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PROFESSOR: To mention,
most everyone did

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extremely well on the quiz.

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But I sense that there's still
some of you who have not yet

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come to terms with
crystallographic directions

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and planes, and you feel
a little bit awkward in

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distinguishing brackets
around the HKL and

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parentheses around HKL.

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And there are some people
who generally get that

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straightened out, but when I
said point group, suddenly

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pictures of lattices with
fourfold axes and twofold axes

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adorning them came in, and that
isn't involved in a point

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group at also.

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Again, a point group the
symmetry about point.

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A space group is symmetry spread
out through all of

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space and infinite numbers.

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So let me say a little
bit about resources.

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I don't know whether you've been
following what we've been

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doing in the notes from
Buerger's book

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that I passed out.

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That was hard to do is because
we did the plane groups, and

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he doesn't touch them at all.

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So now we're back following once
again Buerger's treatment

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quite closely.

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So read the book.

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And if you like, I can tell
you with the end of each

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lecture, this stuff is on
pages 57 through 62.

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The other thing.

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As you'll notice, this
nonintrusive gentleman in the

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back is making videotapes
of all the lectures.

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These are eventually going to
go up on the website as

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OpenCourseWare.

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We were just speaking about
that, and it takes a while

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before they get up, but I have
a disk of every lecture.

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So if there's something you
didn't follow or a place where

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I chewed my lines and you want
to go back over it again-- not

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that that happens very often--

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you are more than welcome to
ask me to borrow and borrow

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the disk if you want
to review it.

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So that's another resource.

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And then I will regularly
throughout the term give

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hard-copy handouts of some of
the things that we're doing,

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particularly when it
involves geometry.

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And when we move on to
three-dimensional geometry,

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unless of the graphics is really
tight and precise, it's

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hard to follow what's
going on.

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So in that vein, one of the
first things I wanted to pass

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out-- the only other
one for today--

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is a demonstration that in fact
in the Group 23 all you

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need is a single twofold axis
oriented along the normal to a

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face and a single threefold
axis coming

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out of one body diagonal.

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And that gives you all
of the axes you are

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going to get into 23.

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So when this comes around, there
are number of steps that

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you can perform letting the
axes work on each other.

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And if you start with just the
single twofold axis and the

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single threefold axis-- which
the symbol suggests is all you

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need, there's only one
kind of each--

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if you look at a cube along its
body diagonal, the twofold

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axis that's coming out of one
face gets rotated into

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directions normal to all the
other faces if you rotate by

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120 degrees.

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So the little diagram in the
upper right hand corner of the

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sheet hopefully convinces
you of this.

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Now we've got three mutually
orthogonal twofold axes and

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one threefold axis coming
out of a body diagonal.

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So the vertical twofold axis
swings that around by 180

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degrees to give you another
threefold axis

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along a body diagonal.

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And I labeled that one
3 prime in the

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middle diagram at right.

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And then if you take the axis
3 prime and repeat it by the

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second of the twofold axes
that we have along face

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normals, that in the middle
diagram on the right hand edge

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takes 3 prime and repeats
it to 3 double prime.

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Then finally, the third twofold
axis that we generated

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repeats the threefold prime
axis to the remaining

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threefold axis along the
fourth body diagonal.

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So the results start with one
twofold axis oriented along

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the normal to a cube and one
threefold axis along the

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diagonal of the cube, you get
axes coming out of all faces

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and all diagonal.

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And there's a staple on that
sheet for reasons that I don't

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understand, but it there was,
probably on the surface of the

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Xerox machine when I
went over there.

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So let me now return to our next
step, and that is to add

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what in the language of group
theory is called an extender,

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a new symmetry operation that
can be added to a preexisting

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group that will generate
new operations.

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And let's see what sort of
theorems we need to describe

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what we should look for and in
which particular orientation.

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We've got these 11 axial
combinations, and these are

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frameworks that we can hang
mirror planes on.

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So let's look at a first
simple combination.

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Suppose we have a twofold axis,
and the only nontrivial

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operation there is a rotation
through 180 degrees.

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So this an axis A pi.

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And again, the ground rules
are that if we're to add a

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mirror plane to this axis,
which along with identity

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constitutes a group, we can't
create any new axes.

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So there are two ways
we can do this.

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One is the three-dimensional
analog of something that we

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have already done, namely to
pass a mirror plane through

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the twofold axis.

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And this is the group that we
found as a two-dimensional

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Point Group, and we
called it 2mm.

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And to do that, we used the
theorem that says that if you

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take a rotation operation A
alpha and combine it with a

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reflection operation that goes
through it, you get a new

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reflection plane, sigma prime,
that's at an angle alpha over

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2 to the first.

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So what gave us the second
mirror line in two dimensions,

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in three dimensions that would
give us another mirror plane

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that's at right angles
to the first.

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And this will give us a
three-dimensional symmetry,

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which is also called 2mm.

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So it's nothing more than taking
the two- dimensional

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Point Group and letting it come
out at the board at you--

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Man, that'll give you nightmares
when these things

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are coming out of the
paper at you--

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and that is a valid group 2mm.

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There's another way that we can
add a mirror plane to a

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rotation axis though which will
not create any new axes,

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and that's to add the mirror
plane-- the reflection

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operation sigma I should say
since it is a combinations of

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operations--

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exactly perpendicular
to the locus of

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the rotation operation.

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And that reflection operation
sigma then just flips the

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rotation axis end to end.

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And the rotation operation just
swirls the locus of the

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reflection plane around within
its own locus and doesn't

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create any new reflection
plane.

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So in the particular combination
A pi, if we add a

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mirror plane perpendicular to
that as the operation sigma,

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this would be then all of the
operations of a twofold axis

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over all of the operations
of a mirror plane.

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So we've got now a combination
of a twofold axis with a

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mirror plane.

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We've got the same thing here.

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To distinguish these two
combinations, we'll write this

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as a fraction 2/m.

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And that literally in words
is the way we've added the

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twofold axis.

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It is sitting over the mirror
plane and gets reflected down

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into its other end when the
mirror plane act on.

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So it's just language.

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It's nice though, as we said
some weeks ago, if our

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language have some descriptive
content to so when we look at

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it we can remind ourselves
of what it means.

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So two 2mm means the mirror
planes are parallel to the

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axis that contain it.

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2/m means the twofold axis
is over the mirror plane.

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It goes through and
pierces it.

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What I would like to
ask though is have

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we got a group yet?

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And let's take a
first object--

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let's call it right handed--

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rotate it by 180 degrees.

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You get a second one, which
will stay right handed.

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And then repeat it by a
reflection operation in the

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mirror plane, and we'll
get a third operation.

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Reflection changes
chirality, so the

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third one is left handed.

