Quotient geometries known in popular culture, such as "flat torus = Asteroids video game"

Question

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:

are there other real-life examples of geometries formed by identifications? We know the cylinder and Moebius strip and there are probably some interesting equivalents of those. Origami are coverings of a punctured torus and I heard that there are crocheted examples of complicated 2-d and 3-d objects. Are there simple, Asteroids-like conversational examples for flat surfaces formed as quotients? Punctures and orbifold points and higher dimensional examples all would be interesting, but I am looking less for mathematically sophisticated than conversationally relevant examples, such as a famous game or gadget that makes a cellphone function as a torus instead of a rectangle.

(edit: Pac-Man, board games such as Chutes and Ladders, or any game with magic portals that transport you between different locations, all illustrate identification of points or pieces of the space, but they lead to non-homogeneous geometries. The nice thing about Asteroids was that it was clearly the whole uniform geometry of the torus.

edit-2: the Flying Toasters screen-saver would have been an example of what I mean, except that video of it exists online and shows it to be a square window onto motion in the ordinary Euclidean plane.)

Answer 1

In (some of) the Civilization games you can decide whether you want to play on a (bounded) plane, or a cylinder, or even a torus, I think.

Answer 2

You could view a kaleidoscope as a picture of a pattern in the quotient of the plane by a suitable group action.

Similarly, some Escher work is a picture of life inside hyperbolic surfaces -- quotients of the hyperbolic plane.

On the more extreme end, $\mathbb R^2$ is a quotient of $\mathbb R$, so presumably you have a stricter question in mind than the one you've actually written, as you can get all kinds of things as quotients. $SO_3$ (rotations in 3-space) are a quotient of the unit quaternions. This is used in computer graphics, among other things.

For the kaleidoscope, you're really viewing the quotient as an orbifold with its natural geometric structure, rather than just as a topological space.

Answer 3

Many role-playing games have toroidal worlds. Especially Dragon Quest and Final Fantasy games. It's a pity that few go through the trouble of implementing a true spheric topology. I don't mind other topologies if they fit the game world, but in RPGs it's mostly understood that the world should be Earth-like.

And then, there is also this marvelous site:

http://www.geometrygames.org/HyperbolicGames/

Answer 4

A running track is a good one-dimensional example. We start with the real line and identify all points {x + n*400meter}, or equivalently we take a 400meter long line segment and glue the ends together. Now, we can sit in the stands and watch an entire 800m, 1600m, or 3200m race, because we have a nice "compact" track!

Once you explain how tiling the real line with 400m segments leads to a running track, it may be easier to explain how two-dimensional tilings of the plane (or sphere or hyperbolic plane) can lead to quotient spaces. Of course, we need to know how to glue the edges, and that point may be a little vague in a casual conversation-- but if Asteroids is not available, I think that tilings are one of the best ways to visualize quotient spaces.

Answer 5

The original Sonic the Hedgehog games (on the Sega Genesis and similar consoles) provide good examples. Many of the levels wrap around vertically, so that there is no bottom or top. They are equivalent to cylinders, I suppose.

Here are some examples from Sonic 3:

http://info.sonicretro.org/File:Ic1map.PNG
http://info.sonicretro.org/File:Mg1map.PNG

The designers used this to make the giant hill at the beginning of the ice cap level that is many times taller than the level itself. Also, the Sonic 3 Special stages are toroidal, just like Asteroids, though in the game they are presented with an apparent spherical curvature, which had me terribly lost and confused until I googled it. In hindsight, I should have realized that you can't cover a sphere with squares, but Sonic doesn't exactly give you a lot of time for thinking.

http://info.sonicretro.org/File:S%26KSS1.png
http://www.youtube.com/watch?v=LpDbSlbhP5M