PROFESSOR: But you
all knew that.
That is quite a mouthful.
So if you like, you can
refer to it as SST.
And today, we're taking off.
So my name, for those of
you who don't know
me, is Bernardt Wuensch.
My room number is 13-4037, The
office that's become a legend
in its own time.
And my extension number
is 3-6889.
And hey, just to get you
sensitized and thinking about
the right things, let me point
out that my extension number
has a point of 180 degree
rotational symmetry right in
the middle.
You can pick it up,
turn it head over
heels by 180 degrees.
And it's mapped into coincidence
with itself.
Just happened to get it.
You might think I would have had
to have fought for years
to get an extension
number like that.
But no, it just happened
to come my way.
OK, some words about the
formalities of the subject.
First of all, the format of
the class is unusual.
We meet four hours a week, but
because most if not all of you
are graduate students anxious
to get some work done in the
laboratory, we do this in
two two-hour chunks.
So we meet Tuesday and Thursdays
for two hours.
Two hours is a lot of time for
anything, however good.
So what we do is to take a
long intermission halfway
through and let you go out and
enjoy what's left of the
lingering summer for
10 or 12 minutes.
And then, come back refreshed
and we will resume.
Most graduate students like this
arrangement because it
gives them a chance to duck out
and make a setting on a
furnace or turn something
off in the laboratory.
And it works out better for them
than having a one hour
time chunk every day of the week
or four of the five days
of the week.
In any case, nobody's
complained about it.
So I assume that will work
satisfactorily for you as well.
The other question that one
immediately asked at the
beginning of the term,
how many quizzes?
And we're supposed to tell
you that straight up.
There will be three quizzes.
No final examination--
do I look like the kind of
Scrooge that would prevent you
from getting a good flight home
at Christmas time or have
you working, cramming for a
final examination a few days
before Christmas?
And for my part, I can remember
the good old days--
or not so good old days--
when I did give a final.
And there I would be, lying down
on my stomach under the
Christmas tree grading
final examinations.
And every time one of my little
kids would come near,
I'd lash out with my foot and
say, get out of here, kid!
Can't you see Daddy's got
papers to correct?
Well, no Scrooge.
No final examination.
We'll have three quizzes.
The quizzes will also be
a little bit unique.
Since we have two hour chunks of
time, I found by experience
that if I give the quiz during
the first hour everybody is
sitting around glassy-eyed,
absolutely brain dead and pay
no attention to the lecture
that follows.
If I give the quiz in the second
hour, everybody is
pretending to pay attention
and then sneaking
surreptitious looks at their
notes just so they can have
everything packed away before
they have to write on paper.
So my quizzes are two hour
quizzes which lets you not
work for two hours, but gives
you all the time you could
possibly want.
And people start leaving after
about an hour and a quarter.
But you can stay for the entire
two hours if you want.
Even at that, I found by
experience that if I give you
two hours for the quiz, after
about an hour and a half,
everybody's looking out the
window, looking back at the
ceiling, not a single
pencil is moving.
Then I will say, OK,
you all done?
Everybody starts writing
again and going through
their papers once more.
And even after two hours I find
that in order to get the
quiz papers, I have to plant one
foot on the edge of your
table, grab hold of your quiz
with both hands and drag it
out of your clutching fingers.
So you can take the
full two hours.
But it should not be necessary
for you to
consume that much time.
The quizzes will come-- and this
is something else we're
supposed to explain to you--
they will come at one third of
the way through the term, two
thirds of the way through the
term, and 2.983/3 of the way
through the term.
And if you're wondering where
that number comes from, this
lets me put the quiz just before
the final week of the
term when we're not supposed
to give examinations.
So there'll be three quizzes.
You will have opportunity
for lots of
practice with problems.
We will have on the order
of 15 problem sets.
And for the most part, they
will be very short, or
modestly short, and designed to
give you some practice in
working with the material.
Because as the nature of this
subject begins to unfold,
you'll see that it involves
a type of mathematics that
you've really perhaps not
had much practice with.
It involves geometrical
relations.
And to really master it, you
have to work with the material
and get some practice.
Another question that's
perennially asked if not
outright raised in private,
what do the quizzes count?
What do the problem
sets count?
The answer to that is that the
problem sets will count a lot
towards your understanding
of the material.
But I'm not going
to grade them.
And I'm not going to factor them
in along with the quizzes
to decide your final eventual
fortunes in this class.
The problem sets, moreover,
there are a lot of them.
