3d equivalent geometric shape of a 2d tiled space

Question

In case anyone remembers the old game Comets, it was about this: You had a spaceship which you could move around the screen and various meteors appeared and you had to shoot them up. When you moved the spaceship up and it reached the top of the screen, it then appeared at the bottom of the screen at the same distance from the left and vice versa. The same went for moving left and right. Correspondingly if you go to the top left corner by moving diagonally towards it and you reached the end of the screen, you would appear at the bottom right corner.

So my question is, is there a real-world equivalent 3d space geometric shape for this sort of tiling 2d space ? If so, what would it be ? A torus ?

Also, what would the perception - concerning a being living in such a world - for the shape of the "place" it is living on, be ?

I'm providing a link to such a "world".

http://picpaste.com/pics/tiled_map-lT9i1TWT.1457704432.jpg

Apparently there is a rectangular red coloured mountain in this world, a cyclical lake and an orange coloured building.

Thank you advance.

Answer 1

This world is what geometers call a "flat torus". Curved tori do exist in ordinary 3-dimensional space: the surface of a doughnut is a good example. Flat tori do not exist in ordinary 3-dimensional space, but they are no less real from a mathematical perspective.

There is a nice book explaining flat tori and related geometric objects: "The shape of space" by Jeff Weeks.