II.
A Pause for Reflection
Sometimes the balloon is booby-trapped.
Grinning little vandal, full of pranks, you jab with your pin. Ouch! It isn’t a balloon at all. It’s a Klein bottle. The pin comes through behind you, butt high.
If you’re obstinate, you play Torquemada with yourself for a long time.
Take a strip of paper. Make it, say, two inches (or five centimeters if you’re metrically minded) wide and fifteen (40 cm. is close enough) long. Give it a half twist, then join the ends. Take a pencil and begin anywhere, drawing a line parallel to the paper’s edge. In time, without lifting your pencil, you will return to your starting point, having drawn a line on both sides of the paper.
The little trickster is called a Moebius Strip. You might use it to win a beer bet sometime.
Now imagine joining the edges of the strip to form a container. What you would create, if this were physically possible, is a hollow object whose inside and outside is all one contiguous surface.
It’s called a Klein Bottle, and just might be the true shape of the universe.
Again, you could begin a line at any point and end up where you started, having been both inside and out.
There is always a line, or potential line, before your starting point and after, yet not infinite. Indeed, very limited. And limiting. On the sharply curved surface of the bottle the line can be made out only for a short distance in either direction. You have to follow it all the way around to find out where it goes before it gets back.