But they will be optional in the
sense that if you do them,
I will carefully correct them,
add words of inspiration and
advice, correct things where
you've gone wrong and then
return them to you as
quickly as possible.
But if you how to do the
problem, you say, ah!
Why does he wants us to do this
and waste our time with a
silly problem like this?
Don't do it.
Don't do it, because if you
know how to do the problem
set, that's fine.
And you've got better things
to do with your time.
And I've got better things
to do with my time if the
feedback is not going to
be of benefit to you.
So I hope you will do them.
And as I said, if you do them,
I will correct them promptly
and thoroughly.
And if you haven't got the
foggiest idea how to do the
problem, do what you can.
And then, write down
a plea of help--
I don't understand what's
going on here!
OK, and then I will take the
time to write out what's going
on, hopefully to your
benefit and use.
Will I turn out solutions?
Only on an individual
basis in the fashion
that I've just described.
I find that if I write out a
solution to each of these
problems, you don't do them.
And you say, oh, so that's how
you do that, throw it in a
file, and not look at it until
the night before the quiz.
So there will not be solutions
handed out other than
correction on an individual
basis on your papers.
Is that all that I
wanted to say?
I think that's about all
for the formalities.
Actually, I should add one
postscript to say the problem
sets don't count anything
toward your final grade.
They do in one minor sense.
When you have a large class and
you plot up the grades and
there are no lumps with gaps
in between, there comes a
point where you have
to separate
one grade from another.
And if you've done well on the
quizzes and you've done well
on the problem sets and there's
just one quiz that's a
little bit, I say, OK,
he or she had a
bad day that afternoon.
And I'll give you the benefit
of the doubt.
And even though I'm not a
vindictive sort, if you're
right on the fence and you
haven't done any of the
problems, then without
malice I say, gotcha!
And you go down [INAUDIBLE]
on the low side of
the barricade.
And I think that's only a
natural indication because
there are some cases where, with
Solomonic judgement, you
have to decide who
gets what grade.
OK, let me say a little
bit about the texts.
There are a number of
books that deal with
crystallography.
For the most part, though,
they consists of an
introductory chapter.
Every single book on the solid
state feels compelled to write
some sort of half baked chapter
on crystal structure
or crystallography.
And usually, these chapters
consist of big tables.
And they say, there
are 14 of these.
There are 17 of these.
There are 32 of these.
There are like 230 of these.
There are 1,170 of these.
And then, that has all the
excitement and stimulation of
reading the telephone
directory.
It's a crazy cast of characters,
but it's awfully
hard to see the plot.
So what we will do all the way
through is derive everything.
So you can not only see how it
turns out, but why it has to
be that way.
And that's the way, in my
opinion, one really learns
this material.
A couple of other very pedantic
comments about the material.
In the early part of the term,
the first half in fact, we're
going to use plain
old geometry.
Now, geometry really
doesn't cut much
mustard around the Institute.
If you can't integrate it or
take its Fourier transform,
that's a mathematics you don't
have to take seriously.
Well, geometry is a perfectly
valid branch of mathematics.
And one can do what we're going
to do in more complex
terms using the language
of group theory.
And we will, in fact, use a
little bit of that later on.
But for the most part, just
diagrams with simple geometry
are going to be one of the
principle tools in the initial
part of the class.
About halfway through, we'll
switch over to something that
is much more mathematical in
the traditional sense.
Here, a little bit of linear
algebra and matrix algebra
will help you.
If you haven't had that or
haven't looked at it for a
while, we'll build it up from
ground zero so that you'll be
able to fully understand it.
We'll hit a few eigenvalue
problems towards
the end of the term.
If that doesn't get your
adrenaline pumping, that will
be developed in a physical
context so that you're doing
the sort of problem before you
even know what it's called.
So it's going to be a user
friendly course that doesn't
rely on something that
you may have had two
or three years ago.
OK, but the other thing that I
wanted to say was that this
class is not like many classes
in that you talk about
something for one week.
And then, you put it aside and
you talk about something
completely different
the next week.
Our first half of the
course will be one
long process of synthesis.
We're going to start out very,
very simply with little
mapping transformations.
This is picked up and rotated
and slid over to here.
And you'll say, ho-hum,
let's get on with it.
Come on, go faster.
But we'll build on this and then
build on what we've just
done to what comes next.
And unlike most of the classes
in science that you take where
you start with general terms and
you zero in on some little
nugget like f equals ma, e
equals mc squared, lambda
equals 2d sin (theta), a little
nugget like a bullion
cube that you can drop
in your pocket.
