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Homeomorphism

[edit]
A classic example of homeomorphism: a coffee mug and a donut are topologically the same.
For discussion.
Reason
It caught my eye, I thought it looked cool :)
Articles this image appears in
Topology, Homotopy, Homeomorphism
Creator
User:Kieff
Nominator
TomStar81 (Talk)
  • Support — TomStar81 (Talk) 04:31, 26 February 2007 (UTC)[reply]
  • Weak support. A little on the simple side, but still interesting and makes the reader want to know more. --Tewy 04:45, 26 February 2007 (UTC)[reply]
  • Weak Oppose. This image doesn't do it for me. it's just too bland. If it were spiced up somehow I would support. Witty lama 04:58, 26 February 2007 (UTC)[reply]
  • weak support - lovely idea, gray is a bit drab, the polygons comprising the solids are a bit simple, i.e. the cylinder part of the mug expands non-smoothly into the handle. Also, it would be nice if the figure preserved its volume throughout. Debivort 08:29, 26 February 2007 (UTC)[reply]
    • Homeomorphisms aren't volume-preserving transformations, so I don't think there is a reason to have the volume preserved. There is something to be said even for not having the volume preserved. Spebudmak 00:58, 28 February 2007 (UTC)[reply]
      • I realize this. I just think it would covey the concept of deformation better if volume wasn't appearing from no where. Debivort 05:03, 28 February 2007 (UTC)[reply]
        • I agree, the bottom of the coffee cup rising to make a solid cylinder at the start of the animation could be a bit smoother with the rest of the deformation. Spebudmak 07:34, 1 March 2007 (UTC)[reply]
          • That was my original idea, but it didn't work the ways I've tried. It's just hard to interpolate a highly concave shape like the mug into a convex shape like the torus the way I did it. Maybe later when I figure a better way to do it, but right now that's beyond my abilities. Sorry. — Kieff | Talk 00:32, 2 March 2007 (UTC)[reply]
  • Comment Good animation. Gray is fine. But I'm afraid it gives a wrong idea of a homeomorphism. It doesn't illustrate which point of the donut goes to which point of the mug. And it is not necessary to have a continuous deformation between the two objects in order to have a homeomorphism: think of a trefoil knot which is homeomorphic with a cylinder for instance. This animation should probably go to homotopy instead. Someone on Talk:Homeomorphism has already made this remark. --Bernard 01:41, 27 February 2007 (UTC)[reply]
  • Comment - Shouldn't this image go in string theory? I remember seeing a film on it and homeomorphism was explained, but I don't remember why. --Iriseyes 19:01, 27 February 2007 (UTC)[reply]
  • Support. The animation is just fine where it is. Just because basic topology is a prerequisite to understand String therory I wouldn't cram it into that article. And I disagree about the move to homotopy too, that's just taking it too abstract. The anim is a proverbial example for topological equivalancy. The fact that it is animated is not the point, it just helps understanding whats going on. How it is animated, whether it conserves volume or not etc. is completely irrelevant to the concept presented. --Dschwen 19:53, 27 February 2007 (UTC)[reply]
  • Weak support This is a good animation and illustrates well enough the concept. But I don't like the part when the cup is "emptied". After all topology is about "deformation" not "removal" of material. Maybe it could be modified to make it more obvious. In that sense, I agree with Debivort's comment. - Alvesgaspar 00:03, 1 March 2007 (UTC)[reply]
  • Comment well, apparently the colors were too faint for anyone else to notice, so I made it bluer. It's the only change I can make right now, with my current tools, time and knowledge. — Kieff | Talk 00:58, 2 March 2007 (UTC)[reply]
  • Oppose. There are several ways that people could get wrong ideas of homeomorphisms from this animation:
    • If you think that a homeomorphism is a continous deformation, you have it wrong. You must understand that the homeomorphism is just the map from the initial state to the final state.
    • If you think that such deformations always exist between homeomorphic objects, and therefore conclude that a trefoil knot cannot be homeomorphic to a cylinder, you have it wrong.
    • Then there are problems about the way the mug is filled. If you visualize it like water being poured in the mug, you have it wrong: you have to deform matter already present, not add some new.
    • If, consequently, you think that the mug should be filled by expanding the inner part of the bottom of the mug, you still have it wrong, because the map is not continuous.
    • If, consequently, you decide to expand the inner and outer parts of the bottom of the mug together, and simultaneously shrink the upper part of the mug, you may still have it wrong, because in the process the inner part of the boundary of the mug (the cylinder part) gets contracted into a circle and the map is no longer injective.

