PROFESSOR: We are thundering rapidly to the
midpoint of the term.
I think a week from this coming Thursday will be
exactly halfway, and the halfway point is when I
usually try, give or take a day, to change gears very
abruptly and start looking at the symmetry of physical
properties and tensors and actual discussion of physical
properties that are anisotropic.
So we'll be finishing up our discussion of
symmetry fairly quickly.
We still have, however, some major derivations to perform,
and we'll get into one of these today.
Last time we had begun the process of deriving what are
called the point groups as opposed to the plane groups.
The plane groups were combinations of symmetry with
a lattice that extended throughout a plane in space.
A point group we've seen in the form of two-dimensional
point groups.
We're now going to today derive the three-dimensional
point groups.
And, again, the name stems from the fact that these are
clusters of symmetry about a common point so that at very
least that one point in space stays put.
So, hence, point groups--
these are symmetries about a point.
And to put some embellishing adjectives on here, these will
be the three-dimensional crystallographic point groups.
Because there are an infinite number of point groups, we're
considering just those symmetries that can be
combined with a lattice and therefore are
permissible to crystals.
I'd like to point out in case you've not been making much
use of Buerger's book that we actually are getting back to
Buerger's treatment.
We'll do some of the derivations over the next few
days with a little bit more detail and with a slightly
different procedure than Buerger.
But the bases covered are the same.
So let me just point out where some of this material is in
Buerger's book.
Buerger does Euler's Construction, and he does this
in a once over lightly.
I think this is not the best part of the book.
This is on pages 35 to 43.
In particular, he sort of leaps ahead quickly and just
looks for one angle between rotation axes and doesn't get
the-- do specifically the ones that are 90 degrees and are
not as interesting.
Some of the combination theorems are in Chapter 6, and
there you'll find the statement that two reflections
at an angle mu is equivalent to a net rotation of mu,
except Buerger doesn't call them sigmas.
He calls them m's.
He uses the same symbol for individual operations and for
the symmetry of them, and I think that's a little untidy.
Also, the theorem that says that if we have a mirror plane
that's perpendicular to a twofold rotation axis, that
gives rise to inversion center at the point of intersection.
And then he launches into a rather condensed version of
the derivation of the point groups on pages 59 to 68.
Then, in Chapter 7, derives the two dimensional lattices
or the plane nets.
This is on page 69 to 83.
We've done that differently and much more exhaustively by
deriving the two-dimensional plane groups and the lattices
are part of the plane groups, and we've gotten them
automatically.
There is a place where in the derivation of the space
lattices, he's a little bit incomplete and actually does
something that's wrong A-ha--
I showed Buerger was wrong at one place in his book.
But anyway, this is where the material that we're covering
now is mentioned in Buerger's book.
Last time, we had hit a new surprise.
We had asked if we take the basic rotational symmetries
and add an inversion to something like three--
so, three is a subgroup, the operation of
inversion is an extender.
We found that there was a new operation that came up, and
this was a rotoinversion operation.
And then the resulting point group, which we called 3-bar--
the symbol itself, 3-bar, is a threefold rotoinversion.
And if we were not clever enough to invent a
rotoinversion operation, we would have stumbled headlong
over it in adding inversion as an extender
to a threefold axis.
And a rotoinversion axis is something that involves in
general rotating through some angle alpha to get from a
first object to a second object of the same handedness
and then not yet putting it down, first inverting it
through axis--
through a point on the axis--
to get a second object.
I shouldn't call this second because this is not rotation
with inversion simultaneously but a two step operation.
So we rotate from 1 to this virtual object also of same
handedness and then before putting it down, we invert it
to get number two, which is left handed.
And the operation of getting there in two steps but in one
repetition is the operation of rotoinversion.
That operation was one of the members of the group that
results when you add inversion as an extender to 3.
But, really, this was not something that was a new sort
of symmetry element.
It was an operation that came up, but we could regard the
point down that we've labeled 3-bar as simply a threefold
axis with an inversion center sitting on it.
But then we looked at another one.
And by anticipation, I said we ought to look at these.
And we looked at what happened when we took a fourfold
rotoinversion operation where we would take the first object
number one, rotate it through 90 degrees, and
not yet put it down.
We'd first invert it through a point on an axis, and we got
one that was 90 degrees away but pointing down in of
opposite handedness.
If we repeated that operation several times, we found that
we got two objects up above a plane normal to the rotation
part of the operation and passing through the point,
which we inverted.
