Talk:Condition for Straight Lines being Parallel

Do $\beta$ and $\beta'$ have to be different? Same question with gamma on Condition for Planes being Parallel. --Cynic 17:10, 16 November 2008 (UTC)

Depends on whether you allow that a line / plane may be parallel to itself. Convenient if a line is allowed to be parallel to itself because then you can neatly assert that parallelism is an equivalence relation and all is well with the kozmos. So: no, they don't have to be different. --Matt Westwood 17:12, 16 November 2008 (UTC)

But then the lines clearly intersect ... don't they? So wouldn't we need to modify the definition of parallel to be either they never intersect or they are the same? --Cynic 17:16, 16 November 2008 (UTC)

Drat, hadn't thought of that. I think we need to add a separate clause to the definition of parallelism such that: "A line is defined as being parallel to itself" or something. If parallelism is defined by the equations as given in this entry, rather than via the Euclidean definition, then the issue goes away, but that doesn't seem right for this site which will likely be used more by people who are familiar with Euclid than those who have gone into all that degree-level linear algebra.

Now: do we add that extra clause into the Euclidean definition of parallelism or into this entry here? --Matt Westwood 17:28, 16 November 2008 (UTC)

I'd say into the euclidean definition. Otherwise it feels like we're creating two separate definitions of parallel, one algebraic and one geometric. This one already allows a line to be parallel to itself. --Cynic 17:32, 16 November 2008 (UTC)

See the drunken waffle I've just added to the "parallel" page. --Matt Westwood 20:01, 16 November 2008 (UTC)