Article on everything2 describing the topology of the Asteroids universe as a torus (donut shape).

…So the topological part is this: when you fly up off the top edge of the screen, you magically appear at the same position on the bottom of the screen, and vice-versa. The same is true of the left and right edges. So consider this: from the pilot’s perspective, he or she is flying around in a 2-dimensional universe with no edge, ie: where every spot the ship is in looks locally like two-dimensional Euclidean space. Mathematicians call this sort of thing a manifold, specifically a 2-manifold. I’m going to represent it like this, as it is represented on the game screen:

The edges ‘a’ and ‘b’ are labelled to indicate that the top and bottom are the same location in space (a), as are the left and right (b). In fact (when you think about it) the four corners are actually the same point! If you were to try to connect this up as a real physical surface (this is called anembedding), you could think about it as a sheet of paper where you first glued edge a-top to a-bottom (giving you a rolled-up paper tube), and then bent the resulting tube around gluing b-left to b-right. You would end up with…wait for it…a donut! Or, in topological jargon, a torus. So when you are playing Asteroids, you are actually playing it on a torus, mathematically speaking. (The advantage to this explanation is that in a bar, there’s always a napkin around that you can use to demonstrate. Sometimes there are even videogames.)…

Read the full article here: http://www.everything2.org/node/746760

Tags: Atari Asteroids, donut, geometry, topography, torus

Categories: Awesome

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