It is likely that some mathematicians, when thinking of homeomorphisms, have in mind something like in the animation, but unfortunately it is difficult to make it into a rigorous argument. I'm curious to know if wikipedians have made the mistakes I describe? It is still a good animation, but should be better explained, probably moved to homotopy, and should not be featured. --Bernard 16:43, 3 March 2007 (UTC)[reply]

Very strongly oppose. See my comments below. --Bernard 23:50, 11 March 2007 (UTC)[reply]
The first three objections are nitpicks. There are many wrong things one can imagine a layman will think from such an animation. The relevant question is whether the essential idea has been conveyed. The last two objections are apparently why BernardH considers this not to demonstrate a homeomorphism, but it does. This is a perfectly good isotopy in fact. --C S (Talk) 17:37, 13 March 2007 (UTC)[reply]
Well, I very well knew all the time that it could demonstrate a homeomorphism, and I wrote it. Salix's solution doesn't surprise me; I could certainly have done something similar if I had wished (it's actually very much like making the two steps of my solution into one). I felt that people, in the discussion above, were at risk of making those mistakes, and I think I was right. Even after I had warned about pitfall 4 two times, someone below still made the mistake. Was I wrong to insist on these problems? I don't think so. My conclusion is that warnings in the image page would be useful. You talk about confusion below but it is not on my side. --Bernard 20:23, 13 March 2007 (UTC)[reply]
  • Support The fact that homeomorphisms are not necessarily continuous deformations does not change the fact that this is an ideal illustration of the often-repeated phrase, in undergraduate classes, of "a donut and a coffee cup have the same topology". As such, per Dschwen, the animation is perfectly adequate at doing what it purports to do. Also, my comments above were just nitpicks and I still think this is a good animation. Spebudmak 04:54, 7 March 2007 (UTC)[reply]
  • Support The fact that is an animation helps the viewer understand how the two shapes are topologically the same. Sure, you don't have to transform between two items with an animation to make them topologically the same, but it sure illustrates the point! Normally I'm a bit skeptical at bland, simple illustrations, but this actually does have a "wow" factor. And it's certainly encyclopedic. Enuja 10:31, 8 March 2007 (UTC)[reply]
  • Support although the caption should be changed from "A classic example" to "The classic example". On Bernard's points above, correct me if I'm wrong, but every single frame in this animation is homeomorphic to every other frame. Also, I find the topologies as physical "matter" objection unconvincing. Since they're both subsets of the R³ they both contain infinitely many points, so there is no matter added even if the object expands. A mug is also homeomorphic to a mug five times its size. ~ trialsanderrors 16:37, 8 March 2007 (UTC)[reply]
    • Thanks for replying to my arguments. It's true that every single frame is homeomorphic to every other, but the reader has to imagine for himself what those homeomorphisms are, and if he takes the trouble to do so, then the animation strongly suggests a transformation that is not a homeomorphism. I'm going to repeat and expand on my last point above: if we expand the lower part of the mug and simultaneously shrink the upper part, it's all the upper, 3-dimensional cylinder part of the mug that becomes contracted into a 2-dimensional annulus. Sure the transformation could be made a homeomorphism, but the animation is not helping. It took me some time to see this problem, and somehow it looks like just a detail, but it still makes the animation either imprecise or mathematically incorrect. That's annoying. I could imagine ways to fix the problem... But anyway I don't like this animation so much. --82.66.235.134 22:25, 8 March 2007 (UTC) --Bernard 22:27, 8 March 2007 (UTC)[reply]
      • OK, I'm not quite if I follow this, your "bottom part" of the mug is the disc at the bottom and the "top part" is the tubular part, and they are for some reason distinct elements? ~ trialsanderrors 08:57, 9 March 2007 (UTC)[reply]
        • That's it. We consider them as distinct elements because we really need to apply different treatments to different parts of the mug. Another possible decomposition would be a radial one, but as I said it doesn't work either. --Bernard 11:17, 9 March 2007 (UTC)[reply]
  • Change to Oppose for reasons unlrealted to mathematics: I just noticed that the lighting is inconsistent between the cylinder and the ring. On the ring, there is a spotlight above the viewer, but this light never appears on the cylinder. Also, as the mug hollows out one of the shadows indicates a light source to the right, but the right side of the cylinder is itself in the shade. ~ trialsanderrors 08:57, 9 March 2007 (UTC)[reply]
    • Actually, you are wrong. In the POV-Ray scene I wrote, there's only one light source just behind the "camera" and a bit to the right. The reason the lighting may look odd is that I'm using orthographic projection and a bit of transparency and ambient light to soften shadows a bit. Also, there are no areas on the surface of the cylinder with a normal vector at the right direction to create a specular reflection in this angle, unlike in the torus, so your criticism doesn't really make any sense, 'mathematically'. Sorry, but there's nothing inconsistent here. — Kieff | Talk 11:12, 9 March 2007 (UTC)[reply]
