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In basic school mathematics, I was taught that a pair of straight lines could be either parallel (maintain the same distance between each other ie. Not cross, or intersecting (crossing) each other.

But it is very easy to find a third situation, where the 2 straight-lines are NOT maintaining equal-distance throughout their length, yet they are not crossing each other. Such as this:----

Exception... CUBE

This condition could be easily imagined- if in my room I imagine a diagonal along the floor floor from South-East Corner (C) to North-west corner (D); and imagine another line, a diagonal along the ceiling from North-East-Corner (B) to South West Corner. The 2 straight lines obtained would never touch in a real and finite world, yet they are not parallel.

This is only one condition; many other similar conditions exist. Such as this one on a hexagonal prism (such as pencil)

Exception- Prism

What should I call them? Should I call them parallel? (since they will not cut anywhere)

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    $\begingroup$ In 2 dimensional space, 2 lines are either crossing or parallel. In 3 dimensional space, 2 planes are either crossing or parallel. Lines in 3 dimensional space can be "neither". $\endgroup$
    – David
    Jan 10, 2017 at 20:11
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    $\begingroup$ ^ And to build on Dave's answer, in 3-d space, if they are not parallel yet do not cross, they are called skew. $\endgroup$
    – The Count
    Jan 10, 2017 at 20:11
  • $\begingroup$ The more dimensions the spaces gives you, the more possibilites to miss each other. 2D planes won't necessarly intersect in four dimensions. $\endgroup$
    – Laray
    Jan 10, 2017 at 20:12
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    $\begingroup$ see this:mathworld.wolfram.com/SkewLines.html $\endgroup$
    – Arnaldo
    Jan 10, 2017 at 20:12
  • $\begingroup$ You may write it as answer @TheCount $\endgroup$
    – Always Confused
    Jan 10, 2017 at 20:12

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In 3-d space, if two lines do not cross yet are not parallel, they are called skew. The lines in your 'pencil' example are skew, for instance.

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  • $\begingroup$ I cnnot accept it within 9 minutes and I'm going to sleep tonight. I'll accept it tomorrow. Thanks it will next-time help me to understand help me the term 'skew' in stereochemistry related text $\endgroup$
    – Always Confused
    Jan 10, 2017 at 20:16
  • $\begingroup$ Oh, no worries. That's fine. Glad I could help, and sleep well! I remember stereochemistry. Enantiomers and all that. Good luck! $\endgroup$
    – The Count
    Jan 10, 2017 at 20:16
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The fact you mentioned:

lines are either parallel or intersecting

Is only true in the second dimension, in the third dimension, it's planes (entire second dimensions) that intersect and parallel. Lines in the third dimension can be "neither", or not intersect or parallel. As @The Count mentioned, these lines are called skew.

This logic applies to any dimension, an object in $n$ dimension is either parallel or intersecting if the dimension we are measuring in $=n+1$, but not if the dimension $\lt n+1$ In your example lines are $1$ dimensional, and you are calculating in the third dimension. because $1+1\lt3$, lines can be skew.

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