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So we are performing the
sequence of operations A pi

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followed by sigma.

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And let me append just so we
make no mistake and not

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confuse it with 2mm that the
reflection operation is normal

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to the mirror plane.

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So question, what is
the net effect?

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And lo and behold, we have
stumbled over-- if we had not

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been clever enough to invented
it and suggested it early on--

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the only way you can get from 1
to 3 in one shot is to turn

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it inside out and change its
chirality by projecting it

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through a point which is the
location where the twofold

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rotation pierces the
mirror plane.

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And what this is going to do is
to change the sense of all

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three coordinates.

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This is going to take the
coordinates XYZ of object

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number 1 and change them into
minus x, minus y, minus c.

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And what we're doing is
inverting the motif, turning

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it inside out as it were,
into an enantiomorph.

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And we have discovered as soon
as we combine a pi with a

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perpendicular reflection plane,
a new operation which

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we'll call in words inversion.

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And the symbol that used to
describe this is a one with a

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bar over it, and we'll
see why later on.

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But this is a onefold axis
with an inversion center

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sitting on it, and that's
what an inversion

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center by itself is.

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The implication is that they're
going to be other axes

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that we can abbreviate such as
3-bar, 4-bar, and so on.

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So this is analogous to a
situation that will come later

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where we really need
a new notation.

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So we've fell headlong over
a new type of symmetry

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operation, and we should
consider taking inversion and

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adding that to the
rotation axes by

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themselves as an extender.

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Obviously, if we take inversion
and add it to a

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mirror plane, we're
going to get A pi.

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If we take a twofold rotation,
add it perpendicular to a

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mirror plane, we
get inversion.

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If we put inversion and put it
on a 180-degree rotation,

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we'll get the mirror
plane back.

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So these things all permute
one to another.

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You may even remember some time
ago we asked in general

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terms when do two operations
permute

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without changing anything.

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And the answer is if these
operations to leave the locus

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of the other one alone.

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And the mirror plane obviously
leaves the locus of the

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rotation operation unchanged.

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The rotation operation spins the
mirror plane around in its

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own plane and doesn't
create a new axis.

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And the inversion center leaves
the mirror plane alone

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and takes the twofold
axis top to bottom.

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So those three operations,
inversion A pi, sigma are the

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three nontrivial operations
that exist in the space.

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The fourth one is the
identity operation.

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So here is the set of elements
in the group that

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we will call 2/m.

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00:13:06,920 --> 00:13:10,460
2/m implies three operations
inversion, a 180-degree

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rotation, reflection in a plane
perpendicular to the

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axis, and the identity
operation.

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And I'll leave it to yourself
for you to convince yourself

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that I can rotate and then
reflect or I can reflect and

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then invert.

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And all of these operations
do not create any new

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motifs in the set.

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The group multiplication
table, in other words,

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contains just the four
elements, 1,

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1-bar, A pi, and sigma.

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So the full pattern that
corresponds to 2/m consists of

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four objects.

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It'll be a fourth
one down here.

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The twofold axis tells you
have this fellow is

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related to this one.

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Inversion tells you how this
one is related to this one.

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00:14:01,880 --> 00:14:04,460
And the mirror plane tells you
how this one, number 1, is

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00:14:04,460 --> 00:14:05,870
related to number 4.

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00:14:05,870 --> 00:14:08,063
And the identity operation
tells you how 1

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is related to itself.

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00:14:09,490 --> 00:14:12,930
So, again, as we've seen in
the set of operations that

244
00:14:12,930 --> 00:14:16,220
constitute a group, there's a
one-to-one correspondence

245
00:14:16,220 --> 00:14:20,460
between the transformations that
are elements of the group

246
00:14:20,460 --> 00:14:22,760
and the number of objects
in the pattern.

247
00:14:30,500 --> 00:14:34,540
So we've made one combination,
and what we found from this is

248
00:14:34,540 --> 00:14:38,950
a new transformation inversion
that involves changing the

249
00:14:38,950 --> 00:14:43,350
sign of all the coordinates in
a space through a point which

250
00:14:43,350 --> 00:14:44,635
is called the inversion
center.

251
00:14:48,313 --> 00:14:49,563
Questions?

252
00:14:58,590 --> 00:15:05,470
If not, let me quickly rattle
off other Point

253
00:15:05,470 --> 00:15:07,840
Groups in this family.

254
00:15:07,840 --> 00:15:13,975
We could take the operation A
pi/2 in 90-degree rotation and

255
00:15:13,975 --> 00:15:18,135
add this perpendicular
to mirror plane.

256
00:15:22,240 --> 00:15:24,840
Let me now say something
that I've said

257
00:15:24,840 --> 00:15:27,260
again many times before.

258
00:15:27,260 --> 00:15:32,200
The pattern of objects that will
result is the pattern of

259
00:15:32,200 --> 00:15:37,270
objects that's produced
by the initial group--

260
00:15:37,270 --> 00:15:39,460
let's say a fourfold axis--

261
00:15:39,460 --> 00:15:41,590
repeated by the extender.

262
00:15:41,590 --> 00:15:45,570
And the operation sigma
perpendicular is the extender.

263
00:15:48,430 --> 00:15:52,530
So the pattern, without making
any big deal about it, is

264
00:15:52,530 --> 00:15:57,340
going to look like this square
of objects reflected down

265
00:15:57,340 --> 00:15:59,390
below the mirror plane.

266
00:15:59,390 --> 00:16:00,840
So it'll be one going
down like this.

267
00:16:00,840 --> 00:16:01,620
One like this.

268
00:16:01,620 --> 00:16:03,300
One like this.

269
00:16:03,300 --> 00:16:06,810
The operation A pi sits
perpendicular to the operation

270
00:16:06,810 --> 00:16:10,360
of reflection, so there will
be an inversion center that

271
00:16:10,360 --> 00:16:13,590
arises at the point
of intersection.

272
00:16:13,590 --> 00:16:17,260
And indeed the square above
can be inverted

273
00:16:17,260 --> 00:16:20,030
through this point--

274
00:16:20,030 --> 00:16:22,350
little hasty repairs there--

275
00:16:22,350 --> 00:16:26,690
and every one up above gets
inverted down to an

276
00:16:26,690 --> 00:16:28,210
enantiomorph below.

277
00:16:28,210 --> 00:16:31,400
So all of these guys up on
top are right handed.

278
00:16:31,400 --> 00:16:35,820
All these guys down below
are left handed.

279
00:16:35,820 --> 00:16:37,340
So this is another group.

280
00:16:37,340 --> 00:16:44,070
This is 4/m in international
notation, a fourfold axis

281
00:16:44,070 --> 00:16:45,810
perpendicular to
a mirror plane.

282
00:16:48,710 --> 00:16:54,030
There is also, unfortunately,
a Schoenflies notation.

