Yayoi Kusama is showing her Infinity Mirror Rooms at the Smithonian. What is the difference between infinity mirror rooms and 3-torus?
Are Infinity Mirror Rooms A 3-Torus?
In a 3-torus you can roam freely around, whereas in the topology implied by a mirror-room an object would collide / intersect with itself as it penetrates the mirror. Thus a mirror-room has a strange topology (non-Hausdorff?).
This means you cannot glue facets of an "elementary" cell together – be it orientable or non-orientable – in such a way that the result is a mirror-room.
To illustrate the previous answer, imagine being inside a cubic room where all faces are mirrors and only one candle is floating.
Now look a the candle, the flame is oriented upwards (to you), imagine now that you slowly sink inside the mirror you're standing on (and let us not worry if you're going through your own self doing so). You are now fully inside the mirror, and the candle you see in the room is now oriented downwards (to you).
If you were inside a 3-torus, the candle would still be upward (to you).