890
breaks, we're going to adjourn
now and have a 40 minute break

266
00:17:11,890 --> 00:17:13,849
so you can get it out
of your system.

267
00:17:13,849 --> 00:17:18,980
And we will resume at exactly
3 o'clock sharp.

268
00:17:18,980 --> 00:17:27,849
And we will launch into an
elucidation of the nature of

269
00:17:27,849 --> 00:17:29,099
screw axes.

270
00:17:40,050 --> 00:17:43,960
And again, for those of you who
were not here at the very

271
00:17:43,960 --> 00:17:48,190
start of class, the reason is
that I have a phone call that

272
00:17:48,190 --> 00:17:52,110
had to come in about this time
of day and no other time.

273
00:17:52,110 --> 00:17:55,300
So rather than keep you
shifting, wondering whether

274
00:17:55,300 --> 00:17:57,950
I'm coming back, we'll start at
three when I should be done

275
00:17:57,950 --> 00:17:59,050
with my business.

276
00:17:59,050 --> 00:18:00,810
OK, so that's it.

277
00:18:00,810 --> 00:18:02,910
You missed all the important
discussion of what's going to

278
00:18:02,910 --> 00:18:06,320
be on the next quiz, and nobody
will tell you because

279
00:18:06,320 --> 00:18:09,420
they want to keep class
average down.

280
00:18:09,420 --> 00:18:12,710
All right, so sorry for the
longer than usual interlude,

281
00:18:12,710 --> 00:18:14,310
but we'll start again
promptly at 3:00.