So there's one right handed one up here, a second right
handed one up here, and then two down below, which were of
opposite chirality, number 3 and 4.
And that is an entirely new beast.
We have no way of describing the relation between these
four objects other than saying rotate, don't put it down,
first invert.
Rotate, don't put it down, first invert again.
So this is something entirely new.
It's a two step operation.
You cannot describe it any more simply than saying take
two steps to do the repetition in the same sense that the
glide plane that we discovered in the plane groups had a step
of translation followed immediately by reflection, and
that was a new sort of operation.
So here's another example of a two-step operation.
The symbol for this point group is 4-bar, and the symbol
for it-- if you want to indicate its locus, is, it's
got a squareness to it because these objects as mutually are
separated by 90 degrees, but it really has only a twofold
symmetry, so the symmetry for a 4-bar axis.
It's not really an axis at all.
It's a 4-bar point, because only a
point is left invariant.
But the symbol for a 4-bar "axis" is this square with a
little twofold axis inscribed in it.
Alright.
We ought to really, then, step back and ask how many
different rotoinversion axes there might be, and add these
to our bags of tricks.
And what I'm going to invite you to do is to examine this
systematically on a problem set which I will pass around,
along with several other problems for your
consideration, as well.
And having stumbled onto that combination, another thing we
might ask to broaden our bag of tricks still further is to
say is there such a thing as a rotoreflection axis?
And we called n-bar a rotoinversion.
I wrote a reflection axis is indicated by an n with this
little squiggle on top.
Anyone who has studied Spanish realizes that the proper name
for the squiggle is a tilde, T-I-L-D-E.
And this would be an operation where there's a locus in which
we'll perform the reflection part of the operation, so this
would be a 2 step operation that involves taking a first
one and rotating it to a virtual object number two but
not putting it down.
Before putting it down, we will
reflect in a planar locus.
So this is number two, and it will be of opposite chirality.
So that's another operation that involves rotation coupled
as one part of a step that involves R, the second of our
three translation free operations, namely reflection
and inversion.
So are these something we should consider?
What I've invited you to do is to draw out for n equals 1 to
8 the patterns produced by rotoinversion and the patterns
created by rotoreflection, and then see if you regard these
as new operations or whether they can be broken down into
the simultaneous presence of more than one of our old
friends such as 3-bar being a plain old threefold axis with
an inversion sitting on it.
The fact that I'm asking you take your time to do this
suggests that yes, you are going to find some
rotoinversion and rotoreflection axes, which are
inherently 2 step operations and can't be decomposed.
What I'd like to do now, then, is to systematically look at
the axial combinations.
These are of the form n one, two, three, four, and six and
the dihedral symmetries.
These are of the form n22, or just 32 in the case of the one
with the threefold axis.
And then the two cubic arrangements 23, twofold axis
out of the face normal to a cube, threefold axes out of
all of the body diagonals, and 432.
And what I'm going to do with you is to use these as
worksheets to guide the logic of what we're doing, and we
won't do every single one of them.
I think once we do a few, the general principle is clear and
the results there are before you.
But what we'll take, then, is the axes n, the axes n22, and
then the two cubic symmetries 23 and 432.
Then, finally, we found 4-bar as a different rotation
axis-like symmetry element, so I'll add that to the list.
Then consider what we can add to these symmetries as
extenders, and these are an additional element that we can
add to what is already a
self-contained nice little group.
We muck things up by introducing another operation,
and then we'll have to take combinations of that operation
with all the symmetry elements that are there in the parent
subgroup and see what new symmetry elements arise.
The ground rules in these additions are fairly simple
but nevertheless profound.
The addition of the extender cannot create any new symmetry
axes by its operation, and the reason for that should be
quite clear.
We obtained these combinations of rotation axes using Euler's
principle, and that was thorough and orderly,
systematic and complete.
These are the only arrangements of
crystallographic rotation axes that are possible.
So if we add an extender that creates, for example, a new
twofold axis in n22, it's either going to be something
that we already have in this list, and therefore not
interesting, or something that just cannot
constitute a group.
We'll take combinations of operations and we will never
find a closed finite set.
So if we add a mirror operation, a reflection
operation as an extender, the addition of the reflection
operation sigma must be either perpendicular to the axis.
And what that's going to do is just flip the top part of the
rotation axis and reflect it down to the bottom axis.
Nothing new was created.
And it could alternatively be parallel to the axis and
passing through it.
