Math 113 Finite Math with a Special Emphasis on Math & Art
Lecture notes. Lun-Yi Tsai, Fall 2012, University of Miami
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3.1
Manifolds
Flatland
We begin with the telling of the story1 of these people that live in a place called “Flatland.” We sometimes call people 2-dimensional if they are really uninteresting and
boring. But these people, whom we call “Flatlanders” are totally 2-dimensional, so for
them there is only left-right and forward-backward–no up and down. They have absolutely no idea that there even is such a thing as up and down; for them sideways and
forward-and-back is all there is to moving around in the world! Almost everyone in Flatland believes they live on an infinite “plane” that stretches on and on in the directions
they know of–left-right and forward-backward.
But there happens to be a small group of rather strange people made up of folks
that think out-of-the-box, I mean out-of-the-rectangle. One of its members, A. Square
decides he’s going to do a little exploring, which this is something that timid Flatlanders
have never done before. Few have ever even left the environs of Flatsburgh, their safe
and comfortable city. A. Square decides to walk straight West and never waver from
this route and as he does so, he lays down red thread to mark his path. Three weeks
later he arrives back to Flatsburgh but coming from the East! Nobody believes A.
Square walked in a straight line wast, they all think he walked in a huge circle and just
happened to come round from the east.
What all Flatlanders thought happened:
Flatsburgh
Undaunted by his critics, A. Square decides to embark on another trip this time heading straight north and laying down blue thread. This time it only takes him two weeks to
arrive back to Flatsburgh; and he comes back from the south! Again, everyone thinks
he veered off his straight path making a big circular detour back to Flatsburgh from the
south. The only thing that puzzles everyone including A Square is that on the whole
two week trip heading north, he never once crossed the red thread!
Group work. If the blue thread doesn’t cross the red thread, can Flatland be a sphere?
But if it isn’t a sphere, what could it possibly be?
1
These lecture notes are based on the work of J. Weeks who has managed to find the most direct route
for non-mathematicians to enter the heart of what mathematicians love to think about. I have also found
inspiration over the years from R. Courant, H. Robbins, H. Poincaré, G. Francis, R. Penrose, W. Thurston,
T. Gowers, E. Abbott, D. Mumford, C. Series, D. Wright, and J. Conway all of whose pioneering works
have blazed a path for the rest of us who want to share higher math with non-math people. Lastly, Bill
Thurston’s recent passing is still much on my mind.
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3.2
The 2-dimensional or flat torus.
We can use a rectangle to describe Flatland, but owing to A. Square’s pioneering
discoveries, we need to make some additions called gluings; by this we mean that the
left and right sides of this rectangle are the same (like they’re “glued” together) and
that the top and bottom are also the same. We don’t actually glue them, rather we
think of it as if we cross the left side, we emerge from the right since these sides are
the same; likewise if we cross the top, we come out on the bottom. This is also called
an abstract gluing as opposed to physical gluing, which means actually rolling up (and
hence deforming) the rectangle to get the sides together and gluing them up. Thus a
rectangle with the above mentioned abstract gluings is called the flat torus.
Example and group work. Flat torus tic-tac-toe. We learn to play a new version of
tic-tac-toe that has some surprising winning combinations as a result of the gluings.
Because of these gluings, it turns out that what seem to be totally different tic-tac-toe
configurations are actually the same! These are called equivalent configurations or
positions.
Group work. Which of the following configurations are equivalent?
3.3
The 3-dimensional torus.
We are working our way to the concept of a manifold, but we are doing so by way
of examples. First, the 2-dimensional plane, which most Flatlanders believe in, is
a manifold; in fact, it’s the simplest 2-dimensional manifold. The flat torus, which is
more interesting, is also a manifold. Now, we move into 3-dimensions. Euclidean 3dimensional space, the topic of linear perspective that we previously studied, is the
simplest 3-dimensional manifold–it’s the idealized version of space that goes out infinitely in the x, y, and z directions.
To construct the 3-dimensional torus, a.k.a., 3-torus, we take a box and perform abstract gluing. Let’s use a classroom, which is usually more or less a box. We say that
the ceiling and the floor are the same. If I make a hole on the ground and jump in,
then I’ll find myself coming in through the ceiling. Furthermore, we say the right and
left walls are the same; and then that the front and back walls are the same. This
would be quite an unpleasant situation for all of us, since it would mean we could
never leave the room–if we went out the door, we’d only find ourselves entering from
the opposite wall!
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A=⇒
A=⇒
Example. If we assume that the walls are see-through, think about what you’d see as
you look at the different surfaces of the room.
Group work. 1) How would you play catch (baseball) with yourself?
2) What if this box were the size of a city? And all north-south streets run one-way, all
east-west streets run one-way, and all elevators go up? Could you get from any point
A to any point B in the city?
