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Puzzling

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  • $\begingroup$ In my puzzle, I did not say that the solution had to be 2-dimensional so I will allow a 3-dimensional solution. I can’t quite visualize your solution. Your first line that wraps around the cylinder might be a problem. I suspect that it is not a straight line in 3-dimensional space. I required lines to be straight. Also squares in 2-dimensional and 3-dimensional space consist of perfectly straight line segments. $\endgroup$
    – Will.Octagon.Gibson
    Commented May 28 at 18:27
  • $\begingroup$ Regardless of whether your solution is valid or not, it is interesting. I hope you or someone else posts an image of it. $\endgroup$
    – Will.Octagon.Gibson
    Commented May 28 at 18:29
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    $\begingroup$ @WillOctagonGibson I can't draw, so I made a model out of string and scotch tape. Although these lines aren't straight in traditional Euclidean 3-space, there is a well-defined vector space where they are indeed exactly straight (and exactly orthogonal, unlike my quick+sloppy model)! I think there's one detail that needs attention and is glossed over in the answer, but it should be easily handled, namely, you have to be careful that you don't make large squares that wrap around the cylinder the "long" way! $\endgroup$
    – Daniel Wagner
    Commented May 28 at 19:35
  • 1
    $\begingroup$ This is 2-dimensional anyway--it's just that one of the dimensions loops. $\endgroup$
    – Hearth
    Commented May 29 at 1:39
  • 2
    $\begingroup$ @DanielWagner Sorry, but this isn't a vector space - a vector space over the reals can't wrap around. It is, at best, a manifold... but if you're fine with lines on any manifold, why not just cut 'wormholes' in the plane to make it one line? $\endgroup$
    – Deusovi ♦
    Commented May 29 at 17:12