In that case, we've already seen the consequences of this
addition in deriving the two-dimensional point groups,
namely to 2mm, 3m, 4mm, and 6mm.
No new axis is created.
One of the additions that is something we mentioned last
time is that in the case of the dihedral groups where we
have an n-fold axis and then twofold axes
arranged in some fashion--
not in some arbitrary fashion.
It's going to be 2 pi over n times 1/2, depending on the
n-fold axis--
we could pass the mirror operation
through the twofold axis.
And that's what we'll call a vertical mirror plane.
But when there's more than one axis present, we could also
put the reflection operation diagonally in between the
twofold axes.
So we're going to refer to this as a diagonal mirror.
It's snaked in between the axes that are present.
So, perpendicular reflection operation as an extender, one
parallel to the axis and passing through it, or one
that is diagonal passing in between them.
And finally, that exhausts what we can do with
reflection, but we could add inversion.
And if we're not going create new axes, and we're going to
get a point group-- a finite set of objects--
the inversion center has to be on one of the single axis,
namely these groups n, or at the point of intersection, if
there's more than one.
So there's the job laid out for us.
And we've got two theorems to aid us in quickly finding the
new operations that arise--
in effect, taking a shortcut to establish the group
multiplication table, and the two theorems are as follows.
We saw that if you take a rotation operation A pi, that
takes a first object that's right handed and rotates it to
a second object that is also right handed.
Then follow that by reflection in a mirror plane that is
perpendicular to the axis.
The net effect of going from 1 to number 3, which is left
handed, is to create an inversion center at the point
of intersection.
So there's a theorem that we can write down once and for
all and say that A pi followed by reflection in a mirror
plane that's perpendicular to axis A has, as a net effect,
the operation of inversion.
And if this is not a twofold axis, it's something like a
fourfold axis or a sixfold axis, that contains the
operation A pi as part of the operations contained in that
axis, and that's going to create an
inversion center as well.
So for any evenfold axis, add a perpendicular mirror plane,
and automatically the inversion center comes up.
And we can permute the order of these operations if we
rotate by 180 degrees and then invert the net effect is
reflection in the perpendicular mirror plane.
So this is something that you permute around into three
different combinations.
Does the order of the operation make a difference?
And the answer is no, the order is not important.
For example, rotating and then reflecting is the same as
reflecting and then rotating it.
It went up at the same point, number 3.
Remember a fancy word for this when we define what's meant by
a group, that these groups are going to be what's called
Abelian, and an Abelian group is where any combination of
operations a followed by b is identical to b followed by a.
I told you that great joke among mathematicians, what is
purple and commutes?
And the answer is an Abelian grape.
Ha, ha, ha--
I think that's stupid, but it drives mathematicians into
guffaws of laughter.
A second theorem, and this is one that we've already seen in
two dimensions, is that if you take an operation A alpha,
pass a reflection operation sigma 1 through it, that to
the effect of that is going to be another mirror plane that
is alpha over 2 away from the first.
Notice I'm a little bit sloppy in talking about reflection
operation and mirror plane because if a reflection
operation is there, that's all I need to have to say that
this is the locus of a symmetry
point, a mirror plane.
So this is another one.
Do you think that this an Abelian combination to say
that A alpha followed by sigma 1 is equal to sigma 2.
Is this Abelian?
AUDIENCE: No.
PROFESSOR: No, it's not.
And how do you answer that?
Not through any means more profound than just drawing it
out and seeing what you get.
Let's take 3n as an example.
So let's do the operation.
In this order, do the rotation A 2 pi over 3 to go from this
one up to this one, and then let's reflect across this
horizontal mirror plane.
So this is 1 right handed, this is 2 right handed
reflect, here's number three and it's left handed.
So that's A 2 pi over 3 followed by
reflection in sigma 1.
And if we do the operations in reverse order, reflect in
sigma 1 to go up to here, so this would be 2 prime and then
rotate by 120 degrees.
We would go to here to here, and that is not the same
location as this one.
So this is not equal to sigma 1 followed by A 2 pi over 3.
How can you tell when is the combination of operations is
going to be Abelian and when is it not?
It sounds rather clumsy to state it in words, but
whenever the two symmetry operations leave each other
unmoved, then you can permute the order.
For example, in this combination here the rotation
leaves the mirror plane unchanged.
It just spins around in its own plane.
The mirror plane leaves the rotation unchanged, it just
reflects it end to end.
Here this rotation axis and these mirror planes obviously
do not leave each other unchanged because the
threefold axis rotates the mirror plane to
two other new locations.