3.4
Topology and Geometry
Suppose everything is made of a rubbery material that can be bent, twisted, stretched,
shrunk, and otherwise deformed so long as it’s not cut, punctured, or ripped. Two
things are then said to be have the same topology if either one of them can somehow
be deformed into the other. Thus, a topological property is anything that is unaffected
by deformation; while a geometric property is totally affected by deformation since
geometry is based on measurement. Recall in our study of geometry, we encountered notions such as length, area, and angle measure–all of which are affected by
stretching, twisting, bending, and shrinking.
Group work. 1) Which of the following have the same topology?
2) In the story of Flatland, is the fact that the red thread and the blue thread never
intersect a topological property or a geometric property?
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In the flat torus, we performed abstract gluing. Suppose now that the rectangle is
made of rubber, which allows us to bend, stretch, and shrink as we please; this will
allow us to perform physical gluing. First, we bend the top rectangle down and glue it
to the bottom forming a cylinder; then we bring the left and right ends of this cylinder
together and glue them to each other making an inner tube shape (here, the outside
will stretch while the inside will shrink–kind of how a “slinky” behaves when you match
up the ends). This “surface of a donut” is usually called a torus, but we will often call it
the round torus in order to distinguish it from the flat torus.
The flat torus has the same topology as the round torus, because the gluings are the
same–one being abstract and the other physical. But because we had to deform the
flat torus into the round torus, they don’t have the same geometry.
3.5
Intrinsic and extrinsic properties
An intrinsic property of a space is one that can be observed inside the space, while
an extrinsic property is one that can only be observed from outside the space. If we
represent Flatland by a flat rectangle, e.g. a sheet of paper, and we bend the paper to
form an arch-shape, then for us “Spacelanders” the rectangle has obviously changed.
However the Flatlander living on the 2-dimensional rectangle will not notice anything
different.
If the distance from a point A to a point B in Flatland were d before the bending, then
after the bending that distance would still be d “inside the space.” Of course, what
before was a straight line segment between A and B is now a curved segment, but the
Flatlander would not notice this, because the curving happens outside the space, i.e.,
extrinsically. Therefore, we say that the flat rectangle is intrinsically the same as the
arched rectangle, but that they are extrinsically different.
Another example is if we take a rubber band, cut it, do a 360◦ twist to one end,
and “glue” it back together perfectly. If this happens overnight, then when the 2dimensional residents of the rubber band wake up the next morning, they would not
notice anything different. We are assuming, of course, that the twist did not result in
any stretching so that, for example, the distances between points on the rubber band
remain the same.
Group work. 1) Can you roll a sheet of paper into a cylinder? a cone? a sphere?
What do your answers say about the intrinsic geometry of these surfaces?
2) In the last example, what can you do different to the rubber band to make it into
something that is intrinsically and extrinsically different?
3) Is the sum of the angles of a triangle an intrinsic or an extrinsic property?
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For us, intrinsic properties are much more important than extrinsic ones. The reason
for this is that we experience 3-dimensional space intrinsically! Just as Flatlanders
can’t get out of 2-dimensions, we can’t get out of 3-dimensions. A perpetual question
that non-mathematicians ask of mathematicians underscores this point: “What’s the
fourth dimension?” Because we’re stuck, we need to learn to think of things intrinsically2 . To practice this, we need to study (2-dimensional) surfaces intrinsically as well
(even though we can easily think of them extrinsically).
A convention. We study all (2-dimensional) surfaces intrinsically, i.e., we ignore the
bend in the sheet of paper and the 360◦ twist in the rubber band.
Our philosophy. A manifold exists in and of itself and need not lie in some higher
dimensional space.
This is a very fruitful idea as it eliminates many irrelevant questions3 that obstruct the
way toward a better understanding of reality. This was the work of many mathematicians in the 19th century, but it was Bernhard Riemann who made it into a powerful
theory. When Albert Einstein was struggling to formulate and express his physical
ideas, it was the work of Riemann that provided the mathematical basis for Einstein’s
General Theory of Relativity.
fig.1 The sum of the interior angles of a triangle on different surfaces.
Something to think about. Imagine living in a 3-torus universe.
3.6
Local and global properties
Local properties are those that are observable within a small region of the manifold;
global properties require the consideration of the manifold as a whole.
Group work. Use this definition to determine whether the following discoveries are
local or global.
1) The angles of a triangle are carefully measured and found to be 61.2◦ , 31.7◦ , 89.3◦ .
2) An explorer set out east and returned from the west, never deviating her path from
a straight line.
3) People discover that the area of their world is finite.
Local geometry and global topology.
