I posited this question to my geometry teacher in highschool many years ago, and it stumped her. I've recently brought it up again in conversations with friends and have not gotten any answer that's satisfactory. I have an intuitive feeling of what I expect, but I don't know enough to really back it up with solid reasoning.
Basically, consider this theoretical situation:
Suppose you have two infinitely long lines (you could say a couple laser beams which never diffuse, but any perfectly straight line would work), and they are mounted so that they intersect at some point along an infinite plane in space. Then, you gradually rotate one of them so that the angle of intersection gets gradually closer and closer to 0, moving the lines closer and closer to parallel. At some point, intuition tells us that there is a position at which these lines will be perfectly parallel and never intersect. However, math tells us that they will always intersect, and the angle at which they intersect will simply approach 0 forever but never reach it.
Theoretically, parallel exists. If these lines were angled away from each other, they will not intersect, because at each increment, the distance between the two lines increases. If one of the lines in this case were rotated closer and closer to parallel, eventually they will angle towards each other and intersect at some distance away.
Math, again, tells us that this intersection point is found by a limit calculation, but it evaluates to infinity.
Considering that theoretical parallel does intuitively exist, when the lines become perfectly parallel, what happens at the point of intersection? There couldn't be a final point of intersection, could there?
EDIT
I am de-emphasizing the following paragraph as it confuses the question, more than anything. It addresses a physical world representation where this question is largely theoretical, and concerns the mathematics behind the concept.
The only way I can wrap my brain around this situation, in my relatively limited understanding of physics and mathematics, is if these lines physically manifested as infinitely large loops, and then behave like intersecting rings. At any frame of reference, there is a point which appears parallel (the tangent lines) but where elsewhere, on the other side of the ring, they intersect. But this then assumes that the universe is curved along itself. Is there a better theory to explain this? I'm sure I can't be the first one to imagine this kind of situation.
Best Answer
If you start with two infinitely long lines, which intersect at a point that is a finite distance in front of you, and straighten them so that they are parallel, then the point of intersection will shoot off to infinity in finite time.
This may seem counterintuitive, but stuff like this happens when you have an infinitely long line and you move it around as a rigid body. For example, if you just think about one line, when you rotate it, a point on the line that is distance "x" away from you sweeps out an arc at a certain speed. Another point at distance "2x" will sweep out an arc twice as long in the same time, so it moves twice as fast. And since the line, by definition of the thought experiment, is infinite, there are points on the line arbitrarily far away from you, which sweep out arbitrarily huge distances at arbitrarily fast speeds when the line is rotated.
This has nothing to do with physical reality because physical reality doesn't contain infinitely long, infinitely rigid physical bodies that respond instantly, all along their infinite length, when you rotate them at the origin. If you point a laser in one direction, and then change the direction, it takes time for the redirected beam to spread out into space, the beam was never infinitely long (because the laser would have had to be switched on forever in order to have created an infinite beam), etc. Anything you actually do will only involve finite distances and changes that take time to travel; you don't need to bring up the curvature of space in order to explain why your thought experiment can't happen in the real world.