And then when you need it later
on, you pull it out and
add hot water.
And then, you have a tool
that you can use.
We will do something
that's completely
different in its structure.
It will start out simple.
It will grow.
It will blossom like an
elegant [? Filigree ?]
structure that gets more and
more complicated and diverges
rather than converging to a
nice, tight, little nugget.
It's going to get very,
very complicated.
And the reason for doing this
gradually and thoroughly is so
that you can understand
the complexity and
where it comes from.
OK, so my moral here
is keep up.
It may seem easy
when you start.
But we're going to assume that
you've got that down cold
before we go on to
the next step.
OK, texts.
Apart from these half baked
treatments which I just keep
[INAUDIBLE] on, one of the very
best books is by an old
MIT guy, Martin Buerger, who
was one of MIT's most
distinguished faculty.
He was the very first faculty
member to be honored with the
title Institute Professor,
the very first one.
Chairman of the Faculty, all
sorts of awards from
professional societies--
he has a book called Elementary
Crystallography.
This is published by Wiley.
There's some who dispute the
term "elementary." But he
really has a book which uses,
at the outset, nothing more
than geometry.
He doesn't throw in comments
like, "It can be shown that,"
or "By further work, it
turns out--." He does
everything for you.
Everything is down there so you
can see how it's done and
what the results are.
To me, it is the best
book on the subject.
That's the good part.
The bad part is that it's
been out of print
for about 15 years.
So what I am going to
do is to make--
now that I know how many of you
are going to be present--
I'm going to make a Xerox
copy for you of the
first half of the book.
What a department!
What a class!
You're going to get a classic
text, 50% of it, without
spending a nickel.
And that'll be the text for the
first part of the class.
We will be doing some
derivation that
are not in this book.
And for that, I will have notes
that I have written out.
And you'll get Xerox
copies of that.
So we'll have lots and lots of
handouts during the course of
this semester.
I'd like to call your attention,
though, to two
other books.
These are not textbooks.
These are reference books.
And you can see from the shape
of this one that this is one
of my favorite volumes.
It's thoroughly worn out.
This is something that is
called The International
Tables for X-ray
Crystallography.
And it is published by an
organization called the
International Union for
Crystallography.
The funny sounding term,
"International Union for
Crystallography," sounds like
an organization under which
diffractionists go out and
strike for higher pay.
But no, this is actually a
federation of all of the
national societies of
crystallography from
all over the world.
And among the useful things that
they do, besides having a
splendid conference every couple
of years, is to publish
these tables.
And volume one is called
Symmetry Tables.
And everything that we will
derive and all of its
properties--
physical and geometrical--
are tabulated in this book.
It is, however, a reference
book and not a textbook.
You don't learn it for the first
time from this book.
But in terms of generating
atomic arrangements from the
data that's present in the
literature, looking at the
arrangement of symmetry elements
in space, and how
they move atoms around, it is
the code book that tells you
how to crack the arcane language
in which diffraction
and structural results are
recorded and find out how to
unravel it.
I call also to your attention,
although it will not be
germane to this class, there
are four other volumes.
Volume two is called
Mathematical Tables.
And this has all sorts
of useful stuff.
If you've ever done diffraction,
you know that
depending on the symmetry of
the crystal, there are some
planes for which h squared plus
k squared plus l squared
divided by 2 pi is not a
reflection [INAUDIBLE]
if the crystal is green, and
other arcane rules like that.
All of these are summarized
in these books.
There are quantities that you
need to calculate, things like
interplanar spacings.
Tables are available there.
So this is a handy thing
primarily for diffraction.
Volume three is called
Physical Tables.
And this is where you find
things like absorption
coefficients for x-rays
and for neutrons.
It's where you find the latest
values of absorption
coefficients, neutron
scattering length.
And since these things are
derived experimentally, the
values improve and change
from time to time.
So this is where you find the
most up to date values of
physical constants and items
that are necessary for
diffraction.
It never ceases to amaze me how
somebody who has the good
fortune of having to use the
diffraction for a thesis will
labor carefully over making
the measurements and
reducing the data.
And then when it comes to using
a wavelength, which is
how the final numbers will
be determined, goes to an
appendix of a book on
diffraction that was published
20 years ago.
And that's not the most
up to date value.
Scattering powers of x-rays by
the electrons on the atoms are
calculated from wave functions,
which constantly
get better from year to year.