283
00:16:54,030 --> 00:16:57,650
The international notation tells
you what you've got, a

284
00:16:57,650 --> 00:16:59,770
twofold axis or a fourfold axis

285
00:16:59,770 --> 00:17:02,060
perpendicular to a mirror plane.

286
00:17:02,060 --> 00:17:05,490
The Schoenflies notation tells
you how you derived it.

287
00:17:05,490 --> 00:17:11,079
And the symbol for a twofold
axis is C2, so this is a

288
00:17:11,079 --> 00:17:13,180
twofold axis.

289
00:17:13,180 --> 00:17:18,930
And what we did was to add an
extender consisting of a

290
00:17:18,930 --> 00:17:20,609
horizontal mirror plane.

291
00:17:20,609 --> 00:17:25,630
Schoenflies calls this one
C2h; Group C2, which is a

292
00:17:25,630 --> 00:17:33,695
twofold axis; a horizontal
m is the extender.

293
00:17:36,900 --> 00:17:39,820
So Schoenflies tells you
how you make it.

294
00:17:39,820 --> 00:17:42,040
The international notation
tells you what

295
00:17:42,040 --> 00:17:45,280
you get as a result.

296
00:17:45,280 --> 00:17:49,200
Schoenflies notation here would
be C4, that's the symbol

297
00:17:49,200 --> 00:17:53,763
for a fourfold axis, and
the extender is an h.

298
00:17:57,840 --> 00:18:00,480
And then without making any big
fuss about it, if I do the

299
00:18:00,480 --> 00:18:06,510
same with a sixfold axis, I will
have six objects related

300
00:18:06,510 --> 00:18:09,730
by a sixfold rotation axis.

301
00:18:09,730 --> 00:18:16,010
If I take that sixfold axis
and put a mirror plane

302
00:18:16,010 --> 00:18:20,960
perpendicular to it, these will
be reflected down to a

303
00:18:20,960 --> 00:18:25,400
hexagon of enantiomorph
equidistant

304
00:18:25,400 --> 00:18:26,650
below the mirror plane.

305
00:18:36,040 --> 00:18:38,100
That's not terribly good, but
it's not terribly bad either.

306
00:18:38,100 --> 00:18:43,693
So this would be called 6/m,
Schoenflies notation C6h.

307
00:18:47,080 --> 00:18:53,240
And the operation A pi exists in
a sixfold axis, so there is

308
00:18:53,240 --> 00:18:56,690
an inversion center but also
arises as a new symmetry

309
00:18:56,690 --> 00:18:58,493
element at the point
of intersection.

310
00:19:03,160 --> 00:19:05,750
So with reckless abandon, you
can continue on here and

311
00:19:05,750 --> 00:19:10,360
derive noncrystallographic
groups for all the even-fold

312
00:19:10,360 --> 00:19:16,740
axes and derive an 8/m
and a 12/m and 16/m.

313
00:19:16,740 --> 00:19:17,760
Lovely symmetries.

314
00:19:17,760 --> 00:19:21,240
I wouldn't want to draw them,
but they're still symmetries.

315
00:19:21,240 --> 00:19:24,210
They all have an inversion
center in them, but they're

316
00:19:24,210 --> 00:19:25,340
noncrystallographic.

317
00:19:25,340 --> 00:19:27,460
So we don't have to worry about
them for prism purposes.

318
00:19:32,910 --> 00:19:41,200
I left one out because it
introduces a complication that

319
00:19:41,200 --> 00:19:43,360
is kind of curious
and interesting.

320
00:19:43,360 --> 00:19:45,040
Any questions on what
we've done here?

321
00:19:45,040 --> 00:19:45,340
Yes?

322
00:19:45,340 --> 00:19:48,770
AUDIENCE: The 2mm, it's
just the same--

323
00:19:48,770 --> 00:19:51,480
Schoenflies notation
is just C2--

324
00:19:51,480 --> 00:19:53,810
PROFESSOR: Schoenflies notation
for three dimensions

325
00:19:53,810 --> 00:19:55,880
is exactly the same as in two.

326
00:19:55,880 --> 00:20:01,540
So the three-dimensional version
where this extends in

327
00:20:01,540 --> 00:20:10,350
a direction that is normal to
the two-dimensional space of

328
00:20:10,350 --> 00:20:11,080
our two dimensions.

329
00:20:11,080 --> 00:20:13,090
Two dimensions it was this.

330
00:20:13,090 --> 00:20:16,220
Now just imagine them coming out
of the blackboard at you.

331
00:20:16,220 --> 00:20:17,970
The symbol for this
one was 2mm.

332
00:20:20,550 --> 00:20:26,740
The symbol for this one is
also 2mm, the same thing.

333
00:20:26,740 --> 00:20:29,590
The mirror plane is a vertical
mirror plane.

334
00:20:32,730 --> 00:20:35,510
So the Schoenflies notation is
exactly the same as what we

335
00:20:35,510 --> 00:20:37,580
used per two dimensions.

336
00:20:37,580 --> 00:20:38,830
It's called C2v.

337
00:20:42,205 --> 00:20:45,190
We've added a vertical
mirror plane.

338
00:20:45,190 --> 00:20:47,090
And again, horizontal
and vertical.

339
00:20:47,090 --> 00:20:50,300
Horizontal is horizontal
with respect to the

340
00:20:50,300 --> 00:20:52,470
axis of higher symmetry.

341
00:20:52,470 --> 00:20:56,750
Vertical is vertical with
respect to the two-dimensional

342
00:20:56,750 --> 00:21:00,240
space of the two-dimensional
Point Group, parallel to the

343
00:21:00,240 --> 00:21:01,800
axis in three dimensions.

344
00:21:01,800 --> 00:21:03,455
So that's the vertical
indication.

345
00:21:07,120 --> 00:21:10,100
So let's, though, tuck that
away for future reference.

346
00:21:10,100 --> 00:21:13,340
We've got two kinds
of extenders.

347
00:21:13,340 --> 00:21:18,060
We've got a horizontal mirror
plane, and we've got a

348
00:21:18,060 --> 00:21:23,570
vertical mirror plane, and these
are extenders that we

349
00:21:23,570 --> 00:21:25,760
should consider adding.

350
00:21:25,760 --> 00:21:31,520
So we've taken care of
2/m, 4/m and 6/m.

351
00:21:31,520 --> 00:21:38,420
2mm, 3m, 4mm, and 6mm are just
the extensions into a third

352
00:21:38,420 --> 00:21:41,570
dimension of what we've seen
and come to love in the

353
00:21:41,570 --> 00:21:42,820
two-dimensional space.

354
00:21:45,990 --> 00:21:49,050
Let me now turn to the
threefold axis.

