ABSTRACT
Exercise 4.1 Write a story in which you travel across the universe to an apparently distant galaxy, only to discover that you’ve made a complete trip around the universe and returned to our own galaxy. When you find Earth, you’re startled to see that it looks like Figure 4.1. What do you see when you land? Describe a walk through your hometown. What do people think of you? □
In the Flatland story, each surveyor came back to Flatland as his own mirror image. To see just how this occurred, study Figure 4.2, which shows a swath of territory similar to the one the surveyors traversed. This swath of territory is a Mobius strip. [A true Mo bius strip has zero thickness. If you mistakenly imag ine it to have a slight thickness-like a Mobius strip
made from real paper-then you’ll run into problems with A Square returning from his journey on the op posite side of the paper from which he started. As long as the Mobius strip is truly two-dimensional (i.e. no thickness) this problem does not arise.]
The question is, in what sort of surface could a Flatlander traverse a Mobius strip? A Klein bottle is one example. You can make a Klein bottle from a
square in almost the same way we made a flat torus from a square. Only now the edges are to be glued so that the arrows shown in Figure 4.3 match up. As with the flat torus, I don’t mean that these gluings should actually be carried out in three-dimensional space; I mean only that a Flatlander heading out across one edge comes back from the opposite one. The top and bottom edges are glued exactly as in the flat torus: when a Flatlander crosses the top edge he comes back from the bottom edge and that’s all there is to it. The left and right edges, though, are glued with a “flip.” When a Flatlander crosses the left edge he comes back from the right edge, but he comes back mirror reversed. A Klein bottle contains many Mobius strips (see Figure 4.4).
