Let's continue our project of exploring the geometry of flat surfaces by living on them. If instead of identifying opposite sides of a square directly, we reverse direction in one of the pairs, we get a new surface, called a Klein bottle.

First, a necessary and fun diversion. In general we'll deal with surfaces which have no edges or boundaries, with one exception: the Möbius band. Identify two edges of a square with opposite orientations (or two short edges of a rectangle, if you want to actually make it with paper), and you'll have a space with a boundary called a Möbius band.

- Suppose you start out going right on the Möbius band as in the illustration. What changes as you go around the band once? (Hint: recall that surfaces, like the Möbius band, have no sides or thickness, the surface itself is the space.)
- What happens when you go around the band twice?
- How many different edges does a Möbius band have?

A Klein bottle is a surface with no boundary but with the same property as you've observed above in the Möbius band: going in some directions returns you to the place you've started from but mirror reflected.

- You're now standing on a Klein bottle. What do you see? (Hint: this is trickier than the case of the torus. Draw pictures, check how light rays travel.)
- Now you run forward, in the direction of the red arrow. What do you see happening around you?
- Tired of running, you slowly head North (green arrow). How does the view change?

Check your answer to the above activity here. Just as in the case of the torus,
the "seams" or edges of the square are not *actually* there.
They are not perceived by the inhabitants of our spaces, they simply
help us define the geometry of these spaces. From personal experience,
I think this fact is a lot harder to accept when it comes to the Klein
bottle; there's a natural tendency to picture a magical boundary that
flips you over when you pass through. As I hope you've seen in the past
exercise, this is not so. This fact is so important in our further
study of geometry, that I'll say it again:

Having said that, as inhabitants of a Klein bottle, we can in fact
draw a *fundamental polygon*, which is just math-speak for the
square containing the whole area of the Klein bottle, like the ones
we've been drawing to define the Klein bottle to begin with. Here's one
way to do that.

- Take some paint and a paintbrush. Dip the brush in some paint and press it to the floor.
- Find one of the closest right-side-up (not reflected) images of yourself, and head straight to it, painting a line on the floor as you go. (You'll be going either "up" or "down".)
- Stop when you get back to the start of your line, turn around, and
head toward the
*next closest*right-side-up image of yourself, one either on your "right" or on your "left", continuing to draw a red line on the floor. - When you get to an existing red line, you're done! Note that the positioning of the square grid and alignment of adjacent columns (though not the grid's orientation!) is arbitrary: you can start this procedure from any point on the Klein bottle.

Here's a pictorial illustration.

- How would you modify this procedure to make adjacent columns of squares align, resulting in a chessboard type grid? (Hint: imagine yourself actually doing the painting; modify the beginning of step 3.)
- Find a Möbius band inside the Klein bottle. Find another one! (Possible answer: here's one.)
- Glue two Möbius bands together along their boundary. What do you get? (Hint: see the picture on the right.)
- Can you make a hexagonal Klein bottle analogous to the hexagonal torus from the last section? (Hint: yes you can!)
- How do things look inside a hexagonal Klein bottle? Draw a picture.

So far we've identified the sides of a square in two ways, either gluing opposite edges directly and getting a torus or reversing the direction of one of the pairs and getting a Klein bottle. To the right you see another identification scheme.

- Let's try to figure out what this space is. First use planar tiling like we did with the torus and Klein bottle.
- Your sketch should look something like this. What's peculiar about it? For instance, what happens as you approach a grid point?
- Now investigate this space by gluing corresponding edges together. What do you get?

As you've discovered above, I hope, this space has a different geometry from the torus and Klein bottle. Both of those had a flat geometry: every point looked like a point in the plane, even the corner point of the fundamental polygon. This space does not look the same at every point; it has four points with an angle deficit, as is illustrated below.

Points such as the corners of the "sphere" square above are called
*cone points*. Their angle does not add up to 360 degrees, so
they can't lie on a plane. Here comes another subtle but very important
point: the geometry of the identified square above and its folded
version, the double cone, is *not* the same as the geometry of
the sphere we've explored in the last section. Your avatar the cyclops
can easily tell the difference between the double cone and the sphere by
looking around. However, topologically the double cone and the sphere
are the same. If the distinction is not clear right now, do not
despair! We'll talk more about geometry and topology in just a little
while. For now, on to a new surface!

If you're
interested in the geometry of cone points, I highly recommand watching
the video Not Knot. While
the video focuses on the geometry of knot complements, the last half of
the first part includes a very enlightening discussion of the geometry
of cone points.

Next: Surfaces Classified