Parallel lines meeting
Date: Around 1995
Do parallel lines meet at infinity?
Could be. Nobody has ever managed to go to infinity to find out. Actually,
the question of what "infinity" means for spaces of more than one dimensions
is a little tricky. You could have one infinity, which is often done with
the complex plane, or you could imagine infinities in all directions - an
uncountable number of them in fact. If there is only one infinity, then
parallel lines must of course meet there, since there is nowhere else to go.
But if there are an infinite number of infinities, parallel lines could go
to different ones, I suppose. Although, since they are headed in the same
direction, you might think they would meet in the same place. In any case,
it is something that is often said, but I am not sure there is that much
meaning to it.
In the context that you are probably using to think about geometry, the
answer is that parallel lines do NOT meet. I say this since almost everyone
interprets geometry as Euclidean Geometry. The true statements in Euclidean
Geometry are those which can be derived from the definitions and postulates
assembled by the Greek mathematician, Euclid, (400 BC). In this context,
parallel lines cannot meet. In fact, one definition of the word parallel is
"lines that do not meet" (in a plane). As Ross Perot might say, "end of
discussion." However, I think you mean something more, e.g., do two common
perpendicular lines to a single line in a plane ever meet? While the answer
is, no, in Euclidean Geometry, there are definitions and postulates that
have been studied for centuries in which the answer is "yes" instead.
Projective geometry is a case in point. You should be able to find informa-
tion in your library about this. There is a way to visualize this situation
so that a Euclidean plane is contained in a projective plane.
Click here to return to the Mathematics Archives
Update: June 2012