So whenever that's the case, if the operations do not leave
each other unchanged, the order makes a difference.
So that's a useful thing to keep in mind.
Let's now turn to these sheets that I passed out, and you
might want to unstaple them because they're designed to go
side by side, and there are three pages.
What I've done across the top of the pages is to give the
different combinations of crystallographic rotation axes
that are possible.
So going from the first sheet across to the last, there's
the rotation axes by themselves--
one, two, three, four, and six.
Then there are the groups of the form n22, 222, 32, 422,
622, and then the two cubic combinations 23 and 432.
So there are the 11 possible combination of
crystallographic rotation axes, and stuck off by itself
at the end is the oddball 4-bar.
For the axes by themselves, the diagonal mirror addition
is not defined.
The diagonal mirror position by definition snakes in
between axes that are there in the axial combination.
If there's only one axis, that's not defined.
But you could add a horizontal mirror plane.
A horizontal mirror planes adds a onefold axis.
It's just a mirror plane.
So that's called m in the international
notation for mirror.
And it's called C sub S, C because it's a cyclic group,
and the S stands for Spiegel.
That's the Schoenflies notation.
If we add a--
take two and add a horizontal mirror plane--
this is the thing that we sketched out here--
this gives a pattern of four objects, two that are up above
the mirror plane of one chirality, two down below the
mirror plane of opposite chirality.
These projections of the patterns are down along the
rotation axis, and when you see a dot in a circle, the
circle represents the point that is down and the dot is
one that's up on top.
So there are two pairs of objects--
two right handed ones that are up and two left-handed ones
that are down.
And that, in international notation, is 2/m the twofold
axis is over a mirror plane.
So that's the way to make sense of the symbol and
remember what it means.
In the Schoenflies notation at C2.
You've added a horizontal mirror plane as an extender.
Going down the remaining ones in that list because they're
all very, very similar--
add a mirror plane to a threefold axis, the triangle
is reflected down to a triangle of motifs of opposite
chirality, so that's 3 over m.
For sure, it's called 6-bar, but let's forget about that.
The Schoenflies notation C3H, and similarly there's a 4 over
m, a square above, and a square of opposite handedness
below the mirror plane.
The horizontal mirror plane in all of these cases is shown as
a bold line.
They've added fainter vertical lines to give
you an angular reference.
Don't be confused by that in the Xerox copy--
the weight of the lines is not distinguished.
So the only symmetry element in 4 over m, for example,
should be shown by this solid bold circle that is in the
plane of the paper.
The two crossed lines are lighter and those are just as
mutual orientations.
And finally, there's another one of this form
which is 6 over m.
On all of the evenfold additions-- that is, to say
everything but 3m--
if you add that horizontal mirror plane, it's going to be
perpendicular to an operation A pi.
Therefore, inversion pops up as one of the operations in
the group, and 2 over m, 4 over m, 6 over m, those
regular polyhedra of objects--
can be inverted through the point of intersection of the
axis in the mirror plane, and that leaves the set unchanged.
So if we go down to the bottom of the list and say, can we
add inversion to these axial arrangements, in the case of
2, 4, and 6--
no, adding inversion doesn't give you anything new, because
that's already there in the groups of the form CNH, or
2/m, 4/m, 6/m.
Diagonal is not defined, so the only additional one that
we could pick up by adding inversion is adding inversion
to a threefold axis, and that's one we
did the other day.
Adding inversion to a threefold axis gives us three
that are up of one handedness, three that are down of
opposite handedness, and the orientation of
the triangle is skewed.
That is called 3-bar.
But it's just nothing more than a threefold axis with an
inversion center sitting on it.
Schoenflies' notation is C3i.
Then the only other additions to the single axes are adding
the mirror plane in a vertical fashion passing through the
axis, and these we've already seen in two dimensions so we
don't have to spend much time on them.
There's 2 mm, which is C2V.
3m--
all the mirror planes result upon adding a single mirror
plane to the threefold axis--
4 mm, and 6 mm.
These are just three-dimensional extensions
parallel to the axis of the symmetry that we already
derived for a plane in space in the
two-dimensional point groups.
So we're almost half done.
It's easy, isn't it?
Any questions at this point?
So the only ones that are really new that we haven't
seen already in two dimensions are the rotation axis
perpendicular to the mirrored plane.
That brings us to roughly the middle of the second sheet,
and the arrangement of axes 222 represents three mutually
orthogonal twofold axes.