355
00:21:49,050 --> 00:21:50,920
And this is a curious one.

356
00:21:50,920 --> 00:21:56,030
Threefold axes require fewer
symbols to indicate the

357
00:21:56,030 --> 00:21:58,700
vertical mirror planes
because there's only

358
00:21:58,700 --> 00:22:00,890
one independent one.

359
00:22:00,890 --> 00:22:02,760
But let's see what
would happen.

360
00:22:02,760 --> 00:22:06,740
And now I'm not going to attempt
to draw these in three

361
00:22:06,740 --> 00:22:07,710
dimensions anymore.

362
00:22:07,710 --> 00:22:11,140
I'm going to use a stereographic
projection.

363
00:22:11,140 --> 00:22:15,700
And what I'll do is use a solid
dot for a point that's

364
00:22:15,700 --> 00:22:20,690
up above the equatorial plane
and an open dot for one that's

365
00:22:20,690 --> 00:22:24,250
down below the equatorial
plane.

366
00:22:24,250 --> 00:22:30,260
So my stereographic projection
of 4mm would look like.

367
00:22:30,260 --> 00:22:31,780
This is the fourfold axis.

368
00:22:31,780 --> 00:22:33,770
This is the mirror plane.

369
00:22:33,770 --> 00:22:40,370
And I've got one up that gets
reproduced by the axis to give

370
00:22:40,370 --> 00:22:41,730
me a set of four.

371
00:22:41,730 --> 00:22:44,450
All of these are, let's
say, right handed.

372
00:22:44,450 --> 00:22:48,490
And then directly below them
is another set of four

373
00:22:48,490 --> 00:22:53,750
repeated by reflection, and
these are all left handed.

374
00:22:53,750 --> 00:22:57,270
And then there's an inversion
center at the point of

375
00:22:57,270 --> 00:22:58,040
intersection.

376
00:22:58,040 --> 00:23:02,500
And I'll indicate that by the
little open circle sitting

377
00:23:02,500 --> 00:23:04,120
right on the fourfold axis.

378
00:23:04,120 --> 00:23:07,310
So there is a projection
of what 4/m looks like.

379
00:23:10,120 --> 00:23:15,740
So let me now do the same thing
for a threefold axis.

380
00:23:15,740 --> 00:23:22,050
And I'll add to the triangle
of points that a threefold

381
00:23:22,050 --> 00:23:24,460
axis would generate.

382
00:23:24,460 --> 00:23:26,800
So these guys are all
of one chirality.

383
00:23:26,800 --> 00:23:28,050
Let's say right handed.

384
00:23:36,500 --> 00:23:41,170
Then I'll reflect them down,
and I'll get three objects

385
00:23:41,170 --> 00:23:42,420
that are down.

386
00:23:46,850 --> 00:23:48,935
And that's what 3/m
looks like.

387
00:23:54,040 --> 00:23:57,360
Is there an inversion
center here?

388
00:23:57,360 --> 00:23:58,456
No.

389
00:23:58,456 --> 00:24:05,440
No, because the operation
of A pi is missing.

390
00:24:05,440 --> 00:24:08,840
And it was the horizontal mirror
plane combined with A

391
00:24:08,840 --> 00:24:15,620
pi that gave us the inversion
center with all of the even

392
00:24:15,620 --> 00:24:17,340
rotation axis.

393
00:24:17,340 --> 00:24:20,980
So one of the things we have to
say here is that there is

394
00:24:20,980 --> 00:24:30,490
no 1-bar that's present, which
means in this instance, unlike

395
00:24:30,490 --> 00:24:34,200
the other ones, we have
another option.

396
00:24:34,200 --> 00:24:39,730
So we can use the operation of
inversion as an extender too.

397
00:24:48,920 --> 00:24:51,430
So we're going to get another
group out of the threefold

398
00:24:51,430 --> 00:24:53,080
axis besides this one.

399
00:24:53,080 --> 00:25:00,060
And this one we will
name 3/m or C3h

400
00:25:00,060 --> 00:25:01,310
in Schoenflies notation.

401
00:25:07,350 --> 00:25:12,150
They're six objects here, so
there should be six operations

402
00:25:12,150 --> 00:25:13,420
in the group.

403
00:25:13,420 --> 00:25:18,590
So let me number these guys
up on top as number 1,

404
00:25:18,590 --> 00:25:22,190
number 2, number 3.

405
00:25:22,190 --> 00:25:26,400
And 1 is related to itself
by inversion.

406
00:25:26,400 --> 00:25:29,770
There's an operation A 2 pi/3.

407
00:25:29,770 --> 00:25:32,950
And that tells me how the one
that's up is related to the

408
00:25:32,950 --> 00:25:36,260
second one that's up and how
the left-handed one that's

409
00:25:36,260 --> 00:25:38,900
down is related to the
one that's directly

410
00:25:38,900 --> 00:25:41,570
below number 2.

411
00:25:41,570 --> 00:25:46,250
There is an operation
A 4 pi/3.

412
00:25:46,250 --> 00:25:50,970
And that's the same as saying
8 minus 2 pi/3.

413
00:25:50,970 --> 00:25:56,220
And that tells us how the things
that are separated by

414
00:25:56,220 --> 00:25:59,370
240 degrees are related,
both up and down.

415
00:26:06,950 --> 00:26:14,490
I know how 3 up is related to 3
down and how 1 up is related

416
00:26:14,490 --> 00:26:18,480
to 1 down and 2 up is
related to 2 down.

417
00:26:18,480 --> 00:26:24,860
This is all with the horizontal
mirror plane, which

418
00:26:24,860 --> 00:26:26,110
I'll call sigma h.

419
00:26:29,450 --> 00:26:32,540
That is a total of one,
two, three, four--

420
00:26:32,540 --> 00:26:36,730
whoops-- one, two, three
four operations.

421
00:26:36,730 --> 00:26:37,980
I need six.

422
00:26:40,590 --> 00:26:49,430
Let's ask how is this one that's
up related to this one

423
00:26:49,430 --> 00:26:50,680
that's down?

424
00:26:54,550 --> 00:26:58,310
I just got rotation operations
and reflection.

425
00:26:58,310 --> 00:27:02,870
The only way I can get from this
one number 1 to this one

426
00:27:02,870 --> 00:27:10,750
number 3 left that's down is
to take two steps to do it.

427
00:27:10,750 --> 00:27:14,590
I can't get from this one up
here to this one down here

428
00:27:14,590 --> 00:27:24,805
unless I rotate 60 degrees
and then invert.

429
00:27:29,820 --> 00:27:32,530
If I move over to 3-bar.

430
00:27:37,950 --> 00:27:45,225
This is A 2 pi/3 with 1-bar
as an extender.

