PROFESSOR: All right, I feel almost as though I should
introduce myself all over again.
It's been a week and a half since we had a lecture.
So let me begin by reminding you of what we were doing.
We had derived all of the plane groups, and for better
or worse had put them behind us.
And then we moved into three dimensions, where things get a
lot more involved and a lot more complicated, and to say
the least, a lot more numerous.
And the first question we asked was to say, when we're
in a three-dimensional space, we can combine a first
rotation operation with a second rotation operation, B
beta If we're to begin by deriving point groups-- that
is to say, the least [INAUDIBLE] point in this
three-dimensional space is not going to move.
[INAUDIBLE]
two axes to intersect a point.
For a space group, they could be parallel to one another.
But that's gonna be an infinite set of symmetry
elements and operations that extends through all space.
Then we asked the question-- rhetorically, because you knew
I was going to answer--
what is the net result of a sequence of rotating from a
first object to a second, and then picking up the second and
rotating it to an angle beta about second axis.
Begin the third one here.
What is the net effect?
And again, we could do that by the process of elimination.
It has to be either translation or another
rotation, because these are the only two generic sorts of
operations which leaves the chirality unchanged.
And I think I convinced you that indeed there was some
third axis, C, which rotated directly from the first to the
third by some different angle, gamma.
So that is the consequence of combining the first two
rotation axes.
What they would anticipate is that the location and also the
value of the rotation [INAUDIBLE] depends on alpha
and beta, and also [INAUDIBLE]
at which we combine them.
And that's what we'll see.
You could do this in a number of ways that if you don't
[INAUDIBLE]
that gamma would turn out to be a
crystallographic rotation.
And then your result would be true, but it would not be a
rotation operation which could exist in a three-dimensional
point group.
So using the genius of Leonhard Euler and a
construction known as Euler's construction.
We set up a little spherical triangle,
which we could analyze.
Let me tell you a little bit about Euler, because he's a
remarkable individual.
The first remarkable feature of Euler is that he's Swiss,
and there are not many world-class famous people who
are Swiss, simply because the population is so small.
The probability of somebody rising to heights is constant
among all populations.
If you have a small population, there are not
going to be very many.
And to demonstrate that point, can somebody identify some
other citizen of Switzerland who rose to great heights, as
world-famous as [INAUDIBLE]?
Think of one other person?
I'm fairly pressed to do so myself.
There's my uncle, but he actually
didn't amount to much.
But there's an artist, Paul Klee, who is world-class.
He was one of the early modern artists.
And Switzerland has just finished constructing a
marvelous museum on the outskirts of the
capital city, Berne.
It's a structure that is supposed to mimic the rolling
countryside of the central part of Switzerland.
So it's a series of cylindrical structures, glass
in front, glass on top.
And it divides the area into three.
Two of them are exhibit spaces and one is a space for
scholars and researchers.
And it's an absolutely marvelous structure.
It appeared in the pages of Time Magazine when it opened
[INAUDIBLE] about two months ago.
Anyway, Euler was born in 1707, so he
operated a long time ago.
And he died in St. Petersburg on September 18, 1783.
That is exactly 350 years and one day after my birthday.
That's another remarkable thing about him.
He was 76 years old when he died.
And that time of primitive medicine and plague, not many
people got to live to their 60s and 70s.
Euler studied at the University of Basel under the
Bernoullis.
I think you've all heard of the Bernoullis.
There's a very famous principle of physics known as
the Bernoulli effect, which stated in its simple practical
form says that if you have the Sunday paper on the front seat
alongside of you, and you drive your car with the
windows down, the paper will blow out the window.
That's Bernoulli's principle in action.
Euler got his doctorate from Basel at age 16.
It sort of leads one to the rhetorical question, how come
you guys have been spinning your wheels for so long?
But then I said, he was an unusual individual.
The Bernoullis went to St. Petersburg in Russia, under
Catherine the First.
Russia was trying very hard at that time to enter the ranks
of the Western world as a full member.
Euler followed them a little bit later on.
And he succeeded one of the Bernoullis as a professor of
mathematics in 1733.
Then unfortunately, two years later in 1735, he lost the
sight of one eye.
And why?
Because at that time, astronomy had been using this
newfangled telescope which had recently been perfected.
One of the hottest things going was studying the heavens
looking through a telescope.
And if you wanted to look at the sun, people knew nothing
about the damaging effect on retinas of the sun's rays.
So he lost the sight of one eye.
1741, he went back to Europe again, to Berlin.
Why?