If we add a horizontal mirror plane, that's going to put an
inversion center at the point of intersection because that
horizontal reflection is going to be sitting normal to a
twofold axis.
And then since the other twofold axes see an inversion
center sitting on it, there's a mirror plane perpendicular
to those other twofold axes.
So adding again to go through that again, adding inversion
to the point of intersection of the twofold axes creates a
mirror plane perpendicular to each of those twofold axes.
So it becomes 2/m 2/m 2/m--
each of the twofold axes acquires a different sort of
mirror plane perpendicular to its orientation.
And 2/m 2/m 2/m, even though I love saying it,
is kind of a mouthful.
So that's very often abbreviated to just mmm, which
really is a nice exclamation when you see what a lovely
symmetry it is.
The pattern shown in the diagram to the left is just
the pattern of 222 with the objects reflected up or
reflected down, and you get a total of 8.
That means that there are eight operations in the group,
and you can add up quickly what they are.
They are the three operations of rotating by pi, the
operation of identity, then three mirror planes because
the mirror planes all indistinct, and then the
operation of inversion.
So there are the eight operations that are present
that generate the objects from a single one.
Add a horizontal mirror plane to 32 and you get a threefold
axis perpendicular to the threefold axis, so you write
that as 3m.
That mirror plane now passes through the horizontal twofold
axes, and a twofold axis with a mirror plane passing through
it wants another mirror plane 90 degrees away, so a new
mirror plane pops up passing through each of the twofold
axes and perpendicular to the horizontal mirror plane.
Those mirror planes are in the same direction
as the twofold axis.
They're not perpendicular to it, and when symmetry elements
are parallel to a common direction you write them out
on the same line, so here's a case of a mixed metaphor.
One mirror plane that's perpendicular to an axis, the
threefold axis, so you write that as 3 over m.
The other twofold axes have mirror points passing through
them, so you write that as 2m and the threefold axes makes
all of those 2m symmetries equivalent to one another.
Pattern, and this is true for all of these symmetries, is
nothing more than the pattern of the subgroup repeated by
the one operation that you've added as an extender .
So if you look at the pattern of 32, object all of the same
chirality, all apparently up and down, and add the
horizontal mirror plane.
The one that is up goes down, the one that's down goes up,
and you get a set of four around each
of the twofold axes.
422 behaves very similarly.
The pattern is just the pattern of 422 reflected in a
plane, so you have a set of 16 objects--
eight of them up of one chirality, those are the solid
dots, if you will, and then another two, four, six, eight
that are down of opposite chirality.
You have a mirror plane that you added as the horizontal
mirror plane perpendicular to the fourfold axis.
All of the horizontal twofold axes see a mirror plane
passing through them.
So they've got to have a mirror plane that's 90 degrees
away, and those are the vertical mirror planes.
Either the point is obvious now or you're not following me
at all, so I'll simply say on the left-hand column of the
final sheet, 622 with a horizontal mirror plane goes
to 6/m, 2/m, 2/m, called D6h in Schoenflies.
I'll leave the cubic ones until last.
They're not nearly as bad as they seem.
But obviously they have a lot of symmetry all over the
place, and we'll want to take a closer look at the patterns.
The next extender is a vertical mirror plane, and
we've already got the groups of the form cnv, because they
are just extensions in a third dimension of one set we've
seen in two dimensions.
If we try adding the vertical mirror plane, which is defined
as passing through the principal axis of high
symmetry and perpendicular to the twofold axes, we've
already encountered those in every group except 32.
2/m, 2/m, 2/m already has the vertical mirror plane.
The same is true of 3/mm2.
The same is true of 4/m, 2/m, 2/m and 6/m, 2/m, 2/m.
The vertical mirror plane comes in automatically when we
add the horizontal mirror plane as the extender, So
nothing new there at all.
And now we come to one that is interesting.
This is the diagonal mirror plane.
And I'll do this one slowly and then in some detail
because it's another example of how we would trip over
something if we were not clever enough to think of it.
Let me begin by drawing the three orthogonal axes of 222.
So we have one object that's up and one object that's down.
The twofold axis perpendicular to the board will take this
one that's down and rotate it to here and take this one
that's up and rotate it to here.
So that is the pattern of 222.
We can snake a mirror plane in between the twofold axes, and
a mirror plane passing through the vertical twofold axis has
to be accompanied by one that's 90 degrees away, half
the throw of the axis, which comes from our theorem that
says A pi dot sigma vertical has to be equal to a sigma
prime that's vertical and pi/2 away from the first.