If we compare a flat torus with a round torus, we can say that they have the same
global topology, but different local geometry. Furthermore, if we compare a 3-torus with
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The difficulty is that most of our thinking (and most of elementary math) is extrinsic. For example, the
notion of the slope of a tangent line of a curve requires that you leave the curve and think of it as sitting
on the xy-plane and then calculate the change in y over the change in x.
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The modern varieties of these questions are: “If there was nothing before the Big Bang, then where
does this Big Bang happen?” or “If the universe isn’t infinite, then what is it sitting in?”
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“ordinary” 3-dimensional space, we can say that they have the same local topology,
but different global topology.
Finally, we have the vocabulary to give a rough definition of a manifold.
3.7
The definition of a manifold
A 2-dimensional manifold is a space with the local topology of a plane.
A 3-dimensional manifold is a space with the local topology of “ordinary” 3-dimensional
space.
Note. This says that all 2-dimensional manifolds have the same local topology; and
that all 3-dimensional manifolds have the same local topology.
Some sobering facts. It turns out that we don’t know very much about our universe:
we know almost nothing about its global topology and very little about its local geometry. This of course means that we know nothing about its global geometry. The foothold
that we do have (and this is our firm belief) is that the local topology of the universe
is that of “ordinary” 3-dimensional space, i.e., that the universe is a 3-dimensional
manifold4 .
3.8
Homogeneous and non-homogeneous manifolds
A homogeneous manifold is one whose local geometry is the same at all of its points.
Otherwise, it’s a non-homogeneous manifold.
Group work. Homogeneous or non-homogeneous?
a. sphere; b. round torus; c. flat torus; d. blob; e. 3-D space; f. 3-torus; g. ellipsoid;
h. paraboloid; i. pseudosphere; j. infinite cylinder; k. plane; l. line; m. 2-holed donut;
n. circle
3.9
Closed and open manifolds
An intuitive definition. Closed means finite and open means infinite.
Group Work. Closed or open?
a. sphere; b. round torus; c. flat torus; d. blob; e. 3-D space; f. 3-torus; g. ellipsoid;
h. paraboloid; i. pseudosphere; j. infinite cylinder; k. plane; l. line; m. 2-holed donut;
n. circle
The problem with edges. It’s important to note that manifolds do not have edges.
Thus a disk with boundary is not a manifold. The reason for this is that if we pick a
point on the edge or boundary of a (2-dimensional) surface, then it won’t have the local
topology of a plane, because it’ll have a sheer drop-off since it’s right on the edge!
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This is actually implicit in our popular science fiction culture: whether it’s Star Wars, Star Trek, Dune,
or whatever, no matter where they warp to, they never warp to a place that isn’t like 3-space.
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The “bad” torus. This is a manifold that looks like a torus except that on one side it’s
got this pointy protrusion that sticks out to infinity. However, this object has finite area;
basically, if we divide the protrusion into centimeter segments then the surface area (in
1 1
square centimeters) of the segments are 12 , 14 , 18 , 16
, 32 , · · · , 21n , · · · . Although this is
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1
1
an infinite sequence, it is easily seen that its sum, 2 + 14 + 18 + 16
+ 32
+· · ·+ 21n +· · · , is
equal to one. Thus the surface area of the entire object must be finite as the remaining
“normal” torus part is finite.
The question then is whether or not this “bad” torus is open or closed. Here our rough
intuitive definition fails us, because the manifold we are looking at is at once finite (it
has finite area) and infinite (it’s got a part that goes off to infinity). In fact, this rather
pathological example of a manifold is not closed. The reason is that when we say
finite, we don’t mean finite in area; instead, finite means finite distance across5 .
A modified definition. A closed manifold in one whose distance across (in all directions) is finite, whereas an open manifold is one whose distance across (in at least
one direction) is infinite.
Important note. From now on, we will deal only with closed manifolds.
3.10
Orientability
Suppose we take a long strip of paper whose ends we want to glue together. But
instead of gluing end to end to form a collar, we do a half twist (a 180◦ twist) before
gluing the ends. This forms what’s famously known as a Möbius Strip6 . Suppose
Flatland has a Möbius strip inside it, then as I showed in class, A Square heading east
would come back from the west mirror-reversed7 .
We should remark that the Möbius strip itself is not a manifold because it has an edge.
The simplest manifold to contain a Möbius strip is called a Klein bottle8 . We form the
flat Klein bottle by taking a rectangle and doing abstract gluing of the top and bottom,
as we did with the flat torus, but instead of saying the right and left sides are the same,
we glue them abstractly with a “twist,” i.e., the upper right side is the same as the lower
left side. Thus, a Flatlander crossing the upper right side would appear at the lower
5
Openness and closedness are in fact topological properties, so it’s natural that we aren’t going to
deal with geometric notions such as area. Here what we really mean is that the manifold be bounded.
