# topologies of spaces in escher games

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?

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Back when echochrome first came out, I showed it to a fellow grad student (one of the best in our year) who exclaimed "Wow, that's some good projective geometry!". Now, I don't know any projective geometry, so I don't know if that's the right notion, but it's a good place to start looking. –  Jason DeVito Jul 14 '11 at 16:04
"asteroids" anyone? (i believe its $RP^2$, but i havent played for many years...) –  yoyo Jul 14 '11 at 18:58
If "asteroids" was what I think it was, it lived on a torus. "Pacman", on the other hand lived on a cylinder. –  Andrea Mori Jul 14 '11 at 21:09
Related... –  Ｊ. Ｍ. Jul 31 '11 at 14:53
Bonus points for anyone who can describe the topology of Portal. –  Jesse Madnick Dec 28 '11 at 4:47

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?).