And the value of the scattering
powers of the
function of angle gets better
from year to year.
So this is where you want to
go if you need any of that
physical data.
And finally, volume four is--
it's not its title, but it's
essentially an update of the
Physical Tables, giving later
values which came out about 10
years later.
OK, this series was getting
out of hand.
So I have to bend my knees
and use two hands when I
pick up this one.
This is a continuation of
the series, essentially.
But this one is called
International Tables for
Crystallography, period, no
x-rays because neutrons and
electrons are just as important
today for doing
scattering experiments.
And this is International Tables
for Crystallography.
No x-ray in there.
And there are now something
like six volumes out.
They're not called one, two,
three, and four, but they're
called A, B, and C to
avoid confusion.
And volume A is one called
Space Group Symmetry.
And then, there are a whole
series of other ones.
As I say, I think there's six
of them that give physical
data and all sorts
of useful guides.
I have mixed feelings about
the new series.
You will see that it is about
three times as large and three
times as heavy, which means it's
nine times as expensive.
And to me, it's almost the
case for most people of a
situation where if it wasn't
broke, you shouldn't fix it.
And what they've done is that
they've put in all sorts of
esoteric theory which probably
is going to be of interest and
use to perhaps 5%
of the readers.
But nevertheless, if you wanted,
you'll find it there,
which is something that could
not be said before.
They've added a few things which
are useful, but a lot of
additional information which
you don't really need.
And you pay for that whether
you want it or not.
Nevertheless, it's been done.
You can't buy the old
volumes any longer.
You have to buy the
new volumes.
So anyway, this is what you'll
find in the library now.
Maybe they do still have the old
volumes, one through four.
This, we will make reference to
in the course of the term.
I will give you some copies of
certain pages in here as
handouts when we need
them for purposes of
illustration or for use.
But I spent the last five
minutes just to make you aware
of the existence
of these books.
And these are really the
penultimate source of
information and numerical
quantities that will be used
in diffraction, one of the
principle applications of
crystallography.
I think I have just
enough enough--
to start things off, I have a
syllabus for the course that
is, in very dense form, exactly
what we will be
covering this term.
And I'd like to lead you by
the hand through this.
All right, what we will be doing
in the first half of the
term is something that is known
as crystallography.
OK, the meaning of the word is
almost self-explanatory.
The first part is crystal.
We're going to be
dealing with the
crystalline state of matter.
To me, amorphous materials,
although they may be
important, have all the interest
of a piece of steak
before it's been cooked.
The atoms in amorphous
materials are fine.
But they really get interesting
when they organize
themselves into an
ordered fashion.
So the name is self-explanatory.
The first part, crystal, means
we're going to deal with the
crystalline state.
What does the graphy mean?
That means mapping
or geometry.
And let me give you an example
of a few other words that have
the same sort of structure.
Geo--
the Earth--
followed by graph, geography,
is the mapping of the Earth.
And there are many other terms
that involve these two
separate parts.
Crystallography, though, is
very often subdivided into
different flavors.
There is something well
defined called x-ray
crystallography.
And this is the experimental
determination of the
crystallography of a material
using diffraction, usually
x-rays because they're
relatively inexpensive and
they're widely available.
But increasingly, neutron
scattering or electron
scattering is used
for this purpose.
And there are a number of very
powerful, very exciting
sources of neutrons, either
from reactor sources of
unprecedented intensity or
from what's called a
spallation source, where an
entire synchrotron is built
just to direct a beam
of particles onto
a heavy metal target.
And those high energy particles
split off neutrons
from the nuclei of the
target material.
Doesn't really matter what
the material is.
It helps if it's
a heavy metal.
The nice thing about these
sources of neutron radiation
is that they're so expensive
they are all national
facilities.
And the consequence of that is
that anybody with a good idea
and a project worth doing
can apply for beam time.
And if it's a good problem,
you get it.
So you're using a facility
that cost $1 billion.
You have people whose sole
function in life is to help
you do the experiment and make
sure you're doing it properly.
And this is a very, very
exciting time to be somebody
working with diffraction using
these neutron sources.
There's another branch of
crystallography which is
called optical crystallography.
And this is the characterization
and study of
crystalline materials using
polarized light.
You can identify unknowns using
their optical properties
if they're transparent about 10
times faster than you can
do with x-ray diffraction.
It's a technique that today
is little used.
But it's a very powerful
technique.
And all it takes is
a microscope, and
you're off and running.
