This is not the best wording, but it's what I have for now: Label the top, left, bottom, and right edges of the square T, L, B, and R, respectively (so T and L are the ones which must be glued to each other, in suitable fashion). Cut the square down the \ diagonal; this produces two pieces: an L\B triangle and a T\R triangle. Mark the \ sides of these pieces as having to be ultimately glued to each other in the appropriate fashion, and now re-attach them in the way in which L and T are already indicated to be glued (in the process, flipping one of them over). Now one has the equivalent of the M diagram from before, and thus a Mobius strip.
I believe the bottom of L (BL) must coincide with the left of T (LT) and the top of L (TL) must coincide with the right of T (RT); however, musn't all four points coincide in order to properly reglue the \ diagonal? Wouldn't this appear as a "crimped Möbius strip" (if it is even possible)?
Another way to get to this, with perhaps wrong reasoning:
1. Deform the figure until the non-arrowed edges are a disc, and the arrowed edges are next to one another, so you have something like a Pac man figure where the arrowed edges are the mouth.
2. This figure clearly has 1 edge (the outer edge) but only 1 side -- to get to the 'back' side, go through the mouth
Since all the matters topologically is number of edges and sides, this is identical to a Mobius strip.
Yeah I think a problem with this article is that hints at what you can do in topology, but it doesn't really explicitly tell you what all the rules of the game are.
I guess to some extent that's the difference between an article intended for a general audience, and an actual proper course on point-set topology. There are limits to what you can cover.