Because the reigning monarch in Russia at that time was
called Ivan the Terrible, and that says reams about why
Euler would want to get out of Russia.
But then 1776, he went back to St. Petersburg under the next
monarch, who was Catherine the Great.
And that says why one would be interested to go back.
Finally, in 1766, he went fully blind.
Did that slow him down?
Not one bit.
He published in his lifetime 800 papers.
You talk to some big cheese around MIT, they've published
maybe 200 or 300 papers, and that with the assistance of an
army of graduate students, and also, one might add, the
assistance of Xerox machines and word processors.
So back in the days when you wrote everything out by hand
with a [INAUDIBLE] quill pen, 800 papers is an absolutely
unbelievable accomplishment.
It took 35 years after he passed away to publish
everything that he'd written.
People had kept busy publishing what he did.
And among his accomplishments, he was one of the first people
to apply real hardcore mathematics to astronomy, to
make it quantitative.
He was one of the first to suggest that light was a wave
form, and that color was a function of wave length.
That was astonishingly precocious.
And then, lest he seem like an egghead who spent all his time
staring through telescopes and working out theorems to use in
crystallography, he also wrote a popular account of science
for the general public, which was published in 1768.
And that book was published for 90 years, three
generations of people kept gobbling up [INAUDIBLE]
pretty good.
He impinged upon our own language and activities in
several important ways.
He was the one who used lowercase i to define the
square root of minus 1.
We can thank Euler for that.
He was the person who used e to define the constant,
2.71828182845904523536.
And he was the person who first used f
to stand for function.
So he contributed not only a lot of good mathematics, but a
lot [INAUDIBLE].
So this does not have to be easy.
Euler was a great guy.
And this geometry of rotations about different axes is
something that also survives in a mechanism that involves
achieving angular core rotation on a axis by rotation
on two orthogonal arcs.
And that's something that's called an Euler Cradle.
And that is geometry that;s used in a great number of
mechanical devices.
In any case, back to instruction for our purposes.
The thing that we would like to do is let alpha and beta
take on all possible crystallographic values,
namely 360 degrees or onefold axis, although we know that
that's not gonna work.
Twofold, 180 degrees.
Threefold, 120.
Fourfold, 90.
Sixfold, 60.
And let that give the values to alpha and beta,
taking two at a time.
And then let us ask the question, at what angle should
we combine these two axes to get gamma to be a
crystallographic rotation axis?
And if it is crystallographic and not something like 37.9234
degrees, what are the remaining axes with interaxial
angles B and A?
So this is the problem that Euler's construction solved.
And I won't go through all the arguments that we
need to set this up.
But what we found after some f sleight of hand when we were
working on the polar triangle with spherical trigonometry,
what we found was the result that said that if we want to
combine two axes, alpha and beta, so that the third one
turned out to be a rotation of gamma, then the cosine of the
angle between A and B should be the cosine of alpha/2,
cosine of beta/2 plus cosine of gamma/2, divided by the
sine of alpha/2, sine beta/2.
So if you pick your alpha and beta, and you decide what you
would want these first two rotations to turn out to be.
And generally it's not gonna work.
But there are a surprising number of cases
where it does work.
So you specify the combination you had.
You also determine the angle between A and C. And you have
to also determine the angle between the axes B and C. And
there are similar sorts of expressions that one obtains
simply by [INAUDIBLE] alpha and beta again.
So then we set it up just by looking at all possible
combinations of twofold, of two different rotation axes,
and a third, which the net effect might be.
We're not interested in permuting A, B, and C. And A
equal to C equal to B is just as interesting or not as A
equal to B equal to C. We don't care about permutations.
And we generated-- just as we [INAUDIBLE]
a week and a half ago-- a set of combinations that we should
consider, what the axis A would be, what the axis B
would be, and them different choices for the axis C. So A
could be 1.
B could be 1.
And we could look for 1, 1, 1; 1, 1, 2; 1, 1, 3;
1, 1, 4; 1, 1, 6.
Those are legitimate combinations?
Those are absurd combinations, because doing nothing about
the first onefold axis, doing nothing about the second
onefold axis could hardly result in the net effect of
the 90-degree rotation by the third axis.
And [INAUDIBLE] suggested is that sitting around and doing
nothing twice was equal to a rotation [INAUDIBLE] its
junctures [INAUDIBLE] we'd find ourselves spinning on our
axis like tops.
Twofold axis.
1, 1, 2, we have here, so we don't have to
consider 2, 1, 1.