Some other flavors of
crystallography, well, I'll
mention the one that
we're going to use.
What we're going to talk about
is something called
geometrical crystallography, to
distinguish it from these
other branches.
And this is synonymous
with symmetry theory.
So that's what we'll do
for the first month
and a half or so.
All right, let me introduce
now some basic concepts.
Geometrical crystallography is
the study of patterns and
their symmetry.
So let me give you an example
of some very simple patterns
that extent in one dimension.
And let me put in a figure.
The thing that is in the pattern
is something that's
called the motif.
And let me use a plump,
little fat comma.
And I'll make a chain
of these things
extending in one dimension.
The nice thing about this fat
little comma is that it is a
figure which, in itself, has
no inherent symmetry.
So it is asymmetric,
without symmetry.
And imagine this is being
repeated without limit in both
directions, both to the
left and to the right.
We then draw another
pattern with a
different sort of motif.
And let me use a rectangle
with one concave side.
OK, and I think you get the
picture of this one.
And imagine that as extending
without limit indefinitely to
the left and to the right.
Then, I'm getting tired of
inventing new motifs.
So let me use the same motif the
second time, but arrange
it in a slightly
different way.
And again, imagine that as
extended indefinitely.
OK, having now generated these
three patterns in two
dimensions but extending
periodically in two
dimensions.
Let me ask the question now.
And even if you have not the
foggiest idea, you have a 50%
chance of being right.
Are any of these patterns
the same?
Or are they all different?
Are any of the patterns
the same?
Or are they different?
Well, that's a-- yeah?
AUDIENCE: [INAUDIBLE].
PROFESSOR: OK, that is an answer
that's right because
the bottom two involve the
same sort of figure.
They have the same sort
of the motif.
They both have the
same rectangle
with one concave side.
And that's a valid answer.
Do you have a different
answer?
AUDIENCE: The first and
third are the same.
PROFESSOR: First and
third are the same.
Why do you say that?
AUDIENCE: They both have
[? rotational symmetry. ?]
PROFESSOR: OK.
This is the point I was
trying to introduce.
And that is your choice of
answering the question, one is
the nature the motif.
And you're absolutely correct.
This pattern and this
pattern are both
based on the same motif.
But in patterns, we are less
concerned with the motif that
is in the pattern than we are
with the relations between one
motif and all of the others.
And in that context, the first
and the third pattern,
although they look entirely
different, are really exactly
the same sort of pattern.
So let's begin to analyze what
sort of operations are in
these patterns that
take one motif--
and obviously, they're
all the same--
and relate it to all
of the others.
First of all, there is an
operation which I'll call
translation for obvious
reasons.
And I'll represent that by a
vector, T, since a translation
has magnitude and direction
but no unique origin.
I could take this pair of
objects sitting nose to nose,
pick them up, slide them over
by T, put them down again.
And I have the relation
that gives me
this neighboring pair.
Pick it up again, move it
to the right by the same
translation in the same
direction, put it down again.
And I've got this pair.
So that is one operation that
can exist in patterns.
This is the operation
of translation.
So let me call that by
a vector relation.
And it has magnitude.
It has direction, but no unique
origin, just like a
plain old vector.
So in other words, I can't say
that the translation moves us
from here to here or
from here to here.
It's all the same thing--
magnitude and direction,
no unique origin.
In fact, all of these patterns
have translational
periodicity.
There's a translation in this
bottom pattern and another
translation from here to here
in the middle pattern.
The thing that makes a crystal
a crystal is that it is an
arrangement of atoms or
molecules which is related one
part to another by the operation
of translation.
If you don't have translational
periodicity, you
do not have a crystal.
So that comes to the
essence of what
crystallography is about.
You can imagine, in one sense,
the generation of this pattern
by a rubber stamp sort
of operation.
Suppose I have a rubber stamp.
And I put on the rubber stamp
the pair of motifs like this.
Pick it up, move
it over, chunk.
Pick it up, move
it over, chunk.
And I can stamp out the pattern
in that fashion.
Notice that my statement about
no unique origin in these
terms can be stated that it
doesn't matter where the two
motifs are on the stamp.
they could be up in the upper
left hand corner, right in the
middle, down in the bottom.
As long as I move the stamp
through the same distance and
the same direction, I get
the same pattern.
Now, that's not bad for
an introduction.
But I want to be more general
than this because when I deal
in terms of a rubber stamp
operation, that is a
transformation that involves
taking one little chunk of a
two dimensional space, picking
it up, and putting it down in
another location to another
unique location in space.
