I did talk some about the Klein bottle puzzle in *Trinity* in my postmortem on *A Beauty Cold and Austere*. That puzzle was one of the primary inspirations for the game.

I also wrote the essay about mathematics and paradoxes in IF that @nilsf links to. I don’t mention the Klein bottle, though; the essay’s all about Zeno’s paradoxes.

Getting back to *The Impossible Bottle*, I don’t believe there’s actually a Klein bottle in that game. The title refers literally to the ship-in-a-bottle that’s featured on the cover art, as well as metaphorically to the dollhouse. The former is definitely not a Klein bottle. The latter isn’t, either, although it’s similar. As @lft says in his postscriptum:

The Impossible Bottle contains explicit references to the Infocom game Trinity and its famous Klein bottle puzzle. Technically, the dollhouse isn’t a Klein bottle because it doesn’t mess with chirality (for instance by reversing east and west, which would only have given players a headache). But it has the same “impossible” property that its inside and outside are one and the same. So who built the dollhouse? Mom wrote a book about the inhabitability of self-intersecting manifolds, so I wouldn’t put it past her. Her name, Felicia Small, is of course a nod to the German mathematician in question.

Finally, if I remember correctly @mathbrush’s research area is topology (the branch of mathematics that studies properties of space, including objects like Klein bottles and manifolds). So I’m really looking forward to seeing what interesting puzzles he can create given the world of *The Impossible Bottle*!