      • On second viewing it's not inconsistent as much as it's unrealistic. A single light source would leave a spotlight even on a cylindric surface, just turn on your desk lamp and point it on your coffee mug. Of course if you stick to a mathematical model of lighting the spotlight is a single point on the upper ring, which creates the impression that the mug is made from different material than the handle. ~ trialsanderrors 21:28, 9 March 2007 (UTC)[reply]
  • Weak support. After some thinking, I've decided to support my own image. I'm not sure, is this against the rules? ... Now, I like this image. I think it shows well enough how the mug and the torus are topologically equivalent, and it just requires a little bit of thought to figure that the bottom of the cup is rising to the top in order to make the overall shape convex for a smooth transition, and it seems that once the average person realizes this the whole concept of topological equivalence seems to "snap" in place (worked with a few friends I showed to, so I'm happy with the results — but don't take my word for it.) So I guess this animation ended up being a good thing after all. Also, if or not the image would be better at homotopy instead of homeomorphism is irrelevant to this nomination, since it's just a matter of moving the image to a different article. My only issue here is that I wish I could add a texture to it, but that's UV mapping and it only works with parametric surfaces on POV-Ray. That'd be extremely difficult. — Kieff | Talk 20:26, 11 March 2007 (UTC)[reply]
    • I will ask you to be more precise about how you think the object should be deformed when the bottom of the cup is rising. If you think that the bottom should be expanded only in the z direction and the rest of the cup left unchanged, then it is just wrong, since the deformation is not coutinuous. I am annoyed that nobody gives accurate answers to my remarks, and I wonder how people can support without doing so. It seems to me that nobody sees the problem: I can assure you, as a mathematician, that there is one. This problem is all the more serious if nobody sees it: it is acceptable to be approximative only if one is conscious of the limitations. --Bernard 23:50, 11 March 2007 (UTC)[reply]
      • I can assure you, as a topologist, there is no such problem as you imagine. --C S (Talk) 17:37, 13 March 2007 (UTC)[reply]
      • My answer to "I wonder how people can support without doing so" is that non-topologists may support this animation without knowing what's "wrong" with it and without understanding the math just as non-taxonismists can support animal pictures without researching to see if the illustrated animal is, in fact, the correct species and a typical member of the species. This makes it important for topologists, taxonomists, and everyone else to make VERY CLEAR what's wrong with articles or pictures. I'm sorry but I STILL don't understand what's wrong with the picture. Since I still think it very clearly shows that mugs and donuts are the same topologically, I still think this deserves to be a featured picture. Enuja 00:43, 13 March 2007 (UTC)[reply]
        • You are right and my comment was a bit abusive. However I don't know how to explain better than I did... I will ask you to be more specific about what you don't understand. I am a little lost here. I had thought my comments would be understandable at least by mathematicians and would have hoped one of them would lurk around here, so if a mathematician reads this he is encouraged to give his opinion on this matter, whether he understands and agrees or not. If non-mathematicians fail to understand... Sorry, but that is also a weakness of the animation. People think they understand but they actually understand very little, I'm afraid. --Bernard 03:09, 13 March 2007 (UTC)[reply]
      • To be honest, I have no idea what would be the best and most accurate way to make this animation, and I'm pretty sure such a thing is beyond my skills at the moment. So I can't say how it should be deformed. The only reasons it turned out this way, with the bottom rising and all that, is because it gave the best aesthetical results and it was withing my skills. I also have a feeling that a mathematically accurate animation would look less convincing than what it this one is. The original point of this animation, in case you're not aware, was just to illustrate the famous idea that the donut and the coffee mug are topologically equivalent, and to this purpose it seems to be good enough. It was never meant to accurately illustrate the mathematical concept of homeomorphism or homeotopy. See this page for more info on where it came from. I'm certain that it is inaccurate in that sense, but I think you're missing the point and expecting too much from the animation. Meanwhile, you are encouraged to suggest a better and more accurate approach for the image, and if it is within my skills I'll certainly give it a try. Also, if you feel the animation is misplaced and lacks an accurate description to clear things up, just be bold and make the changes yourself! I'd really like some constructive criticism here, and I hope you're willing to provide it. Thanks for the comments, looking forward to a reply. — Kieff | Talk 00:44, 12 March 2007 (UTC)[reply]