431
00:27:51,960 --> 00:27:56,540
The pattern would, again,
look like what are

432
00:27:56,540 --> 00:27:59,660
threefold axis does.

433
00:27:59,660 --> 00:28:04,080
But then if I repeat this set of
three by inversion, the two

434
00:28:04,080 --> 00:28:07,410
triangles above and
below are skewed.

435
00:28:07,410 --> 00:28:10,060
The ones down below
are enantiomorphs.

436
00:28:10,060 --> 00:28:13,800
The three that are up are
of opposite chirality.

437
00:28:13,800 --> 00:28:20,310
And this is a new
type of pattern.

438
00:28:20,310 --> 00:28:24,850
And in international notation,
what do we call this?

439
00:28:24,850 --> 00:28:26,290
It's a threefold axis.

440
00:28:26,290 --> 00:28:32,730
But how do we indicate a
symbol for three with

441
00:28:32,730 --> 00:28:36,420
inversion sitting on it?

442
00:28:36,420 --> 00:28:41,090
Let's ask if we know how each of
these objects is related to

443
00:28:41,090 --> 00:28:42,350
each of the other.

444
00:28:42,350 --> 00:28:53,040
So here's 1, 2, and 3; 1 goes to
2 by A 2 pi/3; 1 goes to 3

445
00:28:53,040 --> 00:28:55,130
by A minus 2 pi/3.

446
00:29:03,490 --> 00:29:06,350
Let's put some numbers on here
for the ones down below.

447
00:29:06,350 --> 00:29:13,485
Let's call them 4, 5, and 6;
1 goes to 6 by an version.

448
00:29:19,540 --> 00:29:24,590
How do I get from 1 up
to 4 that's down?

449
00:29:29,210 --> 00:29:33,160
I can do that only by
taking two steps.

450
00:29:33,160 --> 00:29:36,520
Rotate 1 from here
to number 2.

451
00:29:36,520 --> 00:29:38,450
Don't yet put it down.

452
00:29:38,450 --> 00:29:41,000
First invert it.

453
00:29:41,000 --> 00:29:48,990
So 1 to 4 involves the operation
A 2 pi/3 followed

454
00:29:48,990 --> 00:29:50,240
immediately by inversion.

455
00:29:53,960 --> 00:30:05,530
And I go from 1 up to 5 down by
doing the operation A minus

456
00:30:05,530 --> 00:30:09,240
2 pi/3 followed immediately
by inversion.

457
00:30:11,800 --> 00:30:16,640
And then 1 goes to
1 and itself by

458
00:30:16,640 --> 00:30:18,020
the identity operation.

459
00:30:18,020 --> 00:30:21,170
So I have six objects, one, two,
three, four, five, six

460
00:30:21,170 --> 00:30:22,810
operations.

461
00:30:22,810 --> 00:30:23,140
Yes?

462
00:30:23,140 --> 00:30:26,122
AUDIENCE: In the cases where
you're rotating and inverting,

463
00:30:26,122 --> 00:30:30,098
does it matter which
way to the other?

464
00:30:30,098 --> 00:30:32,930
PROFESSOR: No, it shouldn't
because they

465
00:30:32,930 --> 00:30:33,940
leave each other alone.

466
00:30:33,940 --> 00:30:37,085
So I can rotate from here
to here and invert.

467
00:30:37,085 --> 00:30:40,210
Or I can invert from here
to here and then rotate.

468
00:30:40,210 --> 00:30:41,500
It's the same transformation.

469
00:30:41,500 --> 00:30:46,100
Again, they permute if the two
loci of the two operations

470
00:30:46,100 --> 00:30:50,050
leave the other locus alone.

471
00:30:50,050 --> 00:30:52,640
Maybe the enormity of what
we've shown here

472
00:30:52,640 --> 00:30:54,320
has not sunk in.

473
00:30:54,320 --> 00:30:58,090
This is a new two-step
operation.

474
00:31:06,040 --> 00:31:09,220
We can't describe it any simpler
than saying, rotate

475
00:31:09,220 --> 00:31:12,200
and not put it down yet,
follow up by inversion.

476
00:31:12,200 --> 00:31:12,480
Yes, sir?

477
00:31:12,480 --> 00:31:13,896
AUDIENCE: Couldn't we express
that in another way by sort of

478
00:31:13,896 --> 00:31:16,256
extending the three-directional
glide plane

479
00:31:16,256 --> 00:31:21,692
by saying invert, then transform
by some vector

480
00:31:21,692 --> 00:31:24,596
that's parallel to
the glide plane?

481
00:31:24,596 --> 00:31:28,360
PROFESSOR: Maybe they do in
space group, but as soon as we

482
00:31:28,360 --> 00:31:32,460
introduce a glide plane, you've
got an operation that's

483
00:31:32,460 --> 00:31:34,525
half a lattice translation.

484
00:31:34,525 --> 00:31:37,010
And that means you've got to
have a lattice translation and

485
00:31:37,010 --> 00:31:38,030
double the lattice
translation, so--

486
00:31:38,030 --> 00:31:38,990
AUDIENCE: Oh, we don't have
to worry about that.

487
00:31:38,990 --> 00:31:41,180
PROFESSOR: --when we're in
a space group, yeah.

488
00:31:41,180 --> 00:31:42,080
That could be present.

489
00:31:42,080 --> 00:31:44,550
But not for a point group
because the ground rules are

490
00:31:44,550 --> 00:31:47,380
at least one point
has to remain

491
00:31:47,380 --> 00:31:48,815
immutably fixed in space.

492
00:31:57,570 --> 00:32:00,530
So this is a two-step operation,
and what we're

493
00:32:00,530 --> 00:32:02,470
going to call it is
rotoinversion.

494
00:32:07,040 --> 00:32:12,930
It consists of as a first step
an operation by rotating alpha

495
00:32:12,930 --> 00:32:16,720
from 0.1 to a virtual
point number 2.

496
00:32:16,720 --> 00:32:21,510
But before you put it down, you
will invert it to a new

497
00:32:21,510 --> 00:32:25,035
object number 2 which is
of opposite chirality.

498
00:32:28,300 --> 00:32:33,630
So here then are the operation
of the group that results when

499
00:32:33,630 --> 00:32:37,900
you combine a threefold rotation
axis and add to it an

500
00:32:37,900 --> 00:32:45,210
version center as an extended;
A 2 pi/3; A minus 2 pi/3; a

501
00:32:45,210 --> 00:32:50,960
rotoinversion operation through
2 pi/3 and then

502
00:32:50,960 --> 00:32:56,920
inverting; a rotoinversion
operation of A minus 2 pi/3

503
00:32:56,920 --> 00:32:58,280
followed by inversion.

