Wikipedia:Reference desk/Archives/Mathematics/2018 July 26
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July 26
[edit]Klein bottles
[edit]
Novelty item says "Novelty items based on mathematical objects that cannot exist, such as Klein bottles, and Penrose triangles, have been manufactured." Conversely, Klein bottle has photographs of Klein bottles, e.g. File:Acme klein bottle.jpg or the picture at right. Are these novelty items merely poor-quality representations of something whose ideal form (bring in Plato) cannot be constructed, or is there a mistake in the novelty item article? Nyttend backup (talk) 13:45, 26 July 2018 (UTC)
- The former, a Klein bottle in 4 dimensions wouldn't look like it's going through itself. Imagine trying to draw a Mobius strip (which is easy to create in our 3 dimensional world) on a piece of paper (which only has 2 dimension), it's the same effect as what happens when you create a Klein bottle in 3 dimensions. Iffy★Chat -- 13:52, 26 July 2018 (UTC)
- I just edited the article (which is kind of lacking anyway) to remove the "cannot exist" bit. I figured that was probably the route of least resistance without digressing into information on manifolds, orientability, immersions, embeddings, and so on. –Deacon Vorbis (carbon • videos) 14:00, 26 July 2018 (UTC)
- To add a bit to all that though, what's actually impossible is embedding (smoothly placing without self-intersections) a Klein bottle in ordinary three-dimensional space. The Klein bottle itself is a two-dimensional surface, like a sphere. But unlike a sphere, it's not orientable (you can place your finger down on one point, slide it around, and wind up on the opposite side of the surface). The other common example here is the real projective plane. And nonorientable surfaces can't be embedded into three dimensions; they require at least four (and at most four, by the Whitney embedding theorem). –Deacon Vorbis (carbon • videos) 14:13, 26 July 2018 (UTC)
- I'm reverting Vorbis's change; the novelty items are based on the objects that cannot exist, so there's no contradiction, and that was the main point of the sentence. Substituting something more specific like "cannot exist in 3-dimensional space" would be reasonable for the Klein bottle at least, if people want. --76.69.47.228 (talk) 05:05, 27 July 2018 (UTC)
- To add a bit to all that though, what's actually impossible is embedding (smoothly placing without self-intersections) a Klein bottle in ordinary three-dimensional space. The Klein bottle itself is a two-dimensional surface, like a sphere. But unlike a sphere, it's not orientable (you can place your finger down on one point, slide it around, and wind up on the opposite side of the surface). The other common example here is the real projective plane. And nonorientable surfaces can't be embedded into three dimensions; they require at least four (and at most four, by the Whitney embedding theorem). –Deacon Vorbis (carbon • videos) 14:13, 26 July 2018 (UTC)
- I think they are pretty nice models, but if you want to think of it that way they are just poor mistaken representations. Where a tube goes through a wall the wal should not have a hole in it - but in the model it does. This can be done with another dimension by moving the bit of tube a bit into that other dimension and back again aon the other side of the wall. Dmcq (talk) 14:56, 26 July 2018 (UTC)
- Thank you for clearing up the article issue. Since you raise the 4-dimensional bit, another question. I see While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however. Does this mean the three-dimensional models can't exactly work if they're fixed, but they can be expressed by means of change over time, i.e. at any specific point it won't work but once you bring in time, the whole "process" somehow creates it? I'm basically clueless about how time interacts with the other dimensions in this area. Nyttend backup (talk) 15:26, 26 July 2018 (UTC)
- There isn't really any notion of "time" here. Real, physical objects that people call Klein bottles are immersions (which allow self-intersection), but not embeddings (which don't) (links above). If you want to have a Klein bottle in three dimensions, it's going to have to intersect itself; there's just no way around that. Maybe what you're thinking of is the common process of visualizing a four-dimensional object by playing a movie of cross-sections. You could certainly do that here; what you'd see is an animation of curves in three dimensions. But that's really just a visualization method; it's still looking at the bottle in four dimensions. –Deacon Vorbis (carbon • videos) 15:43, 26 July 2018 (UTC)
