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Started on 18:14 on this video(problem 1), A professor mentioned he could make a topological object with a single piece of paper and without glue, how are you able to make it? By the way, how does that have to do with the theory of topology. link: http://www.youtube.com/watch?v=Ap2c1dPyIVo&feature=related

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    @mixedmath - i have no idea how to do it.2012-04-07
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    but have you tried? He tells you that you make exactly 2 cuts, and you fold the paper, and that's all.2012-04-07
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    @mixedmath - i missed the part you mention me in the video, but after i tried i still not getting it.2012-04-07
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    Actually, I think you need 3 cuts.2012-04-07
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    @DejanGovc - How?2012-04-07
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    Oh - you're right. Well seen.2012-04-07

1 Answers 1

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This is an oldie but goodie. Cut along the indicated three lines. Fold one of the dotted lines one way, and the second one in the reverse direction.

enter image description here

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    [beat me to it](http://i.stack.imgur.com/zI6fz.png) :)2012-04-07
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    +1 for excellent answer, but how does that relate to the course of topology?2012-04-07
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    @Victor: Well, once you make the three cuts you get a topological space homeomorphic to a disk. The folds don't matter from the topological perspective. Not sure what point Wildberger was trying to make with this.2012-04-07
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    @t.b. yours is a mirror image of mine. Perhaps that counts as a different answer. :)2012-04-07
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    maybe :) @Victor: I see the point of the exercise on a much more basic level than Jim: in my opinion spatial imagination and intuition is one of *the* very crucial prerequisites for doing (algebraic) topology and all sorts of other math. Train it, look at knots, for example.2012-04-07
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    @JimConant - For three cut you could make any paper into a disk?2012-04-07
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    @Victor: Theo is surely right, but anyway the paper is already homeomorphic to a disk. Making incomplete cuts (that don't cut it in half) will keep it homeomorphic to a disk.2012-04-07
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    I thought in topology you can bend or stretch the object but you cannot cut it?2012-10-25
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    @Danica: you are correct, and that is an astute observation. The point here is that the obvious map from a paper to a paper with a partial cut in it is not a homeomorphism. There is a more subtle map that takes the original paper to the partially cut paper. The boundary of the original paper would now travel up one side of the cut and down the other side. I hope that helps!2012-10-25