So I'm going to now make another
generalization that
operations, which we've
begun to define,
act on all of space.
So I don't want you to think of
this repetition in terms of
a rubber stamp, although we
could get the pattern that way
and it's conceptually
appealing.
But I'm going to say now that
this string of motifs has
translational periodicity if,
when I pick it up, move it by
T in a particular direction,
and drop the whole infinite
chain back down again,
it is mapped into
congruence with itself.
Which leads me to another
definition--
an object or a space possesses
symmetry when there is an
operation or a set of operations
that maps it into
congruence with itself.
In other words, in plain words,
you can't tell that
it's been moved.
OK, is there anything else that
is a transformation which
leaves the set invariant?
OK, if we look at the first
pattern, there are [? rho ?]
sides such as this one here, or
this one here, or this one
here, about which I can rotate
one motif into its neighbor
or, for that matter, pick up the
entire chain and flip it
end over elbow through
180 degrees.
And it will be mapped into
coincidence with itself.
And that is an operation, and
another sort of distinct
operation of transformation.
And this is one that
I could call
rotation for obvious reasons.
And there are two things I
have to tell you about a
rotation operation.
The first one is the point about
which the rotation takes
place, and that's going
to be some point.
And let me call this point here
A. So this will be some
labelled point that is the
location of the rotation axis.
But then, the other thing that
I have to tell you is the
angle through which I'm
going to rotate.
And I'll append to the
A as a subscript
the angle of rotation.
So this particular operation,
called a twofold rotation
because it rotates through half
of a circle, would be the
operation A pi.
This point is A. We rotate
through an angle pi.
This pattern here has also
rotational symmetry.
In addition to the translation,
there is a
rotation operation, A pi,
in the lower pattern.
So the follow who is unfortunate
enough not to have
a seat-- and I should
have given you this
one a long time ago.
I'll give that to you
as your reward for
giving the best answer.
And you get a seat wherever you
would like to place it.
The first and the final pattern
are the same in the
sense that they contain
two operations,
translation and rotation.
This pattern is a much
more interesting one.
This also has a rotational
symmetry, A pi.
It also is based on
a translation.
But now, there's another
operation that we can do to
leave the pattern invariant.
There exists [? rho sides ?]
that pass through the center
of this rectangular figure
across which I could flip an
individual motif, or for that
matter the entire pattern,
from left to right.
It's a reflection sort
of operation.
So this is a new type
of transformation.
So we'll add that to our list.
And the symbol that's usually
used to indicate the locus of
this operation is m, standing
from mirror.
And that does it for these
particular patterns.
Three sorts of operations--
translation, rotation,
and reflection.
And in fact, that is all
you can have in a
two dimensional space--
not necessarily a
rotation that's
restricted to 180 degrees.
If these patterns are
translational periodic in more
than one direction, you can
have higher symmetries.
One of the things I would like
to suggest to you is that you
look around you in everyday life
at the sort of patterns
that enrich your environment.
I see a one dimensionally
periodic pattern there, the
black and white stripes.
It's translationally periodic,
going up and down.
It also has mirror planes
running through the black
stripes and the white stripes.
I see another two dimensional
pattern back there.
That has translation.
But you could rotate--
no, you can't do anything.
That just has translation,
nothing else.
Get a new shirt.
That's not terribly
interesting.
There's another one there that's
so complex I don't
think I can look at it without
climbing all over him and
drawing some translational
vectors and things like that.
But that's a nice periodic
pattern.
That's a good one.
But there's lots of
stuff like that.
Look at the grills in
the ventilators.
They have mirror planes.
They are translationally
periodic in one direction.
We've got floor tiles.
These are lovely because these
have examples of 90 degree
rotational symmetry.
Same is true of the tiles
up in the ceiling,
same sort of pattern.
And there's another pattern with
four-fold symmetry in the
grills that are underneath
the fluorescent fixtures.
So symmetry is everywhere.
It surrounds us.
We wear it.
We walk on it.
We sit on it.
And think how much richer your
life will be when you can
understand this part of
your environment.
Hey, that's a good, chauvinistic
note, overstated,
on which to end.
So why don't we take
our break?
I'll hang around if you have
any questions or get you a
copy of anything that
came around that you
missed getting one of.
And it is now, according to my
Timex watch, about three
minutes before the hour.
So let's take a break and
stretch for 10 minutes.
Ya'll come back because I've
got your name's on a list.
