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I'm a beginner in Computer Graphics and today, I encountered the concept of "manifold". And according to the brief interpretation in Wolfram MathWorld: (http://mathworld.wolfram.com/Manifold.html), if we can "walk around" at any point on an object, then it is a manifold.

I know this question may sound stupid, but according to that interpretation, is everything around us in the real world is manifold? Because even at the tip of a knife, I can "shrink" myself to "walk around" there. Or please give me some non-manifold examples(I mean, in the real world).

Thanks a lot.

Jiang
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  • This heavily depends on what you mean by "is a manifold", in a way that depends on more precision than the definition "if we can 'walk around' at any point". – Travis Willse Oct 18 '14 at 10:18
  • @Travis You mean the definition that "every point has a neighbourhood"? – Jiang Oct 18 '14 at 10:24
  • That's true of any topological space. A manifold must be (1) locally homeomorphic to $\mathbb{R}^n$ (so, you must be able to walk around, in a way locally equivalent to walking around on Euclidean space), (2) Hausdorff, and (by most definitions) (3) second countable. – Travis Willse Oct 18 '14 at 10:33
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    Anyway, probably one should say that many objects in the real world are well-modelled by manifolds, but asking whether there is a real object that is a manifold is like asking whether there are objects that are perfectly circular. There are very circular objects, but on sufficiently small (i.e., quantum) scales, there is always some "roughness". – Travis Willse Oct 18 '14 at 10:36
  • @Travis That's a quite informative metaphor! I guess I still lack of the ability to consider things in a mathematical way, so I will keep learning. Anyway, thank you for helping me out! :) – Jiang Oct 18 '14 at 10:52
  • You're welcome, I'm glad you found it helpful. Feel free to post on math.se again if you have more questions, and good luck with your studies! – Travis Willse Oct 18 '14 at 10:54
  • let alone, that no object around us stops moving. I wonder if there is a random-manifold definition. – TKM Oct 20 '14 at 19:51
  • @TKM Actually, someone has asked the random-manifold thing. Check http://mathoverflow.net/questions/70714/random-manifolds – Jiang Oct 23 '14 at 05:25

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