But we should consider 2, 1, 2; 2, 1, 3; 2, 1, 4; 2,
1, 6, and so on.
If we filled out this whole table, last time you got a
copy of it and some notes, and all that remains then is to
quote [INAUDIBLE]
and see where we get allowable axial combinations.
And not surprisingly, there are so very, very few.
And we showed-- again, when we finished up last time-- that
you can always take any n-fold axis that has a C gamma that's
equal to C 2 pi over n and combine it with twofold axes
at right angles, provided you make the angle between the
twofold axes equal 2 of gamma over 2.
So the crystallographic possibilities for C are, first
of all, C could be a twofold axis, in which case you could
combine with it a pair of twofold axes, and this 1/2 of
180 degrees would also be a right angle.
We could let C be a threefold axis, in which case the
twofold axes have to be at right angles, two- to
threefold axes.
And they should be combined at half the angular throw of the
threefold axes, which is 60 degrees.
Two more possibilities are four [INAUDIBLE] pair of
twofold axes at right angles.
Of the angle between them, half of the throw of a
fourfold axis would have to be equal to 5 degrees.
And the last one is sixfold axis with a
pair of twofold axes.
Add angles to it.
And a third [INAUDIBLE].
So notice the insidious fact that the angle between the
twofold axes is always a half, 1/2 the rotation angle in
principal axis C are not equal to this rotation axis.
The other thing we saw that is that these twofold axes are
different, distinct, symmetry-independent axes.
They're different in that the principal axis C never rotates
this axis into the second one, and therefore demands that
whatever's going on around one twofold axis be identical to
what's going on at the other twofold axis, different in the
sense that they function in different ways in the pattern,
or if you're describing the symmetry of an object.
So probably the best demonstration of this is a
regular prism with a triangular shape or with a
square shape or with a hexagonal shape.
And the adjacent twofold axes for these prisms would come
out of the normal to a face.
And then if the second twofold axis is going to be 45 degrees
away from the first, the other one has to come [INAUDIBLE].
So yeah, they function in different ways in the space.
One is normal to the face of a regular prism, if that's
what's in our space.
The other one comes out of the edge.
Similarly for a sixfold axis, a hexagonal prism has one
twofold axis coming out normal to a face, the adjacent
twofold axis coming out to an edge.
And as advertised, that angle is 33.
So they function in different ways.
The only exception to that, again, is the [INAUDIBLE]
threefold axis.
And the twofold axes there, which were 60 degrees, come
out of one side from a corner of the triangular prism.
On the other side, that was [INAUDIBLE].
So all of the twofold axes were the same thing.
There's only one independent kind of twofold axis, just as
there was only kind of mirror plane in the combination of a
mirror plane passing through a twofold axis.
The names for these are always, as we've done in the
past, a running list of the independent symmetry
operations that are present.
So this general one, n, 2, 2, with the n-fold axis for some
generic sort of a twofold axis.
This one would be 2, 2, 2.
This one would be 4, 2, 2.
And this one would be 6, 2, 2.
This [INAUDIBLE] with a threefold axis is, again,
called not 3, 2, but 3, 2, because there's only one kind
of twofold axis, just as there's only one kind
[INAUDIBLE].
We pause [INAUDIBLE], see if you have any questions.
These are the crystallographic combinations of this
[INAUDIBLE].
There is no reason why you should not in something that
doesn't have to be compatible with a lattice, combine an
n-fold axis of any sort with twofold axes or right angles.
And indeed, if I look at my old friend, the saguaro
cactus, we can add anything like 28-
up to 32-fold symmetry.
This cactus stem had a 28-fold symmetry.
It would be a twofold axis coming out of the string with
one of the ribs, another twofold axis coming out of the
crevice between the pair of these ribs.
And if I took that thing up, very carefully because of the
spines, and with great effort because it weights several
tons, and rotated it about one axis, and then rotate it again
about a second axis, turning it upside-down, [INAUDIBLE]
axis would be rotation to 128 [INAUDIBLE].
Valid symmetry, but not crystallographic.
OK, any comments or questions?
Get to know these results, because the exercise that's
going to occupy us for the next week is going to be
asking how we can decorate these frameworks with mirror
planes and with inversion.
If you want orientation, we could add another operation to
the collection of axes while it pops up.
Where [INAUDIBLE].
Comments or questions?
OK, there are only two other combinations that are
crystallographic that involve directions
that are not simple.
And one of them involves a combination of a threefold
axis with twofold axes that come out of directions at a
normal to the face of a cube.
