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In 2D the most intuitive way is the "Asteroids topology": when you exit from one edge you reappear from the opposite one. This space is flat, finite, and withou
by Sharlin 3y ago
In 2D the most intuitive way is the "Asteroids topology": when you exit from one edge you reappear from the opposite one. This space is flat, finite, and without a boundary. In mathematical terms it’s what you get when you take a rectangle and identify the left edge with the right and the top edge with the bottom one. It is also topologically equivalent to the (surface of) a torus, which is also flat and wraps around the same way. (Note that the embedding of a toroidal surface in three dimensions is curved, but the 2D space itself is flat: the angles of every triangle add up to exactly 180°.)
In 3D, just generalize from the 2D case.
- pdonis 3y ago> the 2D space itself is flat It's not that simple. "Flat" is not a topological property, it's a metrical property. The correct statement is not that "the" 2D torus is flat, but that it is possible to put a flat metric on the 2D topological space that is called a "torus". That's what the "Asteroids" space does. But it is also possible to put a curved metric on the same topological space--for example, just use the obvious metric derived from the embedding in 3D Euclidean space that we're all familiar with. The "torus" topological space in itself has no metric, and both the flat "Asteroids" metric and the curved "doughnut" metric are valid metrics on that topological space.
- Sharlin 3y agoYes, good point, I wanted to include the flatness property (which was the third condition mentioned in the article) but simplified a bit too much in the process.
- cubefox 3y ago> It is also topologically equivalent to the (surface of) a torus Interesting, I thought it would be like the surface of a sphere. What's the difference?
- dllthomas 3y agoConnectivity. When you go off the top, you come in on the bottom. When you go North on a globe, you come in elsewhere in the North.
- cubefox 3y agoAh, this makes sense. I'm thinking of the Mercator projection of the globe.
- dllthomas 3y agoAmusingly this means a lot of old RPGs and such are arguably set on a toroidal world rather than a globe.
- Sharlin 3y agoWhen you wrap a rectangle around a sphere, all points at the top edge are identified – thecedge is compressed into a single point, the "north pole", and similarly with the bottom edge. When you go off the top/bottom edge of Mercator at longitude N, you emerge at another point at the same edge, namely at longitude N+180 (mod 360). (Also, in Mercator it looks like you can approach the top or bottom edge diagonally, but this is an illusion, an artifact caused by the projection. You can only ever approach the north pole directly from the south, and once you cross it, you find yourself having "rotated" exactly 180° and are now facing south, in addition to having jumped to the opposite longitude. And vice versa for the south pole.)
- roywiggins 3y agoThe poles don't get identified with any point in the plane at all, really - the Mercator projection is infinitely tall, less a rectangle than an infinite strip. The equirectangular projection does map the top and bottom edges to their respective poles, but Mercator just keeps going up (and down).
- pavel_lishin 3y ago> the Mercator projection is infinitely tall Well dang, I just realized that maybe I've based my homebrew D&D world on a Mercator projection.