5 ms·
How can you have a 3d space with finite volume and no boundary?
by jugg1es 3y ago
How can you have a 3d space with finite volume and no boundary?
- layer8 3y agoThe same way the surface of the earth is a 2D space with finite area and no boundary.
- alchemist1e9 3y agoMöbius strip: https://en.m.wikipedia.org/wiki/M%C3%B6bius_strip https://en.m.wikipedia.org/wiki/M%C3%B6bius_strip
- jugg1es 3y agoWhat does boundary mean in this context then? Because the mobius strip has a boundary on the sides even if it is continuous. I guess boundary means an edge in all directions?
- simonh 3y agoA mobius strip is an example of a space that is finite but unbounded in a single direction. A two dimensional plane that is finite but unbounded in all directions is called a Kline bottle, but you can’t build them for real in 3D space, only distorted approximations. You don’t have to do such geometric contortions though. The surface of a sphere is a two dimensional space that is finite but unbounded.
- ithkuil 3y agoNot sure why you need a Mobius strip when an untwisted rectangle with two opposite edges joined (outside of a cylinder) has the same property. Unlike a sphere that would be a flat 2d space (but unlike a sphere it would be unbounded only in one direction)
- BLKNSLVR 3y agoWhen you reach one of the boundaries you re-enter the space at a different boundary. It was one of the very few things I was able to understand. There's "continuity" between two points that, in the 3D rectangle representation, don't appear to have continuity. This continuity is possible because spacetime may be all warpy like that... You can't exit the space at the boundaries. The door out of the room walks you into the same room from a different door. eg. exit Face C at some point, and re-enter from Face A, because Faces C and A are aligned at that point. I'm lost beyond that point though.
- prof-dr-ir 3y agoin 2d: take a square and glue the two pairs of opposite sides together. If you do this with one pair you get a cylinder, and then gluing the other pair gets you a torus. No boundary is left. In 3d: take a solid cube and glue the three pairs of opposite sides together. Maybe a bit difficult to visualize, but the idea is that if you live in the cube and try to exit it on one side then you re-emerge into the cube from the opposing side.
- dylan604 3y agoWould this allow for Escher type spaces?
- Filligree 3y agoYes, though not with a didicosm. (Unless your standards for Escher-esque spaces are low.)
- kibwen 3y agoEscher's most fantastical spaces would be classified as non-Euclidean, whereas the article here stresses that the spaces it describes are strictly Euclidean.
- falcor84 3y agos/Euclidean/locally Euclidean/
- cubefox 3y agoI think they are rather forms of parallel projection (where farther objects aren't smaller, unlike perspective projection) combined with forced perspective, where things look locally connected from a specific angle: https://en.wikipedia.org/wiki/3D_projection#Limitations_of_parallel_projection https://en.wikipedia.org/wiki/3D_projection#Limitations_of_p...
- perihelions 3y agoIn the same way a sphere, like the Earth's surface, is a 2d-space with finite area and no boundary.
- Sharlin 3y agoIn 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.
- mike_hock 3y agoTFA tells you how.