# Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

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 It might be out of date in 2010.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.)

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My students claimed to know what pac-man was. – Jack Schmidt Nov 17 '10 at 20:37
Also, a cylinder (rolled up piece of paper) is locally flat, but not globally the euclidean plane. – Jack Schmidt Nov 17 '10 at 20:39
I thought of Pac-man, but it is not evident that the whole screen is a torus rather than identifications of some parts or "portals". The cylinder and Moebius strip are good because they can be made from paper on the spot, but I was hoping for something "cool" and immediately recognizable like the video games. – T.. Nov 17 '10 at 20:40

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– T.. Nov 17 '10 at 20:45

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– Rahul Narain Nov 17 '10 at 21:37
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## 5 Answers

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.

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I have trouble imagining $\mathbb{R}^2$ as a quotient of $\mathbb{R}$. I see $\mathbb{R}$ as a quotient of $\mathbb{R}^2$. Am I reading it wrong, or did you miss-type? If not, It'd be great to have help seeing this. Thanks. – yasmar Nov 17 '10 at 23:26

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@yasmar, yes it's a corker! First step is there is a space-filling curve, by that I mean an onto continuous function $[0,1] \to [0,1]^2$. The $\mathbb Z^2$-translates of $[0,1]^2$ fill $\mathbb R^2$ so you can put together countably-many space-filling curves to construct an onto continuous function $\mathbb R \to \mathbb R^2$ which is a quotient map. – Ryan Budney Nov 17 '10 at 23:39

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Thanks. I figured it had to be either something mind bending like that, or a mistake. I'm glad I asked. – yasmar Nov 18 '10 at 1:46

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/

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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
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