        • It is not so much that this picture is inaccurate, but that it suggests a wrong way to deform the object. I am not asking too much of the animation. I recognized its value. But having a mathematically wrong animation in FP, that is not possible. As I said, I have ideas to fix the problem, but that would not put it in FP realm to me, and would make it more complicated; and after all, why fix a problem that nobody acknowledges? The best would be to just warn about those problems in the image page. Sorry, don't want to work on the animation myself, lost too much energy here (actually, asking people who oppose to do better themselves has several times been viewed as bad style on FPC). I will ask for comments in the page you mentionned. --Bernard 01:52, 12 March 2007 (UTC)[reply]
          • I'm not asking you to do better animation or anything like that, I just pointed out that you could have edited the articles already. Also, I noticed you have said, several times, that you have ideas on how to fix these issues you pointed out, but you never really stated what these changes are. I'm just asking you to explain this further. I'm just curious, really. And I'll learn something more on the subject, and that's always a good thing. :) — Kieff | Talk 02:22, 12 March 2007 (UTC)[reply]
            • It is some work to describe, and I would have done it more happily if people had understood my previous comments. Anyway, here it is, I've added the thumbnail on top of the section. This raises a few other issues: it will be difficult to understand without seeing the interior of the mug; people who don't see the original problem will wonder why it is done this way... --Bernard 15:59, 12 March 2007 (UTC)[reply]
    • I'm a graph theorist, and we don't really deal with those kinds of minutiae, but it seems to me your objection stems from the visualization of the mug as a tube on top of a disk, with the diameter of the disk the same as the outer diameter of the tube. In that case the removal of matter from inside the tube amounts to a reduction to a 2-dimensional annulus. But an alternative visualization is the disk inside the tube. In that case the removal compresses the inner cylinder into a flat but 3-dimensional disk – a volume-reducing but perfect homeomorphic transformation. ~ trialsanderrors 09:14, 12 March 2007 (UTC)[reply]
  • support graphic is fine, illustrates the topological concept of a homotopy perfectly. But I do agree with Bernard that its not the right one for homeomorphism. It actually reinforces the wrong idea about what a homeomorphish is. People will look at the animation and leave with that incorrect impression. --Salix alba (talk) 22:24, 12 March 2007 (UTC)[reply]
Graphically illustrating why the image could be confusing for homeomorphism. There is a homeomorphism between the torus and Trefoil knot, but no homotopy. --Salix alba (talk) 09:34, 13 March 2007 (UTC)[reply]
  • Thank you for giving your opinion on the homeomorphism/homotopy problem. I would like to ask also what you think of the other problem I wrote about, namely that the transformation that is suggested between the torus and the mug is not in fact an homeomorphism (fails to be either continuous or injective)? Whether you understand, agree, and think it is a serious problem or not... --Bernard 12:24, 13 March 2007 (UTC)[reply]
I beleive you can construct a continuous and injective map which mirrors this illustration. What you don't get is differentiability. Consider just a small portion round the top of the cup, before its been pressed in, and just after. A slice through is illustrated below, I've constructed two diagonals lines and divided the interiour into three sets of points: a,b,c.
-------------       |------
aaaa\bb\ccccc       |\ccccc
aaaaa\bb\cccc       |b\cccc 
aaaaaa\bb\ccc   ----|bb\ccc
                aaaaa\bb\cc
The deformation maps each set of points before onto the corresponding points after. Hopefully enough to convince you. --Salix alba (talk) 13:13, 13 March 2007 (UTC)[reply]
I expect many people intuitively visualize something of this sort when they see the animation. I'm rather baffled that it has been a source of confusion, but in hindsight it's somewhat understandable. I remember when I started learning topology that I would overthink these things. There was a tendency to think things really couldn't be as they somehow appeared. If one works a lot in hands-on topology in 3 dimensions, one learns to trust one's intuition again (or at least certain parts of it....). --C S (Talk) 17:37, 13 March 2007 (UTC)[reply]
  • Support. The only real objection I see is that the animation demonstrates an isotopy, which is a stronger condition than homeomorphism between two objects contained in an ambient space. Is this a serious objection? I don't think so. The gist of what topology is about is conveyed more than adequately. It's a great animation. --C S (Talk) 17:37, 13 March 2007 (UTC)[reply]

Promoted Image:Mug and Torus morph.gif --KFP (talk | contribs) 19:42, 13 March 2007 (UTC)[reply]

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