504
00:32:58,280 --> 00:33:01,390
And the symbol that is used to
represent that new two-step

505
00:33:01,390 --> 00:33:06,730
operation is putting a bar
over the top of the

506
00:33:06,730 --> 00:33:08,940
symbol for the axis.

507
00:33:08,940 --> 00:33:11,260
And then, finally, we have
inversion by itself.

508
00:33:11,260 --> 00:33:12,840
So that's a group rank 6.

509
00:33:16,700 --> 00:33:26,790
The Schoenflies notation is
called C3i because we got this

510
00:33:26,790 --> 00:33:31,830
group by adding an inversion
to C3, the threefold axis.

511
00:33:31,830 --> 00:33:37,200
The international notation picks
up on putting a bar over

512
00:33:37,200 --> 00:33:40,240
an axis to indicate a
rotoinversion operation.

513
00:33:40,240 --> 00:33:43,700
So this is called 3-bar in the
international notation.

514
00:33:43,700 --> 00:33:47,300
So there is a new group,
and it is an oddball.

515
00:33:47,300 --> 00:33:51,070
It sort of stands alone
from the other

516
00:33:51,070 --> 00:33:53,135
groups of the form C3h.

517
00:34:26,710 --> 00:34:34,500
This we derived by using the
rotation of the threefold axis

518
00:34:34,500 --> 00:34:37,050
and adding 1-bar
as an extender.

519
00:34:37,050 --> 00:34:38,600
So there's no mirror
plane in this.

520
00:34:38,600 --> 00:34:40,290
AUDIENCE: That's not
the same as--

521
00:34:40,290 --> 00:34:43,239
PROFESSOR: That's the
same as 3/m, no.

522
00:34:43,239 --> 00:34:46,210
3/m is C3h.

523
00:34:46,210 --> 00:34:52,389
3-bar is C3i, different extender
added to the same

524
00:34:52,389 --> 00:34:53,639
subgroup 3C.

525
00:35:00,407 --> 00:35:03,500
AUDIENCE: What was the
definition of 3/m?

526
00:35:03,500 --> 00:35:05,910
PROFESSOR: Oh, we never
really finished that.

527
00:35:05,910 --> 00:35:10,090
That if we need the six
operations that control the

528
00:35:10,090 --> 00:35:13,050
group, we'll have a sixfold
rotoinversion axis.

529
00:35:13,050 --> 00:35:17,490
But this pattern looks just like
the triangle produced by

530
00:35:17,490 --> 00:35:23,480
3, and we add an reflection
operation as an inversion, and

531
00:35:23,480 --> 00:35:25,140
the 3 go down.

532
00:35:25,140 --> 00:35:28,070
So if we ask how every one of
the top is related to one

533
00:35:28,070 --> 00:35:31,550
underneath, that's by this
horizontal mirror plane.

534
00:35:31,550 --> 00:35:34,980
If I want to know how I get
from this one that's up to

535
00:35:34,980 --> 00:35:38,800
this one that's down, then I've
got to rotate through 60

536
00:35:38,800 --> 00:35:40,590
degrees and invert.

537
00:35:40,590 --> 00:35:42,460
So that would be a rotoinversion
operation.

538
00:35:57,370 --> 00:36:04,530
Let us to extend this idea of
a rotoinversion operation.

539
00:36:04,530 --> 00:36:12,670
And we would find this
eventually in adding different

540
00:36:12,670 --> 00:36:16,930
extenders and falling headlong
over this rotoinversion

541
00:36:16,930 --> 00:36:20,360
operation as we did
here with 3-bar.

542
00:36:20,360 --> 00:36:29,440
But let me in this case start by
defining a 4-bar operation.

543
00:36:32,560 --> 00:36:37,030
And this would contain the
operation A pi/2 followed

544
00:36:37,030 --> 00:36:41,970
immediately by inversion.

545
00:36:41,970 --> 00:36:45,770
And we'll call this
step A pi/2-bar.

546
00:36:49,720 --> 00:36:52,280
So let's try to do that
and see what we get.

547
00:36:55,830 --> 00:36:59,790
Start with a first
point, number 1.

548
00:36:59,790 --> 00:37:02,420
And that's up, so it's
a solid dot.

549
00:37:02,420 --> 00:37:04,980
And let's say it's
right handed.

550
00:37:04,980 --> 00:37:10,670
If we combine that with a
rotation of 90 degrees.

551
00:37:10,670 --> 00:37:11,710
Not yet put it down.

552
00:37:11,710 --> 00:37:13,170
That's a virtual motif.

553
00:37:13,170 --> 00:37:16,160
Before putting it down,
we inverted it.

554
00:37:16,160 --> 00:37:21,650
We would get one that's down,
and it would be left handed.

555
00:37:21,650 --> 00:37:23,060
Do the operation again.

556
00:37:23,060 --> 00:37:25,200
I'll put the little
tadpole inside.

557
00:37:25,200 --> 00:37:27,360
Do the operation again.

558
00:37:27,360 --> 00:37:29,530
Rotate 90 degrees and invert.

559
00:37:29,530 --> 00:37:31,680
We're back up again.

560
00:37:31,680 --> 00:37:32,780
So this was 1.

561
00:37:32,780 --> 00:37:34,350
This is 2.

562
00:37:34,350 --> 00:37:36,080
This was 3.

563
00:37:36,080 --> 00:37:37,210
And that's up.

564
00:37:37,210 --> 00:37:38,900
Do the operation again.

565
00:37:38,900 --> 00:37:40,515
Rotate and invert.

566
00:37:40,515 --> 00:37:43,790
And here is number
4, and it's down.

567
00:37:43,790 --> 00:37:46,030
Do it a fifth time, and we're
back to where we started.

568
00:37:48,750 --> 00:37:51,790
So this is a crazy pattern.

569
00:37:51,790 --> 00:37:55,220
It's a pair of objects that's
up and a pair of objects

570
00:37:55,220 --> 00:37:57,750
that's down.

571
00:37:57,750 --> 00:38:00,260
So there's a twofold
axis in there.

572
00:38:00,260 --> 00:38:04,380
That twofold axis A pi leave
the pattern invariant.

573
00:38:04,380 --> 00:38:07,460
But there is no way of
specifying the relation

574
00:38:07,460 --> 00:38:11,500
between the two that are up
and the two that are down

575
00:38:11,500 --> 00:38:15,450
other than doing this two-step
process of rotating 90 degrees

576
00:38:15,450 --> 00:38:17,360
and then inverting.

577
00:38:17,360 --> 00:38:21,680
So there is actually in this
pattern a new type of

578
00:38:21,680 --> 00:38:26,730
operation analogous to 3-bar,
and it's called a 4-bar axis.