- Hm, I thought when we talked about "fourth dimension", we were talking about the "normal" three, and then the fourth dimension was time. Nyttend backup (talk) 17:08, 26 July 2018 (UTC)
- You're confusing the mathematical notion with a physical one. Our physical spacetime is four-dimensional, and you can sort of pick out a "fourth" dimension of time because of the fact that timelike and spacelike directions can be distinguished. But the mathematical notion of Euclidean space can have any (finite) number of dimensions, and none of them necessarily correspond to time. --Trovatore (talk) 18:04, 26 July 2018 (UTC)
- Oh, okay. [Now I'm having more trouble understanding the situation; I'd have to take an advanced mathematics course to understand it :-)] I've seen moving images of tesseracts, e.g. File:8-cell.gif and File:8-cell-orig.gif, and I figured the movement somehow conveyed the way in which the cube changed through time. Nyttend backup (talk) 18:28, 26 July 2018 (UTC)
- It's not a terribly difficult concept as mathematics goes. Just takes some getting used to. Look up Heinlein's story —And He Built a Crooked House—, which is very entertaining, and if you carefully work through the progression of the various rooms, may also help you build intuition. --Trovatore (talk) 18:33, 26 July 2018 (UTC)
- Those animated images show all sixteen vertices all the time. It couldn't do that if time represented one of the four dimensions. —Tamfang (talk) 06:12, 29 July 2018 (UTC)
- Oh, okay. [Now I'm having more trouble understanding the situation; I'd have to take an advanced mathematics course to understand it :-)] I've seen moving images of tesseracts, e.g. File:8-cell.gif and File:8-cell-orig.gif, and I figured the movement somehow conveyed the way in which the cube changed through time. Nyttend backup (talk) 18:28, 26 July 2018 (UTC)
- You're confusing the mathematical notion with a physical one. Our physical spacetime is four-dimensional, and you can sort of pick out a "fourth" dimension of time because of the fact that timelike and spacelike directions can be distinguished. But the mathematical notion of Euclidean space can have any (finite) number of dimensions, and none of them necessarily correspond to time. --Trovatore (talk) 18:04, 26 July 2018 (UTC)
- Hm, I thought when we talked about "fourth dimension", we were talking about the "normal" three, and then the fourth dimension was time. Nyttend backup (talk) 17:08, 26 July 2018 (UTC)
- There isn't really any notion of "time" here. Real, physical objects that people call Klein bottles are immersions (which allow self-intersection), but not embeddings (which don't) (links above). If you want to have a Klein bottle in three dimensions, it's going to have to intersect itself; there's just no way around that. Maybe what you're thinking of is the common process of visualizing a four-dimensional object by playing a movie of cross-sections. You could certainly do that here; what you'd see is an animation of curves in three dimensions. But that's really just a visualization method; it's still looking at the bottle in four dimensions. –Deacon Vorbis (carbon • videos) 15:43, 26 July 2018 (UTC)
- Thank you for clearing up the article issue. Since you raise the 4-dimensional bit, another question. I see While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however. Does this mean the three-dimensional models can't exactly work if they're fixed, but they can be expressed by means of change over time, i.e. at any specific point it won't work but once you bring in time, the whole "process" somehow creates it? I'm basically clueless about how time interacts with the other dimensions in this area. Nyttend backup (talk) 15:26, 26 July 2018 (UTC)
- You might like the old book Flatland which is about a 3-dimensional sphere ("spheres are people, my friend") visiting a 2-dimensional world. The 2-dimensional people have the same trouble understanding the 3rd dimension as we in the 3-D world have in understanding the 4th dimension. The book is now in the public domain and easy to find online. 173.228.123.166 (talk) 05:33, 27 July 2018 (UTC)