a list compiled by Alex Kasman (College of Charleston)
This story was published in the February 1989 issue of Twilight Zone Magazine.
I agree with Vijay Fafat that the author was incorrect to say that a Klein bottle is just like a Möbius strip " just in three dimensions". Perhaps this is a good opportunity for an explanation of how the square, cylinder, Möbius strip and Klein bottle are related.
Picture a square. In fact, let's be more specific and imagine it as being the screen on which you are playing a video game. You have a little dot which you can move around. You can't get that dot to every point on the screen by moving it just horizontally. So, it is not one-dimensional. But you could get it to any point on the screen by moving it horizontally sometimes and vertically other times. It is because two directions of motion are necessary and sufficient that we say the screen is two-dimensional.
In some video games, you just bounce back into the screen if you try to go off one of the edges. If that's the case, then the playing field of the game really is just a square.
But, in other video games, you come back onto the screen in a different location. That changes the topology of the playing field.
For instance, suppose that if you go off the right edge of the screen, you appear at the left edge at the same height (but you still can't go off the top or bottom). This gives the playing field the topology of a cylinder. In fact, if you think of the dot moving on a square piece of paper, roll the paper into a tube and tape the left and right edges together, it would achieve that same effect. (This is topology. Ignore the fact that the paper is no longer flat and think only about where a dot moving on this cylinder would naturally appear if it was going off the right edge of the square which you had now taped to the left edge.)
The Möbius strip topology can also be obtained by thinking of a video game in which you appear at the left edge after going off the right edge, but now you don't show up at the same height. In this alternative version, if you go off the right edge of the screen x inches from the top, then you appear on the left edge x inches from the BOTTOM. In particular, going off just below the top right corner will result in the dot appearing just above the bottom left corner. Again, this can be achieved with a piece of paper, but this time you have to glue the left and right edges so that the top corner of one attaches to the bottom corner of the other. As I promised, this produces the familiar Möbius strip.
Finally, let's consider a game in which going off the right edge x inches from the top sends the dot to the left edge x inches from the bottom (as in the previous paragraph) but also going off the top x inches from the left causes the dot to reappear on the bottom x inches from the right. This is the topology of a Klein bottle. It would be easy to program a computer game to do that, but it is no longer easy to make a version of it out of paper, at least not in the universe as we know it. Technically, we say there is no way to embed the Klein bottle in three-dimensional space (without intersecting itself).
So, to summarize, the cylinder, Möbius strip, and Klein bottle are all 2-dimensional topological spaces which can be made from a square by adding rules about where a dot would reappear if it were to go off the edge. The cylinder and Möbius strip can both be made out of paper, but a real model of the Klein bottle cannot be produced unless you have access to higher-dimensional spaces. I do not see anything about the Klein bottle which would justify calling it a "three-dimensional analogue" of a Möbius strip as Bretnor seems to do. (BTW The Möbius strip and Klein bottle have in common that they are non-orientable, while the cylinder and square are. This is important not only if you want to avoid getting lost while walking around on one, but also if you are trying to compute a flux integral across one!)
|(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)
Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books
let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.
(Maintained by Alex Kasman,
College of Charleston)
(Maintained by Alex Kasman, College of Charleston)