579
00:38:26,730 --> 00:38:30,870
And it's indicated geometrically
by drawing a

580
00:38:30,870 --> 00:38:33,690
square because there's
a 90-degree

581
00:38:33,690 --> 00:38:36,290
angular symmetry to this.

582
00:38:36,290 --> 00:38:41,740
But a twofold axis inscribed
inside of it because this is a

583
00:38:41,740 --> 00:38:43,360
pattern that has a
twofold symmetry.

584
00:38:45,930 --> 00:38:54,410
So something that has this
symmetry is the symmetry of a

585
00:38:54,410 --> 00:38:55,660
tetrahedron.

586
00:38:58,340 --> 00:39:02,310
And if we draw a line from the
upper edge to the lower edge,

587
00:39:02,310 --> 00:39:05,785
this is the locus
of a 4-bar axis.

588
00:39:13,540 --> 00:39:19,250
International notation
this is called 4-bar.

589
00:39:19,250 --> 00:39:21,500
That's how we generated
the pattern.

590
00:39:21,500 --> 00:39:31,610
The Schoenflies notation is an
S, little bit of exotica.

591
00:39:31,610 --> 00:39:35,570
This geometric solid
is something

592
00:39:35,570 --> 00:39:36,820
that's called a sphenoid.

593
00:39:40,040 --> 00:39:41,790
And sphenoid.

594
00:39:41,790 --> 00:39:43,850
Is the Greek word for axe.

595
00:39:46,570 --> 00:39:52,050
And you can imagine a handle
put onto this thing, and it

596
00:39:52,050 --> 00:39:54,090
does look kind of like an axe.

597
00:39:54,090 --> 00:39:56,550
You could splits firewood
with a thing like that.

598
00:39:56,550 --> 00:40:00,070
It looks like a tetrahedron,
but in a tetrahedron, it's

599
00:40:00,070 --> 00:40:03,650
either elongated along the
4-bar axis are squished.

600
00:40:03,650 --> 00:40:07,090
It doesn't have to be regular.

601
00:40:07,090 --> 00:40:12,290
So this is called S4, and the
S stands for sphenoid.

602
00:40:17,600 --> 00:40:19,492
AUDIENCE: Is it part of
a regular tetrahedron?

603
00:40:19,492 --> 00:40:20,930
PROFESSOR: No, no.

604
00:40:20,930 --> 00:40:24,660
A regular tetrahedron would be
something where all of the

605
00:40:24,660 --> 00:40:26,940
edges had equal length.

606
00:40:26,940 --> 00:40:32,050
And what we're doing is taking
one edge and the edge that's

607
00:40:32,050 --> 00:40:38,090
opposite it and either
stretching it or squishing it.

608
00:40:38,090 --> 00:40:41,050
So there are two edges.

609
00:40:41,050 --> 00:40:43,420
This one, and this one, which
are the same length.

610
00:40:43,420 --> 00:40:48,260
And then these four inclined
edges have a different length.

611
00:40:48,260 --> 00:40:50,330
It could be either elongated
or squished.

612
00:40:50,330 --> 00:40:51,895
But it's not a regular
tetrahedron.

613
00:40:54,460 --> 00:40:58,220
If those three distances were
equal, then geometrically it

614
00:40:58,220 --> 00:40:59,270
would be a tetrahedron.

615
00:40:59,270 --> 00:41:01,540
But strictly speaking, a
tetrahedron is not a

616
00:41:01,540 --> 00:41:06,980
tetrahedron, just as a square
prism with eight sides

617
00:41:06,980 --> 00:41:09,740
approximately equal can't claim
to be a cube unless

618
00:41:09,740 --> 00:41:12,620
there's symmetry present that
demands that this be true.

619
00:41:12,620 --> 00:41:17,920
In this case, the 4-bar requires
that these four edges

620
00:41:17,920 --> 00:41:21,590
inclined to the 4-bar axis
be of one length.

621
00:41:21,590 --> 00:41:23,450
And this have to have
the same length.

622
00:41:23,450 --> 00:41:26,570
But there's nothing that
constrains all six to the

623
00:41:26,570 --> 00:41:29,540
edges to be of identical
length.

624
00:41:29,540 --> 00:41:32,780
So it's not a tetrahedron,
so squished or deformed

625
00:41:32,780 --> 00:41:34,030
tetrahedron.

626
00:41:39,600 --> 00:41:47,200
So there is another two-step
symmetry element that we would

627
00:41:47,200 --> 00:41:51,630
not have been clever enough to
think of had we not discovered

628
00:41:51,630 --> 00:41:55,880
this sort of rotoinversion
operation when we added

629
00:41:55,880 --> 00:41:59,340
inversion to a threefold axis.

630
00:41:59,340 --> 00:42:03,960
A 3-bar axis is a step that's
present when you add inversion

631
00:42:03,960 --> 00:42:07,320
to a threefold axis.

632
00:42:07,320 --> 00:42:11,430
So 3-bar, what we call it for
short, is identical to a

633
00:42:11,430 --> 00:42:14,280
threefold axis plus inversion
sitting on it.

634
00:42:14,280 --> 00:42:20,390
A 4-bar is not equal to a
fourfold axis with inversion

635
00:42:20,390 --> 00:42:21,640
added to it.

636
00:42:23,790 --> 00:42:26,540
A 4-bar is something that you
cannot describe any more

637
00:42:26,540 --> 00:42:30,770
simply than saying there is a
two-step operation in there,

638
00:42:30,770 --> 00:42:33,450
and it's a group of rank 4.

639
00:42:39,390 --> 00:42:44,690
Let me finish by setting up the
task of going through this

640
00:42:44,690 --> 00:42:45,940
systematically.

641
00:42:50,540 --> 00:42:57,464
We have 11 axial combinations
1, 2, 3, 4, 6, 222, 32, 422,

642
00:42:57,464 --> 00:43:03,140
622, 23, and 432.

643
00:43:03,140 --> 00:43:05,930
So there are 11 of those.

644
00:43:05,930 --> 00:43:11,550
We want to examine as extenders
a vertical mirror

645
00:43:11,550 --> 00:43:17,690
plane that would be
one extender.

646
00:43:17,690 --> 00:43:19,830
We should add that to each
of these symmetries.

647
00:43:19,830 --> 00:43:21,510
We already done a
lot of these.

648
00:43:21,510 --> 00:43:28,190
We've done pretty
much up here.

649
00:43:28,190 --> 00:43:30,110
We could add a horizontal
mirror plane.

650
00:43:38,700 --> 00:43:44,710
Or we've encountered inversion
when we added a mirror plane

651
00:43:44,710 --> 00:43:46,820
perpendicular to an
even-fold axis.

652
00:43:46,820 --> 00:43:48,615
We could add inversion
as an extender.

