topologies of spaces in escher games

Question

There have been a couple of games released (or in development) in the past couple of years which do some weird topological tricks: Echochrome (video), Crush (video), and Fez (video). Do the spaces portrayed in these games correspond to known sorts of topological structures?

Answer 1

The main focus of these games is not weird topological structures so much as weird transformations of the space.

In Echochrome, the "rotation" transformation connects points (via holes) that were not connected before, and undoes some connections. At any instant, the Echochrome space can be viewed as some sort of minor topological quotient of $\mathbb{R}^2$, but derived via rules from some underlying 3D space (with a quotient?).

In Crush, the space is 3-dimensional, but the crush action is like a projection onto 2-d space. The space in Crush isn't really notable, but what is notable is that uncrush followed by crush doesn't always yield the same 2d-world. The issue is that the camera can be in the middle of the 3-d world in crush, instead of far away.

I admit I haven't played Fez, but I imagine some combination of the Crush and Echochrome comments would cover it.