In terms of the pattern, this second mirror plane that comes
up is going to reflect this one across to here, it's going
to reflect this one across to here, it's going to take this
one and reflect it over to here, and take this one and
reflect it to the right, as well.
So now we've got a total of eight objects.
This one is right handed, and this one is also right handed
because we repeated it by rotation.
Then we reflected that pair, so these two are left handed,
and reflect it across the other diagonal mirror plane.
They're back to right handed again, and these have to be
left handed as well, so there's the pattern.
Is there an inversion center in this pattern?
The answer is no, because there is no mirror plane that
is perpendicular to a twofold axis, so this point group has
no inversion in it.
It's said to be acentric, without a inversion center.
Do we know how everything is related to everything else?
We know how this one is related to this
one by twofold rotation.
We know how this one is related to this one, and
that's by reflection across the mirror plane.
How is the first one related to this one?
How do you get it through 90 degrees and then pop it down
and change the chirality?
AUDIENCE: [INAUDIBLE]?
PROFESSOR: You can't do that in one step.
You've got to do just what I said in words.
Rotate it 90 degrees, don't put it down yet, and reflect
it down in a horizontal mirror plane.
This horizontal mirror plane is not a symmetry element,
it's part of the operation that's necessary to get us
from this guy over to this guy of opposite chirality.
So this is an example a new type of
operation, a 2 step operation.
This is a 90-degree rotoreflection axis.
And again, I feel compelled to put the axis in quotation
marks, because really it's a pair of operations that leaves
only a point unmoved, so it's really a
point symmetry element.
There's another way of describing the same thing, and
that would be rotoinversion.
We could rotate by 90 degrees in the reverse sense, not yet
put it down, and invert, and we get, again,
the relation between--
what did I do?
Rotate it here, yes--
rotate to here and then invert, and we got that one.
So that's another way of defining the relation.
A 90-degree rotoreflection is a minus 90-degree
rotoinversion.
So you pays your money and you takes your choice, and what
the groundbreakers who went before us did was to take
rotoinversion.
as the operation to survive.
So here is, if we were not smart enough to invent it, our
4-bar rotoinversion axis.
So if we hadn't been clever enough-- and it would take a
pretty clever, devious mind to invent a 2 step rotoinversion
operation--
as soon as we added this sort of extender, a diagonal mirror
plane to a group of the form n22, we would have found that
there was a new sort of transformation that could not
be described any more simply than to say take
two steps to do it.
Rotate 90 degrees and then invert.
So we have to add the operation of A pi over 2-bar
to our basic bag of tricks for generating patterns.
The group 32 is also a group to which we can add a diagonal
mirror plane.
And if you do that, again, the bold lines that are mirror
planes in that group and the axes that are faint lines are
easy to mix up.
But if we examine that group without the confusing lines,
here are the twofold axes.
And remember that they are all equivalent to one another by
the threefold axis.
So this axial group is the group 32.
If we put down a mirror plane interleaved between the
twofold axes, notice that each of those mirror is
perpendicular to a twofold axis.
So we can write this as 2/m.
2/m means there's an inversion center at the intersection of
the twofold axes with the mirror planes.
And a inversion center sitting on a threefold axis makes it a
3-bar axis, so this group is called 3-bar 2/m.
And the pattern is what a twofold axis would do.
These would be up-down-right ones.
Reflect those and you get an up-down pair of left ones.
Reflect again and you get an up-down pair of
right-handed ones.
Reflect yet again, and you get an up-down pair
of left-handed ones.
Reflect still once more, and you get a down-up pair of
right-handed ones.
One more time, up-down left-handed ones.
So you get a total of 2, 4, 6, 8, 10, 12 points.
So this is 3-bar 2/m in Schoenflies notation D3--
that's the symbol 32--
with a diagonal mirror plane added.
You indicate that by a d in the subscript.
OK.
I think that's probably a point where you're more than
ready for a break.
We've got very few yet to do.
One surprise is that if you add diagonal mirror planes to
422 and 622, you get perfectly lovely exquisite groups but
they're not crystallographic, so we don't
include them in our list.
But we'll discuss them when we come back, and then take a
look also at the cubic symmetries.
Impossible to draw, but really not all that difficult to
understand.
So let's stop at this point and take a stretch.
People are stretching already.
They need it.
We'll resume in 10 minutes' time.