NOTE: I *know* this is long, but it's very cool material (I think), and is designed to be accessible without really any math background whatsoever, just an active geometrical imagination. So don't let the title scare you away--read on.

The study of topology is difficult to explain to people. After I
had chosen it as an area of specialization in grad school, I would often
get "so what are you studying?" at parties, etc. First I would
get the usual "oh, mapmaking? How interesting!" (no, it's math)
and then the inevitable I was never any good at maths in school. At this point,
if my interlocutor had still not fled in terror and looked at all interested,
I would try to present a simple example to convey a bit of what topology
actually is. So, like most mathematicians I know, I would tell the parable
of the coffee cup and the donut.

The parable goes like this: from the standpoint of homotopy (one
branch of topology), the surface of a coffee cup and that of a donut
are the same, because they have only one "hole" in them. Both
are a 1-handlebody: there's only one place to loop a string through
them and carry them around. Now, mathematicians think this is a way cool
example to use because they drink a lot of coffee, and many of them like
donuts quite a bit. Sadly, most people at parties are of the opinion that
any idiot would know that coffee cups and donuts are entirely different
things, and maybe if this guy would have a decent breakfast once in a
while he wouldn't be so odd.

So I have been searching for a better example since then, and a while
ago it hit me: Asteroids. No, not the big space rocks: the video game.

Remember Asteroids? (I can't believe that I'm dating myself with this.)
One of the original cool vector graphics-based video games, you flew a
little triangular ship around a rectangular screen in an asteroid field,
shooting stuff frantically. (I actually had one of these in my junior year
dorm room, although it belonged to The Custodian. Sigh.)

So the topological part is this: when you fly up off the top edge
of the screen, you magically appear at the same position on the bottom
of the screen, and vice-versa. The same is true of the left and right edges.
So consider this: from the *pilot's perspective*, he or she is flying
around in a 2-dimensional universe with no edge, ie: where every spot
the ship is in looks locally like two-dimensional Euclidean space. Mathematicians
call this sort of thing a manifold, specifically a 2-manifold. I'm going
to represent it like this, as it is represented on the game screen:

a
+---------+
| |
| |
b | | b Game #1: toroidal universe
| |
| |
+---------+
a

The edges 'a' and 'b' are labelled to indicate that the top and bottom
are the same location in space (a), as are the left and right (b). In
fact (when you think about it) the four corners are *actually the same
point!* If you were to try to connect this up as a real physical surface
(this is called an embedding), you could think about it as a sheet of
paper where you first glued edge a-top to a-bottom (giving you a rolled-up
paper tube), and then bent the resulting tube around gluing b-left to
b-right. You would end up with...wait for it...a donut! Or, in topological
jargon, a torus. So when you are playing Asteroids, you are actually playing
it on a torus, mathematically speaking. (The advantage to this explanation
is that in a bar, there's always a napkin around that you can use to demonstrate.
Sometimes there are even videogames.)

So the obvious question to ask is: are there other kinds of Asteroids
screens that we could devise that would be qualitatively different?

*(at this point, you sense math coming and start to gaze furtively
around the room)*

*(No wait! Stick around! Let me get you another drink...)*

Why of course there are. Before we tried to glue it together into a physical
surface, the Asteroids screen was just an idea, an *abstraction*. So
for instance, who says that when I fly my ship past the top edge it has
to come out in the same position on the bottom? What if we change the game
so that when you flew into an edge the ship came out on the *opposite*
position on the other side:

a
+----<----+
| 1 |
| |
b v ^ b Game #2: Real projective plane universe
| |
| 2 |
+---->----+
a

Now when I fly up past the top edge at (1) I re-emerge on the bottom edge
but *on the other side from left to right* at (2), and similarly for all edges. What I have done is changed
the orientation of the sides that we are identifying as the same (and so
we won't get confused I have added arrows above to denote the direction
of orientation). The result above is called Real 2-dimensional projective
space, or (more specifically) the Real projective plane, RP^{2}
for short. There are many formulations for this: it is equivalent to a sphere
where two points are considered the same if the line through them intersects
the center of the sphere (i.e., if the two points are exactly opposite
on the sphere). Try as you might, you won't be able to physically make one
of these, because unlike the torus, RP^{2} it isn't embeddable in
R^{3} (Euclidean 3-space, where we live).