6
This is named after the German mathematician and astronomer August Ferdinand Möbius. who was
a student of Carl Friedrich Gauss, the prince of mathematicians.
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We also emphasized that Flatland and the Möbius strip are truly two-dimesional and don’t really have
any thickness at all (unlike a piece of paper); this means there really isn’t the notion of A Square being
on one side of Flatland or the Möbius strip.
8
Named after the great 19th century German mathematician Felix Klein, who proposed the very radical and modern notion that geometry should be thought of as the invariant properties under a group of
tranformations. We will explore groups in our next topic symmetry.
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left side. Furthermore, this would also result in a mirror-reversed Flatlander whose left
and right sides have been interchanged.
Group work. We now introduce flat Klein bottle tic-tac-toe. Play a game of flat Klein
tic-tac-toe with your classmate.
Sometimes it’s confusing to know how things connect up.
There’s a neat trick of attaching eight tic-tac-toe boards to
the original tic-tac-toe board (each is in fact either a copy of
the original board or a copy of the original flipped upsidedown). This forms a larger square of nine boards, which
shows clearly what’s really happening in Klein bottle tic-tactoe.
Group work. There are in fact many Möbius strips in the (flat)
Klein bottle. The most obvious is a strip that runs right down
the middle horizontally. Try to think of what some others might
look like.
Definitions. A path in a manifold that brings a traveler back
mirror-reversed is called a mirror-reversing path. Manifolds
that do NOT have mirror-reversing paths are called orientable;
manifolds that do are called non-orientable.
Examples. The sphere and torus are orientable manifolds,
whereas the Klein bottle is not.
The non-orientable 3-torus. If we take the 3-torus, and make fig.2 Physical-gluing of the rectangle gives us the
“round” Klein bottle.
the front and back walls mirror-reversing (while keeping everything else unchanged), then we have a non-orientable 3dimensional manifold.
Group work. 1) Assuming the walls are transparent, what do you see when you look
through the front or back wall of the room? How about the other walls?
2) Suppose you constructed a jungle gym in your non-orientable 3-torus. You could
have some kids play an interesting form of tag where not only do they have to catch
someone, but they have to say which is that person’s left hand.
3) Think of some amusing things you might do in such a non-orientable space.
The projective plane is a surface that you get by taking a hemisphere (half of a sphere)
and gluing opposite points on its rim together. The gluing is done abstractly9 , which
results in a surface that has the same local geometry of a sphere, but has a global
topology that is different from that of a sphere.
To talk about the shortest distance between two points (in a manifold), we introduce
the notion of a geodesic: a geodesic is an intrinsic straight line. You can think of it as
a string pulled tight in the space, just like in our example of the arched piece of paper.
In 3-dimensional space, we often think of geodesics as the path traveled by a beam of
light. On a sphere, geodesics are segments of great circles. A great circle is any circle
on the sphere whose radius is the radius of the sphere; this makes great circles the
largest possible circles on the sphere, hence their name. All lines of longitude10 , like
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10
It’s sort of hard to visualized the physical gluing since it will result in the surface intersecting itself.
We are assuming here that the earth is a perfect sphere, which it actually isn’t.
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the prime meridian, are great circles; but the only line of latitude that is a great circle
is the equator (the others are lesser circles as their radii are less than the radius of the
earth).
Some questions.
1) Is the projective plane orientable? What happens when you cross the rim?
2) If you (a Flatlander in the projective plane) live at the South Pole, where are you
furthest from home?
3) If Flatlanders want to build two fire stations. Where might they build them? (For
maximum effectiveness, it makes sense that the two stations should be the furthest
distance apart, right?) Where might they build three fire stations?
4) Intrinsically, how can you tell if you are on the sphere or the projective plane?
5) Intrinsically, how can you tell if you are on the projective plane or the Klein bottle?
6) Fill in the table.
orientable
non-orientable
flat local geometry
curved local geometry
7) Is orientability a local or a global property?
8) Is orientability topological or geometrical?
Of course, there’s a lot more that can be said about manifolds, but now the plot thickens. We can actually “add” the manifolds we’ve learned about to make new manifolds.
It turns out that we have a fundamental set from which we can make all 2-dimensional
manifolds! If you’re interested, you can see John H. Conway’s ZIP Proof, which is the
direct route to this celebrated result11 . We can also “multiply” manifolds. So there’s a
kind of “algebra” of manifolds.
If you found this material interesting, then you now have an idea of what many mathematicians like to think about.
fig.3 Conway’s ZIP proof
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In fact, Conway’s name for the proof says it all: “ZIP” for Zero Irrelevancy Proof. The above accompanying drawing is from J. Weeks and G. Francis’s article, “Conway’s ZIP Proof.”
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