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I have quite a fundamental geometry question to which I'd guess the answer is already known. Is it possible to have a set of straight lines which covers all of three dimensional Euclidean space (i.e. every point in space lies along a line) without crossing each other and without them all being parallel?

I'm imagining there may be a set of twisted straight lines locally resembling a helical structure, but I'm not sure how to formulate this, or prove that no such set of lines can exist.

Thanks in advance for any help.

Chris
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  • MMm could you say a bit more about what you mean by 'covers' three dimensional space? And when you say 'co-linear' do you mean parallel? – Simon Terrington Aug 20 '20 at 10:55
  • https://math.stackexchange.com/questions/371302/cover-mathbbr3-with-skew-lines – Angina Seng Aug 20 '20 at 11:07
  • "Without all being parallel" then something like the tumbling tower would work unless you require none of them being parallel. – cr001 Aug 20 '20 at 11:50
  • You can start with a set of lines, all parallel to each other, which partitions the whole space, and then take any plane made from such lines and rotate that plane in place, so that the lines in that plane are no longer parallel to the rest. – Jaap Scherphuis Aug 20 '20 at 12:06

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