To get a feel for this, the orientation-reversal is exactly the difference
between a strip of paper glued in a loop and a strip glued in a loop with
a half-twist, also called a Mobius strip (also noded on E2 as Moebius
strip, because its namesake was named 'Möbius' with the umlaut).
A Möbius strip is remarkable because it is nonorientable -- it has *only one side*,
due to the twist:

+-------edge---------+ +-------edge---------+
| | | |
a ^ ^ a a ^ v a
| | | |
+-------edge---------+ +-------edge---------+
loop of paper Möbius strip
(glue ends directly) (twist ends, then glue)

(Try making one right now if this doesn't make sense. Now try to color
the two sides differently.) That same sort of non-orientability is going on with the projective plane when it reverses edges, but it's harder to visualize because we can't make a physical one and try it.

OK, any others? If we reverse the orientation on one of the edge pairs
(b) but not the other (a), we get:

a
+---->----+
| |
| |
b v ^ b Game #3: Klein Bottle universe
| |
| |
+---->----+
a

Now we have reversed orientation in only one direction, so when we fly
up or down we emerge in the same place, but left-right we we end up on the
opposite end. This is equivalent to a Klein Bottle. The Klein bottle is
also not embeddable in R^{3}: the way a Möbius strip is a strip
with only one side, a Klein Bottle is equivalent to a bottle whose inside
is the same as its outside. In fact, a Klein bottle is just two Möbius
strips glued along their edges.

How about the sphere? Yes, but it is so simple we don't even need 'a'
and 'b' sides to define it. All we need is:

a
---->----
/ \
+ * Game #4: spherical universe (no left-right, just up-down)
\ /
---->----
a

Here again, the a sides are identified but now the two endpoints are
now distinct. (If this is difficult to visualize as a sphere, think of
it as an opened change purse.)

In fact, we don't need the 'b' sides to represent the Real Projective
plane as we did before either: it looks like the sphere above with the orientation
reversed (which necessitates making the endpoints the same again):

a
---->----
/ \
+ + a simpler projective plane diagram
\ /
----<----
a

If you know a little group theory, you can maybe see where this is
going. We are really enumerating all possible edge-paths, and looking
for equivalences by considering quotient spaces of those paths. Imagine
a small bug crawling counterclockwise around the inside edges of these
figures. When it is traverses the same direction as an edge is oriented,
we will write (for example) 'a', and conversely 'a^{-1}' (a^-1,
ie: a inverse) if it is walking against the arrow. In this manner we
compose a path around the whole figure. When our little bug returns to its
starting point, we will have made a trip equal to the identity, which
we write as 'e'. So over the different manifolds we get:

**Torus**: aba^{-1}b^{-1} = e

**RP**^{2}: abab = e

**Klein bottle**: abab^{-1} = e

**Sphere** aa^{-1} = e

after a little right-multiplication we can rewrite this as:

**Torus**: ab = ba (commutativity)

**RP**^{2}: ab = b^{-1}a^{-1}

**Klein bottle**: ab = ba^{-1}

**Sphere:** a = a

So that's it. A rigorous proof is longer and more formal than that,
but those are all the 2-manifolds there are. (Technically: all compact
2-manifolds are the connected sum of spheres, projective planes and tori,
ie: anything else can be built by gluing copies of these together. Klein
is in fact redundant as a building block, because it is the connected sum
of 2 projective planes). Hence: these are the four different ways you could
design an Asteroids game.

If you made it all the way down here, then huzzah! it's worked. Let's
you and I go out for some beers and test it on other people...