So these turn out to be very, very strange angles which make
no sense at all until you refer them to
directions in the cube.
The direction of the threefold axis turns out to be
correspondent with the angle of a cube.
The direction of the twofold axis corresponds to
the normal two faces.
You can show--
I did show, and I don't think anybody really followed me, so
I had to hand out that [INAUDIBLE]--
that if you start with one twofold axis and one threefold
axis, what you're going to get is a threefold axis coming out
of all of the [INAUDIBLE] diagonals, but they're all
equivalent to this threefold axis.
And twofold axes come out to normal for all the
faces of the cube.
So there's only one kind of twofold axis and one kind of
threefold axis, so even though we got this by combining a
pair of twofold axes--
I'm sorry, a pair of threefold axes--
that's what happens when you stay up late.
You forget about this one.
A pair of threefold axes at the diagonals
and one twofold axis.
And this is the combination that is called 23, because
there's only one sort of twofold axis and one sort of
threefold axis.
And I'd like to point out and I'd like to warn you of traps
when we come across them, make sure we don't [INAUDIBLE]
across them.
Notice the insidious relation of this pair of integers to
the symmetry that we label [? 232. ?]
32 is a threefold axis with a twofold axis normal to it.
23 is this combination that involves
corrections in a cube.
Now, let me pause parenthetically with an aside.
You might say, how can this be?
Here is a cube.
That cube has got a fourfold axis about it.
Don't call that a twofold axis.
A cube has a fourfold axis coming out of it.
OK, let me give you an example in real life.
There is a fairly common mineral, iron disulfide--
pyrite.
This forms nice, shiny cubes, but the cube faces have
striations on them.
If you look at them, there's a set of lines running this way.
And what those lines are if you look at this crystal face
with a magnifying glass, is that these are little steps of
a second face.
And this is a face of the form hk0.
And this oscillates back and forth, and there seem to be
lines scribed on the surface.
So this sort of a [INAUDIBLE]
crystal growth of one face which never really develops is
not that uncommon.
There's a threefold axis coming out of the corner here,
but this is really a surface that is left at variant only
by 183 [INAUDIBLE].
You cannot rotate that surface 90 degrees.
The orientation of the lines have changed.
But there is a bona fide threefold axis coming up
there, so these striations, if I rotate them to this face,
will go in a way like this.
This edge turns into this edge, and therefore the lines
will run down like this.
And if I rotate [INAUDIBLE] n by 90 degrees, the striations
on the adjacent face will run down.
So there's a decorated cube.
And if you rolled it up and say how can I move this cube
around and leave its appearance totally unchanged?
the answer is, rotate it by 120 degrees [INAUDIBLE]
diagonal.
But we can only rotate it 180 degrees around the face.
So there's an example of a crystal [INAUDIBLE]
on the arrangement of rotation axes 23.
The final one, the highest symmetry of all, involves a
fourfold axis coming out of a direction normal to a face, a
threefold axis coming out of the [INAUDIBLE] diagonal, and
a twofold axis coming out normal to an edge.
And if you let these axes work on one another, there's a
twofold axis that comes out of every edge, and a fourfold
axis that comes out of every face.
And this one is named 432, because it's a combination of
a fourfold, a threefold, and a twofold.
That is the symmetry to the cube.
And for crystallographic symmetries, that's about as
complicated as it gets.
Now, if we look at the regular solids that we've encountered
here, with symmetry 23, there is a regular [INAUDIBLE]
consisting of four triangular faces.
That's a tetrahedron.
And for 432, one of the polyhedra that can form from
the crystal [INAUDIBLE] is an octahedron.
These were the lovely solids called Platonic solids, after
Plato, that we used as our trophies early this afternoon.
So this is an octahedron.
Let me finish up before our break by asking is there any
other regular polyhedra that can result from a combination
of rotation axes that are not crystallographic?
Now, that's a tough question to ask.
You instantly scan your knowledge of geometry.
Clearly, there are a lot of prisms [INAUDIBLE]
infinite number of prisms [INAUDIBLE] n22.
But there's only one other combination of axes
non-crystallographic which results in a regular
polyhedron.
And this is a combination, believe it or not, of a
fivefold axis with a threefold axis.
This is a fivefold axis and a threefold axis
and a twofold axis.
And these result in a regular solid called an icosahedron.
And that is so complicated that I won't
attempt to draw it.
But having said so, it looks like this.
It has diamond-shaped faces.
And there are five of these that come together in a
[INAUDIBLE] form.
