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Q: Isn't a Klein bottle a 2-dimensional surface? How is this a Klein bottle?
A: Yes, a Klein bottle is a 2-dimensional surface, a so-called "manifold." The Klein bottle in this little maze is the surface exactly halfway between the "floor" and the "ceiling." As you explore, you'll notice that the "floor" and the "ceiling" are locally defined - one time you might pass the spot with your feet on one side, the next time you pass the spot, your feet are on the other side.
Q: But isn't a Klein bottle a 4-dimensional curved surface? This surface seems flat.
A: A Klein bottle is a topological space. It cannot be represented as a subspace of 3-dimensional space without intersecting itself, so it is often thought of as existing in 4 dimensions. Our maze only looks like it is embedded in 3-space. It is not.
Let's skip back a bit. Have you played the game "Asteroids?" It's a game where the field of play is a rectangle. When you go off the right side of the rectangle, you come back on the left, and when you go off the top side of the rectangle, you come back on the bottom. If you were a pilot in a ship on this field, you'd have no knowledge that there even was an edge.
We might guess this playing field is much like a sphere - after all, if I travel in the same direction on a sphere, I get back to where I started, right?
This guess would be incorrect. On a sphere, if I travel a circuit from point X back to point X, never intersecting my path, the path seperates the sphere into "left" and "right." There is no way to get from the left side of my path to the right side without intersecting my path. That is a topological invariant of a sphere.
On the other hand, in the Asteroids field of play, if I travel "straight up" until I return to my starting place, this does not seperate my surface - a person can get from any point in the surface to any other point without crossing my path.
In fact, topologically, the playing surface of "Asteroids" is a torus - a donut. Imagine you took your playing field and glued together the left hand and the right hand sides - you'd get a cylinder. Now if you wanted to glue the top and the bottom of the cylinder together, you'd get a torus.
Now, imagine what the pilot of the Asteroids ship would see in this universe [Figure 1.] If she looked straight up, and there were no obstructions, she'd be looking at the rear of her ship. If she looked straight left, she would see the right side of her ship. If she looks in any angle, in fact, she'd have an infinite number of views of her ship.
There's another way of thinking about this universe, though. Imagine the infinite plane, tiled by rectangles, such that for every object in the universe, there is a duplicate object relocated to every rectangle [Figure 2.] Our pilot would be looking at not her ship, but an exact duplicate, with an exact duplicate of herself guiding each one. When she turns left, they turn left. When an asteroid leaves her "rectangle," one enters from the rectangle of one of her duplicates. When she leaves her rectangle, each duplicate likewise leaves her rectangle. She might send a radio message to one of her duplicates, but she'd receive exactly the same message from her corresponding duplicate. Say the message was, "Turn left, and I'll turn right." Our pilot would receive the exact same message shortly after sending it - would she turn left or right? Whichever way she'd turn, all her duplicates would choose the same. Since she and her duplicates are identical in every way, we can think of the collection of them as a single entity, of which each one is but an example. As far as she cares, she's on a torus, because the difference between the two universes is non-existent.
The Klein bottle can be "flattened" in a similar way - but in this case, one pair of sides is twisted. For example, in [Figure 3], the small asteroid on the boundary of the rectangle reappears on the *opposite* side of the opposite edge. In this example, I've included vectors for that asteroid, and the direction it is traveling, which also gets twisted on the border.
We can again see this playing field as an infinite plane [Figure 4] with an infinite number of duplicates of every item. The difference is that each duplicate is not "oriented" the same way. But each one is identical. If our ship goes "up", the things that were previously on the left will now be on the right.
In the Klein gallery, only the "middle plane" - halfway between floor and ceiling - is a Klein bottle.
To be done.
Copyright 1999-2002. Thomas Andrews
(thomas@thomasoandrews.com)
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