653
00:43:51,810 --> 00:43:55,970
And to be complete, we should
add to our list of axes in

654
00:43:55,970 --> 00:43:59,700
quotation marks, the 4-bar axis,
having discovered it.

655
00:44:06,030 --> 00:44:08,710
And does that do it?

656
00:44:08,710 --> 00:44:11,940
Is there anything else we could
do to these axes that

657
00:44:11,940 --> 00:44:15,196
would leave them invariant?

658
00:44:15,196 --> 00:44:16,142
.

659
00:44:16,142 --> 00:44:17,392
AUDIENCE: What about 2-bar?

660
00:44:19,930 --> 00:44:25,300
PROFESSOR: 2-bar; 2-bar
would be rotate

661
00:44:25,300 --> 00:44:28,510
180 degrees and invert.

662
00:44:28,510 --> 00:44:35,310
So 2-bar is identical to a
horizontal mirror plane.

663
00:44:35,310 --> 00:44:36,560
So that's nothing new.

664
00:44:39,410 --> 00:44:42,035
We're already running a
little over time so--

665
00:44:42,035 --> 00:44:42,250
Yeah?

666
00:44:42,250 --> 00:44:43,530
AUDIENCE: 3-bar?

667
00:44:43,530 --> 00:44:47,100
PROFESSOR: 3-bar is 3 plus
1, and we call at 3i, so

668
00:44:47,100 --> 00:44:50,180
this one down here.

669
00:44:50,180 --> 00:44:51,520
This is 3-bar.

670
00:44:51,520 --> 00:44:53,800
We describe it for short as
that, but it really is a

671
00:44:53,800 --> 00:44:55,350
threefold axis with
an inversion

672
00:44:55,350 --> 00:44:56,960
center sitting on it.

673
00:44:56,960 --> 00:45:00,070
This thing is distinct because
it's not a fourfold axis with

674
00:45:00,070 --> 00:45:02,130
inversion sitting on it.

675
00:45:02,130 --> 00:45:04,190
A 4-bar is a 4-bar is a 4-bar.

676
00:45:04,190 --> 00:45:09,050
You can't decompose it as a
twofold axis is a subgroup.

677
00:45:09,050 --> 00:45:12,040
That's only half the story.

678
00:45:12,040 --> 00:45:16,510
I don't want to keep you
anxious, not anxious to find

679
00:45:16,510 --> 00:45:18,175
out what the answer is
but anxious to get on

680
00:45:18,175 --> 00:45:19,350
your way and go home.

681
00:45:19,350 --> 00:45:29,170
So let me submit that when we
have more than one rotation

682
00:45:29,170 --> 00:45:49,440
axis, such as 222 or as in 32,
if we put the mirror plane in

683
00:45:49,440 --> 00:45:56,510
normal to the principal axis,
we'll call that a horizontal

684
00:45:56,510 --> 00:45:59,084
mirror plane.

685
00:45:59,084 --> 00:46:04,930
If we add the mirror plane
through the principal axis, we

686
00:46:04,930 --> 00:46:08,550
could pass it through the
threefold axis and the twofold

687
00:46:08,550 --> 00:46:12,100
axis, pass it through the
vertical twofold axis and the

688
00:46:12,100 --> 00:46:13,770
horizontal twofold axis.

689
00:46:13,770 --> 00:46:17,330
We will call this a vertical
mirror plane.

690
00:46:17,330 --> 00:46:20,260
And that's all we could say for
a single axis, the mirror

691
00:46:20,260 --> 00:46:23,200
plane was perpendicular to the
axis or passed through it.

692
00:46:23,200 --> 00:46:26,930
But when there's more than one
axis, another thing we could

693
00:46:26,930 --> 00:46:33,310
do would be to snake the mirror
plane in between the

694
00:46:33,310 --> 00:46:34,860
twofold axis.

695
00:46:34,860 --> 00:46:37,370
In that case, this
twofold axis gets

696
00:46:37,370 --> 00:46:39,100
reflected into this one.

697
00:46:39,100 --> 00:46:42,750
But I haven't created
any new axes.

698
00:46:42,750 --> 00:46:44,980
So that is going to leave
the results of Euler's

699
00:46:44,980 --> 00:46:46,700
construction unchanged.

700
00:46:46,700 --> 00:46:50,500
I can't similarly put a vertical
mirror plane through

701
00:46:50,500 --> 00:46:54,800
this first twofold axis but
in between the other two.

702
00:46:54,800 --> 00:46:58,740
In that case, this is no longer
222 because these two

703
00:46:58,740 --> 00:47:01,220
mirror planes are equivalent
by reflection, so I want to

704
00:47:01,220 --> 00:47:02,820
drop that at very least.

705
00:47:02,820 --> 00:47:04,440
So in any case, without
belaboring

706
00:47:04,440 --> 00:47:05,860
the point, it's late.

707
00:47:05,860 --> 00:47:12,290
I could do for each of the
groups that involved more than

708
00:47:12,290 --> 00:47:19,885
one axis I could add a diagonal
mirror plane, or I

709
00:47:19,885 --> 00:47:21,500
should try to add a diagonal
mirror planes.

710
00:47:21,500 --> 00:47:26,520
And this means interleaved
between axes that are present

711
00:47:26,520 --> 00:47:29,010
in these combinations,
added such that

712
00:47:29,010 --> 00:47:32,370
no new axis is created.

713
00:47:32,370 --> 00:47:35,930
But the addition clearly is
going to be a new disposition

714
00:47:35,930 --> 00:47:38,030
of symmetry elements
arranged in any

715
00:47:38,030 --> 00:47:39,890
different fashion and space.

716
00:47:42,400 --> 00:47:43,620
So the game's afoot.

717
00:47:43,620 --> 00:47:45,920
This is what remains
to be done next.

718
00:47:48,540 --> 00:47:52,050
What I'll do for next time is
prepare a chart that looks

719
00:47:52,050 --> 00:47:55,680
like this that has the results
of all of the unique

720
00:47:55,680 --> 00:48:00,530
combinations shown and then
hand out pictures of

721
00:48:00,530 --> 00:48:02,880
stereographic projections
of all the results.

722
00:48:02,880 --> 00:48:06,910
I think once you know how to
play the game to go through

723
00:48:06,910 --> 00:48:11,020
and do every single
one in detail is

724
00:48:11,020 --> 00:48:12,400
probably not necessary.

725
00:48:12,400 --> 00:48:15,010
If you know how to do some of
them and you know all the

726
00:48:15,010 --> 00:48:19,930
tricks for adding extenders, you
could do it if you had to.

727
00:48:19,930 --> 00:48:20,270
All right.

728
00:48:20,270 --> 00:48:24,350
So, again, sorry we started late
and sorry that we last

729
00:48:24,350 --> 00:48:25,600
long as well.