So here's one of the [INAUDIBLE] diamond-shaped
faces, another diamond-shaped face, and then the two other
ones that come in like this.
So there are 1, 2, 3, 4, 5 faces, so this is the
orientation of a fivefold axis.
The twofold axis comes out of a place where two of these
edges are shared.
And the threefold axis--
[INAUDIBLE] triangular faces.
[INAUDIBLE].
So here's the fivefold axis.
These are the twofold axes [INAUDIBLE].
And these are threefold axes.
But you know all this.
Is there anybody who [INAUDIBLE]
show me that they've never seen an icosahedron?
Anybody ever who has not seen-- have you seen it?
[INAUDIBLE].
Here--
I spared no expense-- is a live icosahedron for those who
would like to look.
AUDIENCE: How many sides does it have?
PROFESSOR: He'll count them for you.
I don't remember.
I know the number of faces and the number of edges, but not
the [INAUDIBLE].
AUDIENCE: [INAUDIBLE]
faces.
PROFESSOR: I think it's--
I'm not sure [INAUDIBLE].
Not sure.
AUDIENCE: But you said you know the number of faces.
PROFESSOR: Yeah, but I can't tell you everything I know
[INAUDIBLE].
AUDIENCE: Yeah, yeah.
PROFESSOR: There's another figure which also has this
symmetry, and it's a regular solid.
And this is called a rhombic dodecahedron.
And, wise guy, this has 12 faces.
And the faces are pentagonal.
And there are three pentagonal faces that come together at
the threefold axis, a fivefold axis out of each of the
pentagonal faces, and a twofold axis out of the edges.
The face that this has 12 faces at regular intervals
leads an entrepreneur who was familiar with injection
molding to cast these little things as a plastic, and then
puts a month of the year on each of the 12 faces and made
a nice little desk calendar.
[INAUDIBLE] something that reminds you of symmetry as
well [INAUDIBLE].
So this has fivefold faces, 12 of them, and that's the
rhombic dodecahedron as opposed to the normal
dodecahedron which is [INAUDIBLE].
AUDIENCE: So are those both [INAUDIBLE]?
PROFESSOR: I'm sorry?
AUDIENCE: Are those both [INAUDIBLE]?
The icosahedron, [INAUDIBLE]?
PROFESSOR: Yeah, this has a fivefold axis coming out of
the pentagonal [INAUDIBLE].
Is that what you're asking?
And the threefold axis, there are three of them that, come
together, and they do something like this.
So here's the threefold, here's the fivefold, here's
the twofold.
Here we have to see it [INAUDIBLE].
Look for somebody who's got one of these desk calendars.
[INAUDIBLE] used to sell them.
[INAUDIBLE].
OK, this sets up the next stage of our game.
We've got these arrangements of axes.
And if you count them up on the fingers of your hands and
one toe, there are 11 of them.
There are the axes by themselves-- onefold, twofold,
threefold, fourfold, sixfold.
There are these so-called dihedral combinations, where
the only thing that changes from one to the other is the
symmetry of the main axis, the [INAUDIBLE] symmetry, and
therefore the dihedral angle between the twofold axes.
And these are 222, 32, 422, and 622.
And then the two cubic arrangements, 32 and 432.
So when we return, we'll ask the question, how can we add
mirror planes for an inversion center to this
combination of axes?
And these are going to give us new symmetries involving not
only rotation, but [INAUDIBLE] help with rotation
inversion as well.
And the constraint in doing this is that we have to add
the reflection plane and the inversion center in such a way
that it doesn't create any new rotation axes, because we have
systematically derived all of the possible combinations of
crystallographic rotation axes.
So if the addition of a mirror plane, creates a new axis,
that's going to be something that you already have with a
combination of a greater number of rotation axes.
For example, if you take a single twofold axis and a
mirror plane of an angle, that generated another twofold axis
90 degrees away.
But we've already got that.
If a mirror plane moves an axis to an angle that doesn't
correspond to one of these arrangements, it's going to be
impossible, because we have systematically derived, using
Euler's construction, all the combinations of rotation
operations that are possible.
So that's going to be the constraint.
We want to add mirror planes or an inversion center in all
possible combinations.
And this means we're gonna need a theorem.
What happens when you add a mirror plane
to a rotation operation?
We're already familiar with one of them.
You take an axis and you pass an mirror plane through it,
you get another mirror plane that is rotated about the
axis, away from the first, by half the rotation
angle of the axis.
So let's take a breather, and let us
resume in about 10 minutes.