I think we’re just thinking of different definitions of “halfway”.

On my diagram I put home on the center line. So you return home after going once around (to the north), and you’ve been inverted.

If you start in the rook A1 space (or EZ on my diagram), and head north once around, you reach the H1 space (or EV). At this point you’re inverted. You could head home east or west from here, and you won’t be able to read your junk mail.

But say instead you keep going north from H1. You go around again, landing on A1, and now you’re double inverted. Which is to say back to normal.

I built a Klein bottle with a limited number of plot points… now I understand it. It’s interesting in how one travels and gets flipped. I’ll see about making an animation to illustrate, but for now this shows little sticks standing up on the surface, traveling along one axis on a Klein split in half…

…and notice how you never reach the other side by traveling along the axis. It takes 28 steps to return to where you are. However, if we travel long the center spine, we travel 28 steps and return to where we started…

…but we also travel both sides of the surface (like Andrew’s diagram showed. So cool! With the other half of the Klein, it’s like we have two polar axes, like the poles of a planet in a weird 4th dimensional way.

I’ll see about making an animation (with better colours) that plots the grid beside the 3D model and see if that makes things more clear. You can occupy the same space, but be on the other side of the surface.

Let’s say you’re not on the surface of a Klein bottle, but the surface of a torus.

Since you’re on the surface, it’s a two-dimensional space, and you’ve got two directions you can go in. We can call these X and Y for simplicity.

If you go far enough in either the X or the Y direction, you end up back where you started. From the perspective of someone on this surface (imagine an ant walking on a bagel), they’re on a grid that wraps around, like in the old arcade game Asteroids:

The top is “glued” to the bottom, and the left is “glued” to the right. (And if you could see past the edges of the screen, you’d see the whole thing repeating like a grid.)

A Klein bottle would be the same, except one direction (let’s say the X direction) flips everything around when you wrap around, and the other direction doesn’t.

Imagine a long strip of paper with the ends glued together. Now you still have two dimensions to move in (since it’s a piece of paper), but the Y direction has limits (if you go too far, you just hit the edge of the paper), and the X direction wraps around.

Now imagine twisting the paper before gluing the ends together, so you get a Möbius strip. As before, the Y direction has limits, but the X direction wraps around…except when you go around in the X direction you also flip the world around! You end up on the back of the paper instead of the front. If there are words written on the paper, they’re all backwards now.

A Klein bottle is like a Möbius strip with the other edges joined together too, so that you can wrap around in the Y direction instead of just hitting the edge of the world.

(So really the one in Trinity is more a Möbius strip than a Klein bottle, since you can’t move in what I’m calling the Y direction. To make it topologically a Klein bottle, you should be able to go west or east, climbing across the walls and the ceiling and ending up in the same room again with nothing changing.)

Although what if we add the up and down directions ? If I build a house on one side of the Möbius strip, then we could see that the world is not really flipped around, it’s just that the house is now “underground”, right? (I mean, yeah, OK, it is flipped around in the sense that east/west or whatever now mean the opposite in some sense, but we are not really “mirrored” otherwise ?)

I guess it’s because I’m thinking again in terms of extrinsics coordinates, but I’ve got the feeling that in the context of an interactive fiction, we aren’t really flat entities, are we?

The most studied ones are ‘closed’ 3-manifolds, if you go far enough way, you come back to where you started, and there are no edges. You can also have ones with edges, called ‘3-manifolds with boundary’.

Anyway, there are at least 2 ways to make the Moebius strip 3d and something you can walk on.

Take a long strip (like a big plank of wood but made out of spacetime). Rotate one end 180 degrees and glue it to the other side. If you look at the ‘center plane’ of this plank, it’s a Moebius strip! But, strangely enough, this 3d version loses its ‘reversing’ properties, and can be deformed, without tearing, into a regular old solid torus.

Take Zarf’s chessboard picture, and make it ‘thick’ in a third direction (i.e. coming out of the screen), but keep his gluing scheme. This gives a 3-manifold with boundary where the center plane is a Klein bottle, and it keeps the ‘flipping’ mechanism. However, you no longer rotate when you walk around, and you wouldn’t see an upside down house when you come around.

My Phd Research was actually centered on what you’d see visually in such spaces. If there are no physical objecs in a 3-manifold to obstruct sight, you would see yourself over and over. I’d make fractals that show where those copies of yourself would occur:

These are the 8 possible kinds of 3-manifolds without boundary. The first two don’t have nice patterns and you can ignore those pictures. The third one doesn’t have any copies of you floating around, although you can see the back of your head. In the fourth one, you see a line of yourself forever.
The fifth one is related to the klein bottle and torus. The sixth and seventh look the same because they are ‘quasi-isometric’, and both are hybrids of hyperbolic and euclidean geometry.

The eighth is by far the most common, and is hyperbolic geometry.

I have no idea. How do we get off this crazy train?!

And why is the text on the paper backwards?

When standing on the surface looking at your feet, you can see the other side. An entity can occupy the same coordinates, but be on opposites sides of the surface.

And where is the fourth dimension?

The fourth dimension (just an extra dimension, not time), I believe, comes from the impossible intersection of surfaces if we were limited to 3 dimensions. Notice how one surface intersects the other surface, yet in the reality of a Klein dimension, those 2 surfaces do not interfere with each other.

It also important to realize that the Klein bottle that we see made from glass is not a functioning Klein shape. It fails where the 2 surfaces intersect… because of our stupid 3 dimensional limitations.

Did you read that part of the Trinity playthrough? There’s a very nice little puzzle involving a screw-threaded gnomon.

And where is the fourth dimension?

Our chess examples are written in the style of Flatland. The character has no thickness, and lives in the plane, not on top of or underneath it. They can’t perceive the fourth (third) dimension directly; they can only see how the world is connected up as they travel through it.

Coincidentally, I just bumped into Swanglass from an IF Art Show, which apparently concerns Klein bottles. (I don’t know if it explores their topology at all, though.)

The tutorial mode of HyperRogue (Wikipedia) also good for this sort of thing.
Apparently its available geometries include Klein bottles.